Matheology
§ 001 A matheologian is a man, or, in rare cases, a woman, who believes in thoughts that
nobody can think, except, perhaps, a God, or, in rare cases, a Goddess.
§ 002 Can the existence of God be proved from mathematics?
Gödel proved the existence of God in a relatively complicated way using the positive and
negative properties introduced by Leibniz and the axiomatic method ("the axiomatic method is
very powerful", he said with a faint smile).
http://www.stats.uwaterloo.ca/~cgsmall/ontology.html
http://userpages.uni-koblenz.de/~beckert/Lehre/Seminar-LogikaufAbwegen/graf_folien.pdf
Couldn't the following simple way be more effective?
1) The set of real numbers is uncountable.
2) Humans can only identify countably many words.
3) Humans cannot distinguish what they cannot identify.
4) Humans cannot well-order what they cannot distinguish.
5) The real numbers can be well-ordered.
6) If this is true, then there must be a being with higher capacities than any human.
QED
[I K Rus: "Can the existence of god be proved from mathematics?", philosophy.stackexchange,
May 1, 2012]
http://philosophy.stackexchange.com/questions/2702/can-the-existence-of-god-be-proved-frommathematics
The appending discussion is not electrifying for mathematicians. But a similar question had been
asked by I K Rus in MathOverflow. There the following more educational discussion occurred
(unfortunately it is no longer accessible there).
(3) breaks down, because although I can't identify (i.e. literally "list") every real number between
0 and 1, if I am given any two real numbers in that interval then I can distinguish them. – C
GERIG
If you are given two numbers, then both can be given, i.e., belong to the countable set of finite
expressions. – I K RUS
I voted down to close as "subjective and argumentative". Claiming that the well-ordering axiom
implies that someone can order the reals is really inane, in my opinion. – ANGELO
I agree. It is really inane. But most mathematicians don't even know that this belief is inane. We
should teach them: It is really inane to believe that all real numbers "exist" unless God has a list
of them. – I K RUS
God is not the subject of proof. Either you believe or not, but this is only a matter of faith. It
would be too simple if a proof of existence or non-existence existed. We should not have any
choice. – D SERRE
God is the subject of Gödel's proof. God is the subject of my proof. And I am very proud that I
have devised a proof that can be understood by a cobblers apprentice (as Euler requested).
That will pave my way into the paradise. We know, without God there is no paradise, not even
Hilbert's.
You rightfully remark, "we should not have any choice." And we have no choice - unless we
have the axiom of choice. Now I will no longer respond to questions and comments and will
withdraw into my hermitage. Bless you God. – I K RUS
Although I agree with the closing of your question, thanks for bringing up that webpage - it is
interesting and useful. Knowledge can come from many sources. – F GOLDBERG
Yes, but unfortunately in MathOverflow it seems to be not always appreciated. This instructive
question and discussion have been closed as spam and deleted immediately.
§ 003 Is the analysis as taught in universities in fact the analysis of definable numbers?
In October 2010 this question had been asked in MathOverflow by user ANIXX. The following is
substantially shortened. For full text see here:
http://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-theanalysis-of-definable-numbe
All numbers are divided into two classes: those which can be unambiguously defined by a
limited set of their properties (definable) and such that for any limited set of their properties there
is at least one other number which also satisfies all these properties (undefinable).
It is evident that since the number of properties is countable, the set of definable numbers is
countable. So the set of undefinable numbers forms a continuum. ...
But the main question that bothered me was that the analysis course we received heavily
relied on constructs such as "let's a to be a number that...", "for each s in interval..." etc. These
seemed to heavily exploit the properties of definable numbers and as such one can expect the
theorems of analysis to be correct only on the set of definable numbers. ... – ANIXX
The naive account continues by saying that since there are only countably many such
descriptions ϕ, but uncountably many reals, there must be reals that we cannot describe or
define.
But this line of reasoning is flawed in a number of ways and ultimately incorrect. The basic
problem is that the naive definition of definable number does not actually succeed as a
definition. ...
I am taking this opportunity to add a link to my very recent paper "Pointwise Definable Models of
Set Theory", J. D. Hamkins, D. Linetsky, J. Reitz, which explains some of these definability
issues more fully. – J D HAMKINS
As I understand you say it can be postulated in ZFC that undefinable numbers simply do not
exist. – ANIXX
No, this is not what Joel was saying. He did not say that it is consistent to "postulate in ZFC that
undefinable numbers do not exist". What he was saying was that ZFC cannot even express the
notion "is definable in ZFC". – A BAUER
The Preprint by J D HAMKINS et al.
http://de.arxiv.org/abs/1105.4597
contains the following phrases, starting smugly:
One occasionally hears the argument—let us call it the math-tea argument, for perhaps it is
heard at a good math tea—that there must be real numbers that we cannot describe or define,
because there are are only countably many definitions, but uncountably many reals. Does it
withstand scrutiny?
Question 1. Is it consistent with the axioms of set theory that every real is definable in the
language of set theory without parameters?
The answer is Yes. Indeed, much more is true: if the ZFC axioms of set theory are consistent,
then there are models of ZFC in which every object, including every real number, every function
on the reals, every set of reals, every topological space, every ordinal and so on, is uniquely
definable without parameters. Inside such a universe, the math-tea argument comes ultimately
to a false conclusion.
The number of descriptions is countable, and there is a one-to-one function mapping
definitions to the objects they define in a pointwise definable model (so all such models are
countable).
Okay lets take standard analysis. It follows that the number of reals is uncountable (inside this
model) while the number of definable numbers is countable. How can we be confident the
analysis theorems that employ definablility actually true for all reals? –
And yes, I do not say undefinable numbers do not exist. Their existence follows from Axiom of
choice and in theory we can uniquely define each undefinable number by specifying infinite
number of its properties. The problem is that the theorems of analysis as thought in universities
sufficiently rely on the properties of definable numbers. – ANIXX
A definable real number r is a number that can be defined, i.e., r can be identified and
communicated by a finite sequence of bits in real life, just where mathematics takes place. This
makes the set of simultaneously (in a given language) definable numbers countable. Therefore
all real numbers that can appear in the language of mathematical analysis belong to a countable
set.
Independent of real-life conditions it is impossible to distinguish, in the universe of ZFC or
elsewhere, real numbers by infinite sequences of bits. This claim is proven by the possibility to
construct all infinite sequences of digits by means of a countable set of infinite sequences of
digits as follows:
Enumerate all nodes ai of an infinite binary tree and map them on infinite paths pi such that
ai œ pi. There is no further restriction. The mapping need not be injective. Then construct from
this countable set of paths another binary tree. Mathematical analysis is not able to discern
which paths were used for construction.
This shows that outside of a platonist ZFC-universe there are not uncountably many real
numbers. Real numbers created by Cantor-lists are not defined unless the Cantor-list is welldefined, i.e., every entry of the list is known. That requires a Cantor-list constructed by a finite
definition. But there are only countably many finite definitions of Cantor-lists.
The existing real numbers of analysis cannot be listed. But that does not make their set larger
than any countable set. – USER
This last answer, however, enjoyed only a very short lifetime before it has been locked and
deleted. (Also the original question had been closed very soon. It has only be reopened by
intervention of J D HAMKINS.)
§ 004 On March 10, 2009 I asked in FOM (an automated e-mail list for discussing foundations
of mathematics): Who was the first to accept undefinable individuals in mathematics?
http://www.cs.nyu.edu/pipermail/fom/2009-March/013464.html
Until the end of the nineteenth century mathematicians dealt with definable numbers only. This
was the most natural thing in the world. An example can be found in a letter from Cantor to
Hilbert, dated August 6, 1906:
"Infinite definitions (that do not happen in finite time) are non-things. If Koenigs theorem was
correct, according to which all finitely definable numbers form a set of cardinality ¡0, this
would imply that the whole continuum was countable, and that is certainly false."
Today we know that Cantor was wrong and that an uncountable continuum implies the existence
of undefinable numbers. {{This was Wolfgang in sheep's clothing. In fact I should have written:
The implied necessity of undefinable numbers proves the non-existence of uncountable sets,
because numbers are definitions and undefinable definitions are nonsense. But then I could
have spared sending off my text and could have glued it above my desk instead.}} Who was the
first mathematician to deliberately accept undefinable individuals like real numbers in
mathematics?
This question appeared the same day (obviously someone hadn't paid enough attention) and
raised some resonance. Details can be found in
http://www.cs.nyu.edu/pipermail/fom/2009-March/
But, in any case, definable in what language? ... Surely Cantor was wrong only in the sense that
he didn't point out that the notion of definability cannot be absolute, but depends upon the
language. - B TAIT
Unfortunately all my intended answers like the following were "deemed inappropriate by the
moderator":
One fact is independent of any model: Every calculation is limited to a finite amount of bits.
Every set the elements of which can be treated as individuals is countable. The set of finite
words over a finite alphabet is countable. The set of meanings of these words, i.e., the set of
languages, is countable. The set of finite alphabets is countable. The Cartesian product of these,
and possibly some further features, is countable. These facts are independent of the physical
model and independent of the logical stand point.
The repeated rejections annoyed me, and I directed a complaint to the list administrator.
Is this referee a university teacher who is supposed to know something and to explain what he
knows?
That, in turn, obviously annoyed the referee. He wrote to the moderator: Mueckenheim adds to
his previously rejected nonsense the false assertion that the Cartesian product of the set of finite
alphabets is countable. As for university teachers being supposed to explain what we know, I've
wasted far too much time trying to explain mathematics (or even common sense) to people who
insist on some incorrect "proof." I don't consider it my duty or yours to waste more time in that
direction.
Best regards, ...
Nevertheless it is fact that the Cartesian product of finitely many countable sets is countable even if the referee cannot understand that. Nothing hinders the scientific progress as much as
the presumptuousness of incompetent referees. So I addressed the moderators again, and the
following discussion evolved:
Referee to FOM: He's right that some statements about definability don't need details of the
language, but I think he's wrong to infer (in his first sentence) "an absolute meaning of
undefinability."
WM: I do not infer an absolute meanig about undefinability. But I infer that only countably many
real numbers can be defined.
Referee to FOM: Does he perhaps think there's a particular real number that is undefinable in
any (countable) language?
WM: That's a tricky question. Of course I cannot define an undefinable number. ... I would need
to make an infinite sequence of decisions 0 or 1 in order to determine the binary representation
of a real number without a finite defining equation. Of course that is impossible for me and for
anyone else. Anyhow, if there are 2¡0 real numbers, then there are 2¡0 - ¡0 = 2¡0 of such
undefinable real numbers.
After some exchange of arguments I wrote:
Dear Sirs,
I am surprised that some of you seem to be believe that there are uncountably many languages
or that a Cartesian product of three countable sets could be uncountable. If you understand that
a language including thesaurus is nothing but a finite definition, then you will agree that all
definitions including all languages and all theorems, proofs, numbers, symbols etc. are present
in the following list of all finite words:
0
1
00
01
10
11
000
...
This list is the list of everything. Of course some of its words may have different meanings,
according to the language applied. (Definitions of languages can be found in some later parts of
the list.) The only item that is missing is a diagonal word. It is easy to see that a diagonal word
cannot be constructed. And even if it could, it would not mean anything because all meaningful
words have to be finite.
If however you are really unable to understand that this simple logical chain is as justified as
what you may think {{a purely polite phrase; what they think is not in the least justified}}, then I
have to consider FOM as an abbreviation of Fools Of Matheology instead. And I would kindly
ask you to delete me from your list as it would be only waste of time to read what you write.
Regards, WM
After all I received an email with the subject: "You have been unsubscribed from the FOM
mailing list" without love and kisses and any further text. The moderators were disgruntled. I can
understand that. I would be disgruntled too if someone claimed that my life's work was humbug and if he could prove that.
§ 005 In order to save the idea of the set of uncountably many real numbers, these numbers
must be distinct by some property. As there are not enough finite properties (definitions, names,
strings of digits), it is assumed in matheology, that the elements of the set — carry infinite names
such that they can be distinguished in principle or by a God or, in rare cases, by a Goddess. But
that idea can be contradicted by the Binary Tree.
A due question has been asked in MathOverflow by ANO NYMUS on May 5, 2012: A
constructivist’s puzzling argument: I enumerate all nodes ai of an infinite binary tree and map
them on infinite paths pi such that ai œ pi. There is no further restriction. The mapping need not
be injective (and cannot be surjective). But I don’t tell you which paths pi I have chosen. Then I
construct from this countable set of paths another binary tree. You are not able to find out which
paths I have used. You cannot even determine whether or not I used the path of the rational
number 1/3. That proves that real numbers cannot be defined by digits alone, contrary to
Cantor’s diagonal argument, where the diagonal number is defined by digits, in that it differs
from every other number by at least one digit.
Result: Real numbers need finite definitions. But there are only countably many. – ANO
NYMUS
He does not distinguish between "infinite" and "arbitrarily large finite" – B KJOS-HANSSEN
Just this is the question: How can "infinite" and "arbitrary large" be distinguished other than by
finite definitions? – ANO NYMUS
Wait, isn't 1/3 defined by a digit? Or if you prefer a decimal system, isn't 1/10 defined by a digit?
Hooray, these are not real numbers! – ASAF KARAGILA
These are real numbers, defined by three and four symbols, respectively. I would really prefer an
answer instead of polemics. – ANO NYMUS
No matter how finitely many digits 3.14159265358979 I give you, you can't be sure what real
number those are the first digits of. From this it seems he concludes that numbers are not
defined by digits alone.
The number π has a finitely long definition (involving circumferences and diameters etc.). It's
just that the definition does not consist in listing some of the digits. – B KJOS-HANSSEN
The argument is Mückenheimian philosophy, so I'm pretty sure ano nymus is yet another
sockpuppet of WM:
hs-augsburg.de/~mueckenh/MR/Paradoxes.pdf
– M GREINECKER
{{My argument is not philosophical but purely mathematical. If somebody claims that
uncountably many real numbers can be distinguished by infinite strings of bits, but is unable to
discern which of countably many infinite strings make up the complete infinite Binary Tree, then
it's about time that he should try to get accustomed step by step to the faint idea that he was
dreaming. I can only encourage everyone to look into the recommended document.}}
Obviously π has many finite definitions. One of them is "π". – ANO NYMUS
You only think this is a real question. It is not. – M GREINECKER
That's a very good argument!? I must confess, I think it is a real question, and that puzzles me.
From the answers I got here and from the "closing as spam", in particular as one of the closers
obviously has not understood what is asked for at all, I have been shocked. If you do not think it
is a real question, you should say why. And why should my question be spam??? – ANO
NYMUS
After few hours this question and discussion have been closed as spam and immediately
deleted. An answer could not be given, as had to be expected.
§ 006 Is there a well-ordering of the reals, measurable or not?
This question was asked by a user of Math.StackExchange in April 2012: I just stumbled on
these two claims: "Nice" well-orderings of the reals and, in the answer, "no well-ordering of the
reals is Lebesgue measurable". And I am surprised. Is there a well-ordering of the reals after all,
measurable or not? - USER
Assuming the axiom of choice, yes. Any well ordering of the reals, as a subset of the plane, is
not measurable. - ASAF KARAGILA
I am only asking whether there is a well-ordering of the reals. I think I remember that somebody
proved it impossible. But I am not sure. - USER
It depends entirely on your set theory. If you assume the axiom of choice, then there is a wellordering of every set, and in particular of the reals. - BRIAN M SCOTT
It seems you dispute (incorrectly) that uncountable sets can be well-ordered. {{No, no, no. I
swear by Cantor, Zermelo and Gödel that I truely believe in matheology. (Otherwise this
question and discussion certainly would have been closed and deleted immediately as is
customary in media run by religious fanatics or dictatorial regimes despising human rights and
freedom of word.)}} ... You do not even need axiom of choice. - SDCVVC
I am curious how the real numbers can be well-ordered without attaching an expression to each
of them. And that makes me doubt a real well-ordering, if it is not given by a formula that allows
to test it. A proof based on axioms would not be useful. For instance if we use the axiom "every
sum of two natural numbers is even", we can prove that result. Nevertheless ... - USER
From the axioms of ZFC we can prove the existence of a well ordering of any set. In particular
the set of real numbers. ... Such well ordering, though, is not nice in the sense that any formula
defining it would be rather complicated (compared to other sets which we know, like open sets).
- ASAF KARAGILA
You say: "any formula defining it would be rather complicated". So it is not excluded that there
could be a formula defining it? Then the proof that I read is wrong? - USER
In the constructible universe there is a formula of complexity Σ12 defining a well ordering of the
real numbers. The same formula, in larger universes might not define a well-ordering at all. ASAF KARAGILA
I am talking about real numbers as they are defined in a high school math course. Is it possible
to well-order them by a formula in our universe? Can you give a formula where the relative
positions can be read? - USER
The real numbers are not defined in high school. If you have seen a definition of the real
numbers, you're probably one of the lucky few. Regardless of that we can define these numbers
as sets in set theory. I should also explicitly say that when I say formula I am not talking about
"x3 - ey" sort of formula but rather something very very different. - ASAF KARAGILA
I have learned equivalence classes of Cauchy-sequences, Dedekind-cuts and interval-nesting.
Only the definition of Weierstrass is unknown to me. I know that the rational numbers can be put
in bijection with the naturals and the real numbers cannot. Further I know that all finite words
belong to a countable set. That's why I wonder how reals are well ordered without having a
name for each one. I would be interested in seeing a formula which does that. - USER
Historically before the axiom of choice and the work with infinities began, mathematics dealt with
things "we can name" for the most part. After the axiom of choice was formulated, and infinities
began to take form and shape, we began researching things which we can deduce their
existence but not name. In a way this is similar to black holes (but this analogy does not go very
far). We cannot see them, we can only prove their existence due to some laws of physics which
we believe hold. Same here, we cannot write down a well-ordering but we can prove it exists. ASAF KARAGILA
Black holes are a nice example. But to prove, that numbers (which cannot be well-ordered,
because they cannot be named), can be well-ordered nevertheless, is really a joke. Are people
get paid by tax payers for that "research"? And do they know what you are doing? - USER
Similarly, note that you cannot really name every rational number (since your brain is limited in
size and ability to comprehend longer and longer words) but you can still prove the infinitude of
rational numbers between zero and one. - ASAF KARAGILA
We can in principle name each one. That is quite different with the real numbers. So this
example of yours fails. - USER
You obviously misunderstand what does "prove" mean. To prove means to deduce by a finite
series of steps, from certain assertion, a certain conclusion. To prove something does not mean
"finding a name for it". - ASAF KARAGILA
Mathematicians have proven that, assuming "mathematics" (i.e., ZF) {{that is the usual
usurpatory abuse of the word "mathematics" by matheologians}} is consistent, if you want to
assume "— can be well ordered", you can without adding any contradictions. Somewhat
surprisingly, if you want to assume "— cannot be well ordered", this is equally ok - no
contradictions will be added. Practically, this means you cannot prove "— can be well ordered" or
else you wouldn't be able to assume it can't be without contradiction. Likewise, you cannot prove
"— cannot be well ordered", for the same reason. - J DEVITO
Your comment on taxes betrays a rather deep misunderstanding of the matters which you are
attempting to discuss, and a rather large and misguided axe to grind. This is the place for
neither. - A MAGIDIN
User did not answer, and without any authorization or consent from my side this discussion has
been faked in that "user" has been replaced by my name.
http://math.stackexchange.com/questions/137657/is-there-a-well-ordering-of-the-realsmeasurable-or-not
This is a violation not only of copy right and personality laws. But taking legal action would not
be worthwhile. So let me deputize for the user, taking the advantage of answering the last three
contributions as kind of co-author:
"Proving" in matheology seems to boil down to results like this: "If cowshit tastes like honey, then
we should try horse droppings." But I am not interested in such "proofs". I am asking without
fuss or quibble: Would you try horse droppings?
§ 007 In Math.StackExchange the following question was put by a user: What is the difference
between the made up case A and the real case B?
A) We use the axiom: Every sum of two perfect numbers is even. Based on the number 6 we
prove from this axiom that there is no odd perfect number. In case an odd perfect number could
be found, this axiom must be abolished.
B) We use the axiom of choice and prove from it that every set of real numbers can be wellordered. From the fact that there are not enough names available and that ordering of numbers
without names is impossible, we can conclude that the axiom must be abolished.
So what is the difference? Why is has this axiom not yet been abolished?
To prove something does not mean "finding a name for it". But to prove that it is possible to
order every name (and real numbers belong to the set of names), does mean that it is possible
to have every name. - USER
Very soon this question received some amazing answers:
Could you spell out what you mean by "ordering of numbers without names is impossible"? That
seems to be the crucial point. In the usual treatment of the subject, this is not true. – JORIKI
Your claim that "ordering numbers without names is impossible" is simply unjustified. – M
SUÁREZ-ALVAREZ
Likewise "real numbers belong to the set of names". Numbers are not names. – C EAGLE
The axiom you need to abolish is "an immaterial object cannot be put in any well-ordering unless
you can refer to it". – SDCVVC
{{But is the reverse true without axiom? I think, matheologians should add as the eleventh axiom
to ZFC: 10. Ordering does not require knowledge of the ordered elements. At least the axiom of
extensionality
Given any set A and any set B. If for every set C, C is a member of A if and only if C is a
member of B, then A is equal to B.
should be reformed slightly, because we cannot determine whether C is a member of a A unless
C "is given", i.e., C is definable in a finite time by a signal of finite (although arbitrarily high) bitrate.}}
You must also object to our summing numbers which we cannot name, and multiplying them. –
M SUÁREZ-ALVAREZ
{{I have never summed numbers which I cannot name - at least in principle and when
disregarding MatheRealism.}}
In any case, please remember that this is not a discussion site: you ask questions and others
answer, but if you want to debate the answers you should find another venue. – M SUÁREZALVAREZ
{{Done. Too much discussion is bad for any religious branch.}}
How do you hate people that you don't know as individuals? Yet so many people in the world
hate to the bone groups of other people for their sex, the color of their skin, their religion, their
personal beliefs, their sexual preference, their musical preference, etc. etc. – ASAF KARAGILA
I think one cannot hate unknown people, even less one can put them in an order. One may
dislike certain habits of a group of people, as one may know certain properties of a set of
numbers like the reals. But it is a bit of ridiculous to defend the obviously self-contradictory idea
of unthinkable thoughts and unspeakable words by such analogies. – B ZARKIN
OP's claim in the question is that it is impossible. If it has to stand, it should be proven. It is
fallacious to claim it is "apparently impossible" and switch the burden of the proof. That's not
how mathematics works. ... mathematics uses ZF(C) everywhere by default, and failure of
intuition is a problem of a human, not a problem of mathematics. –SDCVVC
{{So it belongs to the realm of "intuition" to believe that I cannot order what I cannot distinguish.
In this manner matheology can be defended till the end of days. Then every contradiction can be
avoided by observing that some manipulations have been applied that are not axiomatized in
ZFC.}}
It is not intuition to conclude that 10 objects cannot be distinguished by 5 labels. But it is
intuition, and in my opinion, it is very good intuition, to conclude that set theorists have never
thought about this point or have repressed it, because every question either is answered by
ridiculous analogies or is closed "as not a real question". You can be sure that many, many
people including many "working mathematicians" would like to have an answer. – B ZARKIN
I completely agree. 10 objects cannot be represented by 5 labels. Are you aware to the fact that
both numbers are finite ? Countably many things can define uncountably many things (e.g. the
rationals define the real numbers), and regardless to that well ordering is not a list of labels. It is
a property of a relation. – ASAF KARAGILA
{{And a Cantor list can contain uncountably many entries, because every counter-proof relies on
finite numbers for enumeration.}}
You claimed to have studied about Dedekind Cuts, this is a way to define the real numbers from
the rationals. Equivalence classes of Cauchy sequences as well. – ASAF KARAGILA
Every Dedekind-cut and every Cauchy-Sequence is a finite string of symbols, some of them
denoting rational numbers. It is impossible, however, to define a real number by an infinite string
of rational numbers. To believe so is only a common mistake. – B ZARKIN
For this discussion no source is available because it has been closed and deleted soon as "not a
real question".
§ 008 Mathematical logicians often joke that the diagonal method is the only proof method that
we have in logic. This method is the principal idea behind a huge number of fundamental results
... [J D HAMKINS, Nov 22, 2010]
http://mathoverflow.net/questions/46970/proofs-of-the-uncountability-of-the-reals/46979
This assertion is correct. And it is a tragedy!
The fundamental mistake of matheology is the conclusion from
1) A finite expression (or number) defines an infinite sequence.
to
2) An infinite sequence defines a finite expression (or number).
This reversal of implication is of some naivity. But the defenders of that mistake call themselves
logicians and admire each other as very intelligent persons.
Nevertheless, even disregarding this error does not help, because the complete infinite Binary
Tree can be constructed by countably many infinite paths. That means uncountably many infinite
paths cannot be distinguished. The proof is very simple. It is based on the impossibility to find
out which paths have been used for construction.
And the notion of uncountability fails for a third reason: Every Cantor-List is capable of providing
one or more diagonal numbers. Preconditions for the creation of a diagonal number b are only:
1) The list is well-defined, i.e., all entries ak and all digits akj are known.
2) The replacement rule akk Ø bk is well defined too.
Now, it is possible, in principle, to enumerate all Cantor-lists starting from the first one
constructed around 1890 to the last one, probably being constructed about 2020. (But even
many more could be enumerated because every elementary cell of a temporally and spatially
infinite universe belongs to a countable set.) That proves: All diagonals of actually constructed or
possible Cantor-lists belong to a countable set. Why can't they be enumerated? They simply
don't exist yet!
Would all real numbers be existing and addressable already, then each one could turn up in a
Cantor-list, as an infinite bit sequence, at the first place or the second or any other one. Then all
these lists could be combined, and diagonalization would contradict the presupposed existence.
Hence it is not countability that is disproved by the diagonal argument but simply existence and
as such the prerequisite of the notion of complete set as is necessary for set theory.
§ 009 Open letter to a non-mathematician
Georg Cantor's ideas of finished infinity and a time after never, have infected many
mathematicians. Their love of these ideas is so strong that they try to punish everyone who
dares to present counter arguments. The following case occurred in mathematics
stackexchange.com,
http://math.stackexchange.com/questions
a forum for discussing simple mathematical questions including homework and the like. Usually
even contributions of low quality enjoy a long lifespan. Not so, however, if someone tries to
convince freshmen of the irrationality of Cantor's ideas. Moderators and "experienced users"
mercilessly delete any heresy. A recent example is this:
Question
When Dag Duck receives a heap of dollar coins he counts them all in order to make sure it's
the correct amount. In reality this is often impossible. How much has to be counted in order to be
fairly sure about the result?
If you get a string of information but can't wait until the end-signal is given, how sure can you
be to have received the correct meaning?
My answer
There is no chance to determine the result unless you have counted the last coin. Similarly,
the information is not transferred until transfer has been completed by the end-signal. Otherwise
a negation could appear in the subsequent part of the string.
All the more it is astonishing that mathematicians are satisfied with Cantor's "enumeration" of
the rational numbers. If you count an infinite set like – you never count a share of more than
limnض 2-n = 0 because every natural number is followed by infinitely many others. How can you
obtain anything sensible from the result? ("All natural numbers" means all elements of a set of
which you cannot have all elements. Infinite means never, not "arrived at after all".)
Same is true for infinite "words". They cannot be used for communicating information.
Therefore the following reversion of implication, usually applied in set theory, is wrong:
A finite formula defines an infinite sequence. ñ An infinite sequence defines a finite formula.
We can never obtain a finite formula like 1/9 or ◊2 from an infinite sequence unless we know
the last term (which is impossible by definition). An infinite sequence (like Cantor's diagonal)
never defines a number. Up to every digit it defines an interval out of countably many. In order to
know the limit, you need information about the infinitely many digits to follow. That requires a
finite formula.
There is nothing in this answer that could be accused to be wrong or difficult to understand.
Nevertheless the answer was deleted after a short time of existence by Henning Makholm who
qualified it as "an anti-Cantorian rant with no connection to the question at all" (he was in error,
as I know) and Asaf Karagila. The vote of the latter is understandable, because he is preparing
his MSc-work in set theory. If it turns out that set theory does not belong to science (as, several
decades before, it happened to astrology), his efforts would have been in vain.
Meanwhile I have been completely excommunicated from math.stackexchange (as before from
mathoverflow and some other centers of what I therefore call matheology). In fact, I had
expected, and a little bit provoked, the outcome. It proves with breathtaking evidence what
everybody not yet infected by Cantor's ideas should know: Many mathematicians are addicts of
finished infinity. They refuse to listen to simple truths, like those discussed in my answer above,
and are trying everything to prevent to be cured.
They are not dangerous. Cantor's ideas have not the slightest chance of any application outside
of mathematics (because they are blatant humbug) - and even inside they merely cause
confusion, namely if actual infinity is actually taken literally, cp. numbers 1035 to 1044 of
http://www.hs-augsburg.de/~mueckenh/KB/KB%201001-.pdf
So be prepared: Many mathematicians believe in finished infinity and may get very angry if you
doubt that belief.
§ 010 The present paragraph proves that matheologicans prefer to accept mistakes if
otherwise the truth about their nonsense cannot be suppressed.
In Mathematics StackExchange Seamus had asked the question: Is there a known well ordering
of the reals?
http://math.stackexchange.com/questions/6501/is-there-a-known-well-ordering-of-the-reals
So, from what I understand, the axiom of choice is equivalent to the claim that every set can be
well ordered. A set is well ordered by a relation, R if every subset has a least element. My
question is: has anyone constructed a well ordering on the reals?
First, I was going to ask this question about the rationals, but then I realised that if you pick
your favourite bijection between rationals and the integers, this determines a well ordering on
the rationals through the natural well order on Ÿ. So it's not the denseness of the reals that
makes it hard to well order them. So is it just the size of — that makes it difficult to find a well
order for it? Why should that be? - SEAMUS
To be or not to be ... There was much blathering about the existence of a well-ordering of the
real numbers.
Goedel explicitly constructed a subset of the reals and a well order on the subset such that (in
ZF) it is consistent that the subset is all reals. But subsequently Cohen showed it is also
consistent that the subset is not all reals. – G EDGAR
I assume you know the general theorem that, using the axiom of choice, every set can be well
ordered. Given that, I think you're asking how hard it is to actually define the well ordering. This
is a natural question but it turns out that the answer may be unsatisfying. ... However, there is
not even a formula that unequivocally defines a well ordering of the reals in ZFC. ... Worse, it's
even consistent with ZFC that no formula in the language of set theory defines a well ordering of
the reals (even though one exists). That is, there is a model of ZFC in which no formula defines
a well ordering of the reals. – C MUMMERT
No, it's not just the size. ... On the other hand given AC one can obviously write down a wellordering in a non-constructive way (choose the first element, then the second element, then...).
– Q YUAN
First a small error: Ÿ in its natural order is not well-ordered, because Ÿ and many of its subsets
do not have a least element. But of course Ù is. And all rationals can be put in bijection with Ù.
Second: It is just the size which prevents the well-ordering of —, because a sensible
mathematician would refuse to well-order what cannot be distinguished. Take the first, then the
second, then the third, ... will not lead further than the countably many possible names reach.
You can be sure that never anybody will be able to give a well-ordering of the reals (except
those countably many which can be identified). And here "never" means never.
Regards, WM
Not necessary to mention that my answer did not live for long. The obvious mistakes ("the
natural well order on Ÿ" and "given AC one can obviously write down a well-ordering in a nonconstructive way (choose the first element, then the second element, then...)" however, remain
there until this very day.
§ 011 God knows a list of all natural numbers [*]. In which way does he memorize the real
numbers?
[*] As an example I refer to the totality, the incarnation of all finite integer positive numbers; this
set is a thing by itself and forms, apart from the natural sequence of the involved numbers, a
fixed and in all parts defined quantum, an aphorismenon, which obviously must be called larger
than every finite number. ... Compare St. Augustin's concurring perception of the integer
sequence as an actually infinite quantum (De civitate Dei. lib. XII, cap. 19): Against those who
say God could not know infinite things.
Als Beispiel führe ich die Gesamtheit, den Inbegriff aller endlichen ganzen positiven Zahlen an;
diese Menge ist ein Ding für sich und bildet, ganz abgesehen von der natürlichen Folge der
dazu gehörigen Zahlen, ein in allen Teilen festes, bestimmtes Quantum, ein aphorismenon, das
offenbar größer zu nennen ist als jede endliche Anzahl. [...] Man vgl. die hiermit
übereinstimmende Auffassung der ganzen Zahlenreihe als aktual-unendliches Quantum bei S.
Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui dicunt ea, quae infinita sunt, nec Dei
posse scientia comprehendi.
http://agios.org.ua/la/index.php/Aurelius_Augustinus._De_Civitate_Dei_Contra_Paganos._Pars_
1#LIBER_XII
[Ernst Zermelo (Hrsg.): "Georg Cantor, Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem
Briefwechsel Cantor - Dedekind. Nebst einem Lebenslauf Cantors von Adolf Fraenkel." Georg
Olms Verlagsbuchhandlung, Hildesheim (1966) p. 401]
http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN237853094
§ 012 For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the
philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however,
hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class
sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for
several minutes, while he puzzled out a problem. He often asked his listeners questions and
reacted to their replies. Many meetings were largely conversation. These lectures were attended
by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing. [Cora
Diamond (ed.): "Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge 1939
from the notes taken by R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies",
The University of Chicago Press, Chicago (1975)]
http://www.amazon.com/Wittgensteins-Lectures-Foundations-MathematicsCambridge/dp/0226904261/ref=sr_1_1?ie=UTF8&s=books&qid=1248181058&sr=8-1#reader
Imagine set theory's having been invented by a satirist as a kind of parody on mathematics. –
Later a reasonable meaning was seen in it and it was incorporated into mathematics. (For if one
person can see it as a paradise of mathematicians, why should not another see it as a joke?)
If it were said: "Consideration of the diagonal procedure shews you that the concept 'real
number' has much less analogy with the concept 'cardinal number' than we, being misled by
certain analogies, inclined to believe", that would have a good and honest sense. But just the
opposite happens: one pretends to compare the "set" of real numbers in magnitude with that of
cardinal numbers. The difference in kind between the two conceptions is represented, by a skew
form of expression, as difference of extension. I believe, and I hope, that a future generation will
laugh at this hocus pocus.
The curse of the invasion of mathematics by mathematical logic is that now any proposition can
be represented in a mathematical symbolism, and this makes us feel obliged to understand it.
Although of course this method of writing is nothing but the translation of vague ordinary prose.
"Mathematical logic" has completely deformed the thinking of mathematicians and of
philosophers, by setting up a superficial interpretation of the forms of our everyday language as
an analysis of the structures of facts. Of course in this it has only continued to build on the
Aristotelian logic.
[Rhees, von Wright, Anscombe (eds.): "Ludwig Wittgenstein, Remarks on the Foundations of
Mathematics", Wiley-Blackwell (1991)]
http://www.amazon.com/Remarks-Foundation-Mathematics-LudwigWittgenstein/dp/0631125051/ref=sr_1_1?ie=UTF8&s=books&qid=1248181778&sr=1-1
§ 013 USER asked in MathOverflow: Did Zermelo and Goedel know about
countability of labels? Cantor did not believe that all pieces of information, i.e., all words that can
be used for information transfer and all labels that can be used to distinguish objects of thought,
belong to a countable set. Is there any evidence that Zermelo or Goedel were aware of that
fact?
These kinds of posters are known, in other forums, as "trolls". And, as the saying goes, "Don't
feed the trolls". Best to close their questions down and not give them the attention they crave. –
L MOSHER {{People who rise above a certain level of fanatism or fall below a certain level of
intelligence think (as far as they can) that everything they don't understand is trolling.}}
The OP asks "Is there any evidence that Zermelo or Goedel were aware of that fact?" I would
ask rather: Is there any evidence for that fact? Specifically, could you please cite a passage in
Cantor's writings where he writes about "pieces of information" and "information transfer" and
"labels"? – A BLASS
You should know that Cantor spoke and wrote German. Therefore your question is nothing but
polemics. And since the present question has been closed, this discussion should cease also.
But if you are really interested in objective historical facts of mathematics and not in stirring up
hatred, then you may bounce your fellows into translating and not deletíng the question that I will
pose immediately. – USER
Letter from Cantor to Hilbert. What did Cantor want to say? On August 8, 1906 Cantor wrote in a
letter to Hilbert:
"Unendliche Definitionen" (die nicht in endlicher Zeit verlaufen) sind Undinge. Wäre Königs
Satz, daß alle „endlich definirbaren" reellen Zahlen einen Inbegriff von der Mächtigkeit ¡0
ausmachen, richtig, so hieße dies, das ganze Zahlencontinuum sei abzählbar, was doch
sicherlich falsch ist. Es fragt sich nun, welcher Irrthum liegt dem angeblichen Beweise seines
falschen Satzes zu Grunde? Der Irrthum (welcher sich auch in der Note eines Herrn Richard im
letzten Hefte der Acta mathematica findet, welche Note Herr Poincaré in dem letzten Hefte der
Revue de Métaphysique et de Morale mit Emphase herausstreicht) ist, wie mir scheint, dieser:
Es wird vorausgesetzt, dass das System {B} der Begriffe B, welche eventuell zur Definition von
reellen Zahlindividuen herangezogen werden müssen, ein endliches oder höchstens abzählbar
unendliches sei. Diese Voraussetzung muß ein Irrthum sein, da sich sonst der falsche Satz
ergeben würde: "das Zahlencontinuum hat die Mächtigkeit ¡0."
What did he mean with this paragraph? - USER
This may be an early attempt to come to grips with the issue of definability, which was still vague
at that time. Formal languages and their semantics had not been introduced, and informal
languages lead to paradoxes (like the first natural number not definable in fewer than 100
English words) if one takes definability in such languages seriously. Cantor seems convinced
that every real number must be definable in some sense, but the sense is not clear. In the first
sentence, he dismisses the idea that a definition could itself be an infinite object. He concludes
that an uncountable number of concepts must be involved in definitions of real numbers, in order
to provide each real number with a finite definition. This contradicts not only our intuitive idea of
definability (in a finite or countable language) but also the apparently similar ideas of König,
Richard, and Poincaré that Cantor cites. (Note that "herausstreicht" can mean "emphasizes" or
"crosses out"; from the context, I infer the former meaning, since Poincaré was not in a position
to delete anything from Richard's paper in a different journal.) It would be interesting to see what
else Cantor had to say about these uncountably many concepts; presumably he would find it
difficult to communicate them. (Does it perhaps lead to theology?) - A BLASS
I cannot yet accept your answer because you say: "Cantor seems convinced that every real
number must be definable in some sense, but the sense is not clear." This is incorrect. The
sense is crystal clear. It must be possible, in finite time (and obviously using a limited bitfrequency), to communicate the complete knowledge concerning an object of mathematics.
Example: I say ◊2 and every reader of MO immediately knows what I mean (and can, to any
desired digit, find the decimal expansion - but that is not important). – USER
The Cantor quote that you posted is indeed somewhat surprising to me, since he seems to
presuppose that all real numbers must be definable in some sense. ... Perhaps the quote
shouldn't have been surprising, since clear notions of formal languages and definability arose
only later. – A BLASS
Mathematics concerns thinking and communication and is impossible without that. Items that
cannot be thought or communicated in finite time do not belong to mathematics. Therefore
modern "languages" and the like cannot compensate for the fact that there are not more than
countably many words or labels that can be used in mathematics to identify objects. And
believing in an order of possibly divine objects that cannot be identified by any finite means is in
fact a religious attempt with no relation to math. That's why some call this branch of transfinite
set theory "matheology". – USER
Cantor was wrong (we would say) but König used informal reasoning. It may not be well defined
in our language which expressions define a real. Also, perhaps the well ordering can not be
completely described in the chosen language. - A MEYEROWITZ
"Cantor was wrong", we would not say. Mathematics concerns thinking and communication and
is impossible without that. Items that cannot be thought or communicated in finite time do not
belong to mathematics. I can assure you: Young, intelligent students having not yet been drilled
in transfinity, understand with not a single exception that well-ordering of objects that cannot be
identified is an absolutely absurd idea. Ask anybody not yet infected with transfinity. But from the
scores of your and Andreas' answer compared to that of my question I can conclude - O, too
little space. – USER
It may not be well defined in our language which expressions define a real. But it is very easy to
prove that there are not more than countably many expressions. And people loving indirect
proofs like Zermelo's well-ordering of the reals should not hesitate to accept also that one as a
fact. – USER
There are also less than 36 100 positive integers definable using under 100 letters and
numbers. – A MEYEROWITZ
I know that it is hard to define what definability is. Therefore I don't even try. But it is easy to
prove what is undefinable, because: "In mathematics description and object are equivalent."
(Ludwig Wittgenstein) – USER
So instead of using mathematical ideas you draw from natural language which, as discussed
between Cantor and Hilbert, is informal and can be interpreted in several ways. Then you use
this informality to claim that set theory is ill-defined. I think that your internal logic is somewhat
offset. – A KARAGILA
On the contrary. Just the formal definition leads to countability. Unfortunately I have only a
German text again: Definiert man die reellen Zahlen in einem streng formalen System, in dem
nur endliche Herleitungen und festgelegte Grundzeichen zugelassen werden, so lassen sich
diese reellen Zahlen gewiß abzählen, weil ja die Formeln und die Herleitungen auf Grund ihrer
konstruktiven Erklärungen abzählbar sind. [Kurt Schütte: "Beweistheorie", Springer (1960)] Look
up in your library: K. Schütte, proof theory. (Schütte was a pupil of Hilbert.) – USER
§ 014 On Jan. 31, 2012 I answered in MatheOverflow a question concerning the infinite Binary
Tree:
The set of all finite paths (from the root-node to any other node) in the complete infinite Binary
Tree is countable. Therefore the complete infinite Binary Tree has countably many paths that
can be identified by nodes.
It is impossible to identify an infinite path by nodes, because
1) every node belongs to a finite path, and
2) there is no identification unless it has been finished.
Therefore an infinite path can only be identified by a finite expression like "always turn left", or
"0.111...", or "the path which represents 1/π", or simply "1/3". However, the set of finite
expressions has countable cardinality.
Therefore the set of all paths in the complete infinite Binary Tree has countable cardinality.
Regards, WM
For a real number r > 1 wouldn´t you consider "the path which represents 1/r" to be a path?
Aren´t there uncountably many real numbers r > 1? – R DE LA VEGA
For a real number r > 1, that can be defined by a finite expression, 1/r represents a path. But
there are not more than countably many finite expressions. – WM
Further questions were not admitted. My answer had been deleted.
§ 015 Belsa Zarkin asked in Mathematics.StackExchange:
All rational numbers of the unit interval [0, 1] can be covered by countably many intervals, such
that the n-th rational is covered by an interval In of measure 1/10n. There remain countably many
complementary intervals of measure 8/9 in total.
Does each of the complementary intervals contain only one irrational number? Then there
would be only countably many which could be covered by another set of countably many
intervals of measure 1/9.
Is there at least one of the complementary intervals containing more than one irrational
number? Then there are at least two irrational numbers without a rational between them. That is
mathematically impossible.
There is no interval left behind. Every interval contains at least one rational number, so if you
remove all rational numbers (let alone a bunch of intervals containing all of them), there can be
no interval left over. – BRUNO {{Intervall, Teilmenge I einer totalen Ordnung (M, ≤).
Insbesondere sind also die leere Menge « und M selbst Intervalle. ["Lexikon der Mathematik",
Spektrum (2003)]}}
There is a mistake in your argument, which follows from the fact that the complement of this
union contains no interval. ...The irrational numbers form a totally disconnected space, namely
every connected component is a singleton.- A KARAGILA
In a totally disconnected space there must be points or intervals disconnecting it. The number of
these points or intervals is countable and in bijection with the remaining intervals. The bijection is
the same as that from Ù to –. (For instance at every step n the configuration of intervals could
be determined in principle.) If you deny the validity of this bijection for the limit, why don't you
deny the validity for the limit of the bijection from Ù to –? – B ZARKIN
The definition of a totally disconnected space is a space in which every connected space is a
singleton. E.g. a discrete space. Are you saying that every discrete space is countable? – A
KARAGILA
This discussion lasted for a while. Meanwhile it has been deleted.
Correct mathematics is this:
There are countably many intervals In of measure 10-n such that In covers the rational qn. Then
1/9 (or less) of the unit interval is covered. In the remaining 8/9 (or more) there are uncountably
many irrationals. But every two irrationals have a rational between each other. That implies two
irrationals have at least one interval In between each other (because there are no rationals
outside of intervals In). That implies two irrationals have at least one of ¡0 endpoints In1 or In2 of
intervals In1 between each other. These endpoints can be considered structuring and
enumerating a Cantor-list. The only difference is that the enumeration does not follow the natural
order of Ù. But the number of naturals does not change by reordering. So we have uncountably
many irrationals separated by countably many endpoints. That is a contradiction similar to
uncountably many entries in a Cantor-list or uncountably many terms in a sequence.
§ 016 If we define the real numbers in a strictly formal system, where only finite derivations and
fixed symbols are permitted, then these real numbers can certainly be enumerated because the
formulas and derivations on the basis of their constructive definition are countable.
Definiert man die reellen Zahlen in einem streng formalen System, in dem nur endliche
Herleitungen und festgelegte Grundzeichen zugelassen werden, so lassen sich diese reellen
Zahlen gewiß abzählen, weil ja die Formeln und die Herleitungen auf Grund ihrer konstruktiven
Erklärungen abzählbar sind. [Kurt Schütte: "Beweistheorie", Springer (1960)]
http://www.amazon.de/Beweistheorie-KurtSch%C3%BCtte/dp/B0000BNKI7/ref=sr_1_1?s=books&ie=UTF8&qid=1286292242&sr=1-1
§ 017 Thesis XIII in Brouwers Dissertation (Appendix): "Over de grondslagen der wiskunde"
(Februari 1907, Dutch) simply reads: "De tweede getalklasse van Cantor bestaat niet",
translated: Cantors second number class does not exist. {{That is an acceptable foundation of
acceptable mathematics.}}
http://www.archive.org/details/overdegrondslag00brougoog
He quickly discovered that his ideas on the foundations of mathematics would not be readily
accepted. {{His ideas would devastate matheology.}}
http://www-history.mcs.st-andrews.ac.uk/Biographies/Brouwer.html
§ 018 Feferman and Levy showed that one cannot prove that there is any non-denumerable
set of real numbers which can be well ordered. Moreover, they also showed that the statement
that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be
refuted.
[Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy: "Foundations of Set Theory", North
Holland, Amsterdam (1973) p. 62]
http://www.amazon.de/gp/product/0720422701/ref=sib_rdr_dp
§ 019 Cantor's 'paradise' as well as all modern axiomatic set theory [AST] is based on the (selfcontradictory) concept of actual infinity. Cantor emphasized plainly and constantly that all
transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to
persuade us to believe that the AST does not use actual infinity. {{Many don't even know the
difference.}} It is an intentional and blatant lie, since if infinite sets, ◊ and Ù, are potential, then
the uncountability of the continuum becomes unprovable, but without the notorious uncountablity
of continuum the modern AST as a whole transforms into a long twaddle about nothing ...
[Letter from A. A. Zenkin to D. Zeilberger]
http://www.math.rutgers.edu/~zeilberg/fb68.html
Prof. Dr. Alexander A. Zenkin (1937 - 2006) was leading research scientist of the computer
center of the Russian Academy of Sciences.
§ 020 Epistola Pentecostes MDCCCLXXXVIII
I have no doubt concerning the truth of the transfinitum, which I have recognized with the help of
God. [...] I happen to be a somewhat familiar not only with mathematics but also with several
other sciences. Therefore I am able to compare theorems, here and there, with respect to their
objective certainty. From no other subjects of the created nature I have a safer and, if this
expression is allowed, a more certain realization than of the theorems of transfinite number- and
type-theories. That's why I am convinced that this theory one day will belong to the common
property of objective science and will be confirmed in particular by that theology which is based
upon the holy bible, tradition and the natural disposition of the human race - these three
necessarily being in harmony with each other.
If one chooses this foundation for the doctrine of actual infinity, one stands firm and is, I might
almost say, easily able to reject all the objections which have been devised over millenia against
the infinite numbers, and to reduce them to their apparent reasons.
I completely agree that [...] all the finite (and, to a much higher degree, all the transfinite), from a
diversity of aspects, points to the Absolute, i.e., the existence of the Absolute can necessarily be
proved by a dialectical rational conclusion, in accordance with Bonaventura's sentence:
Invariable rules (of human reason) are rooted in the eternal light and lead to it.
"Couldn't God, after having created an infinite set of stones or angels, create further angels?"
{{asks Durandus de Sancto Porciano, OP.}} Of course he can do that, must be answered.
When he then continues to conclude: "Therefore the angels created at first were not infinitely
many." so is this conclusion utterly wrong, because the supposed set of created angels is a
transfinitum that can be increased as well as decreased.
[Georg Cantor, Letter of Pentecost 1888 to P. Ignatius Jeiler, OFM {{that does not mean "Online
Football Manager" but "Ordo Fratrum Minorum", order created by Francis of Assisi}}, Praefect.
Coll. S. Bonav., quoted in C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische
Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner
Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 410ff]
http://www.steiner-verlag.de/programm/fachbuch/geschichte/universitaets-undwissenschaftsgeschichte/reihen/view/titel/54670.html
§ 021 Epistola Pentecostes MDCCCLXXXVIII
Understanding of the theory of the transfinite does not require scholarly preparation in newer
mathematics {{modern matheologians hold another opinion: who does not believe in the
diagonal proof cannot have understood it and lacks mathematical skills}}; it could be rather
damaging than helpful, because modern mathematics [...] has been perverted to materialistic
one-sidedness and has been blinded for any objective metaphysical recognition and, therefore,
for its own foundations too. Whereas the theory of the transfinite for its foundations does not
need the so called infinitesimal analysis (calculus), the latter can neither lay its own groundwork
nor proceed to completing its edifice without the former. Under the first aspect I mention the
theory of irrational numbers which need the transfinite for a proper foundation, further there is
the advanced theory of functions that already today raises questions which without the aid of the
theory of transfinite order types and cardinal numbers cannot even be articulated, let alone be
solved.
Yet it is possible for everyone, in particular for an educated philosopher, to scrutinize the
principles of transfinite number theory and to become convinced of its correctness {{and those
who do not?}}.
You emphasize with full right, Reverend Father, that according to nearly all teachings of the old
school (for me the only authoritative one) the divine intellect recognizes with respect to "objecta
extra ipsum cognita [...] infinita actu et categorematice“, not always "a parte rei" but always in
divine recognition "simultatem habent in esse cognito". If only this quite safe and unshakeable
sentence had always been realized in its full contents (i.e., not only in general but also in its
special meaning, in concreto) then one would have recognized without much effort the truth of
the transfinite, and quarrel and errors of thousands of years would have been avoided.
Applying this sentence to a special class of objects of the divine recognition, we arrive at the
elements of transfinite theory of numbers and types.
Every single finite cardinal number (1 or 2 or 3 etc.) is contained in the divine intellect in form
of an exemplary idea as well as a unified form for the recognition of uncountably many
composed things that belong to that very cardinal number. Hence, all finite cardinal numbers are
separately and simultaneously present in God's mind. (Cp S. Augustin, De civitate Dei, lib. XII,
cap. 19: contra eos, qui dicunt ea, quae inf. sunt, nec Dei scientia comprehendi) ("Against those
who state that the infinite could not be comprehended by the knowledge of God")
In their totality they form a diversified unity, a thing, separated from the remaining contents of the
divine intellect, that in itself again is a subject of divine recognition. But since recognition of a
thing requires a uniform shape by which the thing can be recognized, there must be in God's
mind a certain cardinal number that in the same way is related to the set or totality of all finite
cardinal numbers as, for instance, the cardinal number 7 is related to the set of tones of the
scale [C, D, E, F, G, A, H] of an octave.
[Georg Cantor, Letter of Pentecost 1888 to P. Ignatius Jeiler, OFM, quoted in C. Tapp:
"Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen
Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag
(2005) p. 414ff]
http://www.steiner-verlag.de/programm/fachbuch/geschichte/universitaets-undwissenschaftsgeschichte/reihen/view/titel/54670.html
§ 022 All rational numbers of the unit interval [0, 1] can be covered by countably many
intervals, such that the n-th rational is covered by an interval of measure 1/10n. There remain
countably many complementary intervals of measure 8/9 in total.
Does each of the complementary intervals contain only one irrational number? Then there
would be only countably many which could be covered by another set of countably many
intervals of measure 1/9.
My question: Can this contradiction be formalized in ZFC?
[user31686, April 15, 2012]
http://math.stackexchange.com/questions/132022/formalizing-an-idea/150674#150674
It has been mentioned already that the irrationals ξα of the set Ξ of the remaining part of
measure 8/9 (or more), that is not covered by your intervals, form a totally disconnected space,
so called "Cantor dust". Every particle ξα œ Ξ is separated from every other particle ξβ œ Ξ by at
least one rational qn, and, as every qn is covered by an interval In, it is separated by at least one
interval In. Since the end points an and bn of the In are rational numbers too, also being covered
by their own intervals, the particles of Cantor dust can only be limits of infinite sequences (an) or
(bn) of endpoints of overlapping intervals In. (If they don't overlap, then the limits must come
earlier, but in any case infinitely many endpoints are required to form a limit.) Such an infinite set
of overlapping intervals is called a cluster. In principle, given a fixed enumeration of the
rationals, we can calculate every cluster Ck and the limits of its union. Since two clusters are
disjoint (by their limits), there are only countably many clusters (disjoint subsets of the countable
set of intervals In). Therefore, every irrational ξα can be put in bijection with the cluster lying right
of it, say, between ξα and its next right neighbour ξβ. (Note that there is no next irrational to ξα
but there is a next right ξβ œ Ξ to ξα.) So, by this bijection we prove that the set of uncovered
irrational numbers ξα œ Ξ is countable.
[Stentor Schicklgruber, StackExchange (2012)]
http://math.stackexchange.com/users/32353/stentor-schicklgruber
Let all rational numbers qn of the interval (0,¶) be covered by intervals In = [sn, tn] of measure
|In| = 2−n, such that qn is the center of In. Then there remain uncountably many irrational
numbers as uncovered "Cantor dust". Every uncovered irrational xα must be separated from
every uncovered irrational xβ by at least one rational, hence by at least one interval In covering
that rational. But as the end points sn and tn of the In also are rational numbers and also are
covered by their own intervals, the irrationals xα can only be limits of infinite sequences (sn) or
(tn) of endpoints of overlapping intervals In. In principle we can calculate the limit xα of every
such sequence (sn) or (tn) of endpoints of overlapping intervals. Therefore, every irrational xα
can be put in bijection with the infinite set of intervals lying right of it, say, between xα and its
right neighbour xβ. There are countably many disjoint sets like {t | t œ (tn)} of elements of the
sequences (tn) converging to one of the xα. By this bijection we get a countable set of not
covered irrational numbers xα. Where are the other irrational numbers that are not covered by
intervals In? Nowhere. Uncountability is contradicted.
[Quidquid pro quo, MathOverflow (2012)]
http://mathoverflow.net/users/24011/quidquid-pro-quo
§ 023 The true reason for the incompleteness that is inherent in all formal systems of
mathematics lies in the fact that the generation of higher and higher types can be continued into
the transfinite whereas every formal system contains at most countably many. This will be
shown in part II of this paper. {{Part II was never published.}} In fact we can show that the
undecidable statements presented here always become decidable by adjunction of suitable
higher types (e.g., adding the type ω to system P). Same holds for the axiom system of set
theory.
Der wahre Grund für die Unvollständigkeit, welche allen formalen Systemen der Mathematik
anhaftet, liegt, wie im II. Teil dieser Abhandlung gezeigt werden wird, darin, daß die Bildung
immer höherer Typen sich ins Transfinite fortsetzen läßt [...] während in jedem formalen System
höchstens abzählbar viele vorhanden sind. Man kann nämlich zeigen, daß die hier aufgestellten
unentscheidbaren Sätze durch Adjunktion passender höherer Typen (z. B. des Typus ω zum
System P) immer entscheidbar werden. Analoges gilt auch für das Axiomensystem der
Mengenlehre.
[Kurt Gödel: "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter
Systeme I", Monatshefte für Mathematik und Physik 38 (1931) p. 191]
§ 024 Simultaneously I send, as printed matter, a curious booklet by Bolzano, which I let you
have, since I happen to possess another one. Although there is much, perhaps the most, in
error, it was very stimulating for me, in particular because of the opposition which it has aroused
in me. [Cantor to Dedekind, Oct. 7, 1882]
I never fully agreed with Dedekind's paper. [Cantor to Jourdain, July 18, 1901]
In vol. XXII, no. II of the Annalen pag. 249, Klein has his paper about function strips printed
again which I always have considered the non plus ultra of higher nonsense, although Mr. Klein
refers to the wisdom of Mr. Kronecker. If you have read my essay Grundlagen attentively, you
will find, that I, on pag. 9, 10, 11 and on pag. 19 and 20, attack and condemn just these opinions
of Kronecker in strongest terms; on pag. 20, in another matter, I make him a compliment,
together with Dedekind however (that will very much annoy him). [Cantor to Mittag-Leffler, Sept.
9, 1883]
The first old auntie said, we may
Now try to fix it swift.
Our little Sophy, at Saints day,
Shall get from us a gift.
Of course, the second aunt said keen,
I think we should acquire
A summer dress in bright pea green.
That's not at all her desire.
The third old aunt agreed and said:
With yellow stripes, old style.
I know she'll be annoyed and yet
She'll have to thank us and smile.
[Wilhelm Busch: "Die erste alte Tante sprach", Kritik des Herzens (1874)]
(Well, more or less.) The German original can be found here:
http://de.wikisource.org/wiki/Die_erste_alte_Tante_sprach
In the same year Cantor published his first paper on set theory: About a property of the set of all
real algebraic numbers [Crelles Journal f. Mathematik, vol 77, 258-262 (1874)].
Many thanks for your information about the French translation of Paul du Bois Reymond's
miserable paper. [Cantor to Mittag-Leffler, Aug. 4, 1888]
And the "Acta" shall be abused to disseminate this dirty rubbish? He will not use his own journal.
[...] for my own works I demand bias, but not for my transitory person, but bias for the truth which
is eternal and with superior contempt looks down upon the subversives (among others
Kronecker), who dare to imagine, they could lastingly change it with their dreadful scribblings.
[Cantor to Mittag-Leffler, Jan. 26, 1884]
I am quite an adversary of Old Kant, who, in my eyes has done much harm and mischief to
philosophy, even to mankind; as you easily see by the most perverted development of
metaphysics in Germany in all that followed him, as in Fichte, Schelling, Hegel, Herbart,
Schopenhauer, Hartmann, Nietzsche, etc. etc. on to this very day. I never could understand that
and why such reasonable and enabled peoples as the Italiens, the English and the French are,
could follow yonder sophistical philistine, who was so bad a mathematician. And now it is that in
just this abominable mummy, as Kant is, Monsieur Poincaré felt quite enamoured, if he is not
bewitched by him. [Cantor to Russell, Sept. 19, 1911 (English by Cantor)]
With mathematicians, no cheerful relationship can be obtained. [Goethe to Zelter, Jan. 18, 1823]
Originally Goethe wrote (but that does not fit so well): "With philologists and mathematicians, no
cheerful relationship can be obtained.".
For original German texts see "Das Kalenderblatt 090727":
http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf
§ 025 Modern mathematics as religion
[...] Most (but not all) of the difficulties of Set Theory arise from the insistence that there exist
'infinite sets', and that it is the job of mathematics to study them and use them. In perpetuating
these notions, modern mathematics takes on many of the aspects of a religion. It has its
essential creed - namely Set Theory, and its unquestioned assumptions, namely that
mathematics is based on 'Axioms', in particular the Zermelo-Fraenkel 'Axioms of Set Theory'. It
has its anointed priesthood, the logicians, who specialize in studying the foundations of
mathematics, a supposedly deep and difficult subject that requires years of devotion to master.
Other mathematicians learn to invoke the official mantras when questioned by outsiders, but
have only a hazy view about how the elementary aspects of the subject hang together logically.
Training of the young is like that in secret societies - immersion in the cult involves intensive
undergraduate memorization of the standard thoughts before they are properly understood, so
that comprehension often follows belief instead of the other (more healthy) way around. A long
and often painful graduate school apprenticeship keeps the cadet busy jumping through the
many required hoops, discourages critical thought about the foundations of the subject, but then
gradually yields to the gentle acceptance and support of the brotherhood. The ever-present
demons of inadequacy, failure and banishment are however never far from view, ensuring that
most stay on the well-trodden path. The large international conferences let the fellowship gather
together and congratulate themselves on the uniformity and sanity of their world view, though to
the rare outsider that sneaks into such events the proceedings no doubt seem characterized by
jargon, mutual incomprehensibility and irrelevance to the outside world. The official doctrine is
that all views and opinions are valued if they contain truth, and that ultimately only elegance and
utility decide what gets studied. The reality is less ennobling - the usual hierarchical structures
reward allegiance, conformity and technical mastery of the doctrines, elevate the interests of the
powerful, and discourage dissent. There is no evil intent or ugly conspiracy here - the practice is
held in place by a mixture of well-meaning effort, inertia and self-interest. We humans have a
fondness for believing what those around us do, and a willingness to mold our intellectual
constructs to support those hypotheses which justify our habits and make us feel good.
[N J Wildberger: "Set Theory: Should You Believe?"]
http://web.maths.unsw.edu.au/~norman/views2.htm
§ 026 {{Good mathematicians are precise and clear. Nevertheless, leading exegetes of
matheology do not seldom find the correct meaning, often the contrary of the written text,
between the lines. The following paragraph shall easen their effort.}}
(i) Infinite totalities do not exist
in any proper sense of the word
(i.e., either really or ideally).
More precisely, any mention,
or purported mention,
of infinite totalities is,
literally meaningless.
{{This text is not at all put into any perspective by the following paragraph (here the extra lines
will hardly interest the matheologian and, therefore, have been omitted).}}
(ii) Nevertheless, we should continue the business of Mathematics "as usual," i.e., we should act
as if infinite totalities really existed. [...] I must regard a theory which refers to an infinite totality
as meaningless in the sense that its terms and sentences cannot posses the direct interpretation
in an actual structure that we should expect them to have by analogy with concrete (e.g.,
empirical) situations. This is not to say that such a theory is pointless or devoid of significance.
{{Of course this is neither to say the contrary.}}
[A. Robinson: "Formalism 64" in W.A.J. Luxemburg, S. Koerner (eds.): "A. Robinson: Selected
Papers", North Holland, Amsterdam (1979)]
§ 027 By hindsight, it is not surprising that there exist undecidable propositions, as metaproved by Kurt Gödel. Why should they be decidable, being meaningless to begin with! The tiny
fraction of first order statements that are decidable are exactly those for which either the
statement itself, or its negation, happen to be true for symbolic integers. A priori, every
statement that starts "for every integer n" is completely meaningless.
[Doron Zeilberger: "'Real' analysis is a degenerate case of discrete analysis"]
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf
§ 028 The expression "and so on" is nothing but the expression "and so on" [...] the sign
"1, 1+1, 1+1+1 ..." is to be taken as perfectly exact; governed by definite rules which are
different from those for "1, 1+1, 1+1+1", and not a substitute for a series "which cannot be
written down".
There is no such thing as "the cardinal numbers", but only "cardinal numbers" and the concept,
the form "cardinal number". Now we say "the number of the cardinal numbers is smaller than the
number of the real numbers" and we imagine that we could perhaps write the two series side by
side (if only we weren't weak humans) and then the one series would end in endlessness,
whereas the other would go on beyond it into the actual infinite. But this is all nonsense.
"This proposition is proved for all numbers by the recursive procedure". That is the expression
that is so very misleading. It sounds as if here a proposition saying that such and such holds for
all cardinal numbers is proved true by a particular route, or as if this route was a route through a
space of conceivable routes. But really the recursion shows nothing but itself, just as periodicity
too shows nothing but itself".
After all I have already said, it may sound trivial if I now say that the mistake in the settheoretical approach consists time and again in treating laws and enumerations (lists) as
essentially the same kind of thing and arranging them in parallel series so that one fills in gaps
left by the other.
[L. Wittgenstein: "Philosophical Grammar", Basil Blackwell, Oxford (1969)]
§ 029 One of the characteristic features of Cantorism is that, instead of rising to the general by
erecting more and more complicated constructions, and defining by construction, it starts with
the genus supremum and only defines, as the scholastics would have said, per genus proximum
et differentiam specificam. Hence the horror he has sometimes inspired in certain minds, such
as Hermitte's, whose favourite idea was to compare the mathematical with the natural sciences.
For the greater number of us these prejudices had been dissipated, but it has come about that
we have run against certain paradoxes and apparent contradictions, which would have rejoiced
the heart of Zeno of Elea and the school of Megara. Then began the business of searching for a
remedy, each man his own way. For my part I think, and I am not alone in so thinking, that the
important thing is never to introduce any entities but such as can be completely defined in a
finite number of words. Whatever be the remedy adopted, we can promise ourselves the joy of
the doctor called in to follow a fine pathological case. [pp. 44f]
There is no actual infinity. The Cantorians forgot this, and so fell into contradiction. [p. 195]
[Henri Poincaré: "Science and Method", Translated by Francis Maitland, Nelson, London (1914)]
http://archive.org/stream/sciencemethod00poinuoft#page/194/mode/2up
§ 030 We can create in mathematics nothing but finite sequences, and further, on the ground
of the clearly conceived "and so on", the order type omega, but only consisting of equal
elements {{i.e. numbers like 0,999...}}, so that we can never imagine the arbitrary infinite binary
fractions as finished {{Brouwers Thesis, p. 143}}. [Dirk van Dalen: "Mystic, Geometer, and
Intuitionist: The Life of L.E.J. Brouwer", Oxford University Press (2002)]
Certainly we can also create other infinite sequences by finite expressions like 0,101010... that
would read 0,AAA... in hexadecimal notation. In that way also every other rational number can
be written.
But irrational numbers in fact are not available as sequences of digits.
Proof: Construct the Binary Tree by all finite paths that have an infinite sequence of
000...appended. Construct the Binary Tree by all finite paths that have an infinite sequence of
111... appended. Construct the Binary Tree by all finite paths that have an infinite sequence of
010101... appended. Construct the Binary Tree by all finite paths that have an infinite sequence
of all bits of 1/π appended. The reader will not be able to distinguish the constructed Binary
Trees and to determine what infinite paths are missing.
This fine proof has the advantage that matheologians cannot dispute it. They simply are not able
to distinguish numbers by sequences of bits or digits.
§ 031 Let us distinguish between the genetic, in the dictionary sense of pertaining to origins,
and the formal. Numerals (terms containing only the unary function symbol S and the constant 0)
are genetic; they are formed by human activity. All of mathematical activity is genetic, though the
subject matter is formal.
Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral
by prefixing it with S.
Now imagine this potential infinity to be completed. Imagine the inexhaustible process of
constructing numerals somehow to have been finished, and call the result the set of all numbers,
denoted by Ù.
Thus Ù is thought to be an actual infinity or a completed infinity. This is curious terminology,
since the etymology of “infinite” is “not finished”.
We were warned.
Aristotle: Infinity is always potential, never actual. Gauss: I protest against the use of infinite
magnitude as something completed, which is never permissible in mathematics.
We ignored the warnings.
With the work of Dedekind, Peano, and Cantor above all, completed infinity was accepted into
mainstream mathematics. Mathematics became a faith-based initiative.
Try to imagine Ù as if it were real.
A friend of mine came across the following on the Web:
www.completedinfinity.com
Buy a copy of Ù!
Contains zero - contains the successor of everything it contains - contains only these.
Just $100.
Do the math! What is the price per number?
Satisfaction guaranteed!
Use our secure form to enter your credit card number and its security number, zip code, social
security number, bank’s routing number, checking account number, date of birth, and mother’s
maiden name. The product will be shipped to you within two business days in a plain wrapper.
My friend answered this ad and proudly showed his copy of Ù to me. I noticed that zero was
green, and that the successor of every green number was green, but that his model contained a
red number. I told my friend that he had been cheated, and had bought a nonstandard model,
but he is color blind and could not see my point.
I bought a model from another dealer and am quite pleased with it. My friend maintains that it
contains an ineffable number, although zero is effable and the successor of every effable
number is effable, but I don’t know what he is talking about. I think he is just jealous.
The point of this conceit is that it is impossible to characterize Ù unambiguously, as we shall
argue in detail. [...]
Over two and a half millennia after Pythagoras, most mathematicians continue to hold a
religious belief in Ù as an object existing independently of formal human construction.
[Edward Nelson: "Hilbert’s Mistake" (2007)]
http://www.math.princeton.edu/~nelson/papers/hm.pdf
§ 032 Although Zermelo-Fraenkel set theory (ZFC) is generally accepted as the appropriate
foundation for modern mathematics, proof theorists have known for decades that virtually all
mainstream mathematics can actually be formalized in much weaker systems which are
essentially number-theoretic in nature. [...] not only is it possible to formalize core mathematics
in these weaker systems, they are in important ways better suited to the task than ZFC [...] most
if not all of the already rare examples of mainstream theorems whose proofs are currently
thought to require metaphysically substantial set-theoretic principles actually do not; and set
theory itself, as it is actually practiced, is best understood in formalist, not platonic, terms, so that
in a real sense set theory is not even indispensable for set theory. [...] set theory should not be
considered central to mathematics.
Probably most mathematicians are more willing to be platonists about number theory than
about set theory, in the “truth platonism” sense that they firmly believe every sentence of first
order number theory has a definite truth value, but are less certain this is the case for set theory.
Those mathematicians who are unwilling to affirm that the twin primes conjecture, for example,
is objectively true or false are undoubtedly in a small minority; in contrast, suspicion that
questions like the continuum hypothesis or the existence of measurable cardinals may have no
genuine truth value seems fairly widespread.
Some possible reasons for this difference in attitudes towards number theory and set theory
are (1) a sense that natural numbers are evident and accessible in a way that arbitrary sets are
not; (2) suspicion that sets are philosophically dubious in a way that numbers are not; (3) the
existence of truly basic set-theoretic questions such as the continuum hypothesis which are
known to be undecidable on the basis of the standard axioms of set theory, and the absence of
comparable cases in number theory; and (4) the fact that naive set theory is inconsistent. The
classical paradoxes of naive set theory particularly cast doubt on the idea of a well-defined
canonical universe of sets in which all set-theoretic questions have definite answers.
One philosophically important way in which numbers and sets, as they are naively understood,
differ is that numbers are physically instantiated in a way that sets are not. Five apples are an
instance of the number 5 and a pair of shoes is an instance of the number 2, but there is nothing
obvious that we can analogously point to as an instance of, say, the set {{«}}. [...]
Unfortunately, the philosophical difficulties with set-theoretic objects platonism are extremely
severe. First, there is the ontological problem of saying just what sets are.[...]
Perhaps the most influential philosophical defense of set theory is the Quine-Putnam
indispensability argument. According to this argument, mathematics is indispensable for various
established scientific theories, and therefore any evidence that confirms these theories also
confirms the received foundation for mathematics, namely set theory. But as a result of work of
many people going back to Hermann Weyl, we now know that the kind of mathematics that is
used in scientific applications is not inherently set-theoretic, and indeed can be developed along
purely number-theoretic lines. This point has been especially emphasized by Feferman.
Consequently, contrary to Quine and Putnam, the confirmation of present-day scientific theories
provides no special support for set theory. [...]
This raises the possibility that the use of set theory as a foundation for mathematics may be an
historical aberration. We may ultimately find that ZFC really has no compelling justification and is
completely irrelevant to ordinary mathematical practice.
[Nik Weaver: "Is set theory indispensable?"]
http://www.math.wustl.edu/~nweaver/indisp.pdf
§ 033 The clear understanding of formalism in mathematics has led to a rather fixed dogmatic
position which reads: Mathematics is what can be done within axiomatic set theory using
classical predicate logic. I call this doctrine the Grand Set Theoretic Foundation. [...]
Even in the 1940's, with the growth of abstract algebra, axiomatic set theory was not regarded
as a central doctrine. It was not until about 1950 that the Grand Set Theoretic Foundation was
finally complete and officially accepted under the slogan which might have read: "Mathematics is
exactly that subject which can be developed by logical rules of proof from the Zermelo-Fraenkel
axioms for set theory." This foundation scheme had its popular version in the "new math" for
schools. It also had its philosophical doctrine, a version of Platonism, that the world of sets is
that constructed in the standard cumulative hierarchy of all ranked sets. [...] The ZermeloFraenkel axioms are then (a selection of) the facts true for all sets in this hierarchy. This is
sometimes claimed to describe the ultimate Platonic reality which underlies all mathematics:
Perhaps the Zermelo-Fraenkel axioms do not describe everything, but with a little more insight
we will understand all the axioms necessary and then at least in principle all mathematical
problems can be settled from the axioms.
It is my contention that this Grand Set Theoretic Foundation is a mistakenly one-sided view of
mathematics and also that its precursor doctrine (Dedekind cuts) was also one-sided.
[Saunders Mac Lane: "Mathematical models: A sketch for the philosophy of mathematics", The
American Mathematical Monthly, Vol. 88, No. 7 (1981) 462-472]
http://en.wikipedia.org/wiki/MacLane
Don't miss his story: The man who could have changed history!
http://www.ams.org/notices/199510/maclane.pdf
(p. 1136, first paragraph)
§ 034 Announcing some congresses that may be of highest interest
http://ufocongress.com/grant-cameron-added-to-the-2013-iufoc-speakers/
http://www.impan.pl/~set_theory/Conference2012/
http://www.clevelandufo.com/?p=458
http://www.uacastrology.com/schedule2012.shtml
http://www.creationicc.org/
to the participants - but not to anybody else.
Therefore it is no disadvantage that some have ceased already. (Further it is claimed that
congresses on paranormal activities can also be attended in the past - by means of paranormal
activities, of course.)
§ 035 Borel's often-expressed credo is that a real number is really real only if it can be
expressed, only if it can be uniquely defined, using a finite number of words. It's only real if it can
be named or specifed as an individual mathematical object. {{How else could it be? Numbers do
not acquire existence other than that lent to them by (human) minds. Minds who mean to be
able to lend meanings that they, in principle, cannot, and who, in addition, mean to be able to
prove that impossibility as possible, appear to be mean minds - with respect to the spirit of
truth.}} And in order to do this we must necessarily employ some particular language, e.g.,
French. Whatever the choice of language, there will only be a countable infinity of possible texts,
since these can be listed in size order, and among texts of the same size, in alphabetical order.
This has the devastating consequence that there are only a denumerable infinity of such
"accessible" reals, and therefore [...] the set of accessible reals has measure zero. So, in Borel's
view, most reals, with probability one, are mathematical fantasies, because there is no way to
specify them uniquely. This is a more refined version of Borel's idea [Borel, 1960] of defining the
complexity of a real number to be the number of words required to name it. Why should we
believe in real numbers, if most of them are uncomputable? Why should we believe in real
numbers, if most of them, it turns out, are maximally unknowable like Ω?
The latest strong hints in the direction of discreteness come from quantum gravity [...], in
particular from the Bekenstein bound and the so-called "holographic principle." According to
these ideas the amount of information in any physical system is bounded, i.e., is a finite number
of 0/1 bits. But it is not just fundamental physics that is pushing us in this direction. Other hints
come from our pervasive digital technology, from molecular biology where DNA is the digital
software for life, and from a priori philosophical prejudices going back to the ancient Greeks.
According to Pythagoras everything is number, and God is a mathematician. This point of view
has worked pretty well throughout the development of modern science. However now a neoPythagorean doctrine is emerging, according to which everything is 0/1 bits, and the world is
built entirely out of digital information. In other words, now everything is software, God is a
computer programmer, not a mathematician, and the world is a giant information-processing
system, a giant computer.
[Gregory Chaitin: "How real are real numbers?" (2004)]
http://arxiv.org/abs/math.HO/0411418
§ 036 An argument against so called cranks (i.e., those who neither do believe in finished
infinity nor in undefinable definitions) is the claim that they do not only vehemently deny set
theory but as vehemently contradict each other. That may happen. But it is not different in
matheology and, therefore, cannot serve as an argument in favour of the superiority of the latter.
Here are some examples:
Hitherto logicians have only operated with chains consisting of a finite number of conclusions.
But since Cantor mathematicians operate with infinite chains. [A. Schoenflies: "Über die
logischen Paradoxien der Mengenlehre", Jahresbericht der Deutschen MathematikerVereinigung, (1906) 19-25.]
When paying attention you can find the mathematical literature flooded with inconsistencies and
thoughtlessness that in most cases are caused by the infinite. For instance, if, as a restricting
requirement, it is emphasized that in strict mathematics only a finite number of conclusions is
admissible in a proof - as if anybody ever had succeeded in drawing infinitely many conclusions!
[D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) 161-190]
These are Cantor's first transfinite numbers, the numbers of the second number class, as Cantor
calls them. We reach them simply by counting across the ordinary infinite. [Hilbert, loc. cit.]
Here Hilbert even has contradicted himself, because counting is nothing else but concluding
from n on n + 1. And there is another self-contradiction in one and the same paper:
No one shall drive us from the paradise which Cantor has created for us. [...] his theory of
transfinite numbers; it appears to me as the most admirable blossom of mathematical spirit and
one of the supreme achievements of purely intellectual human activity. What are the facts of the
matter? [...] the infinite is nowhere realized; neither is it present in nature nor is it admissible as a
foundation of our cerebral activity - a remarkable harmony between being and thinking. [Hilbert,
loc. cit.]
Kant already has taught us - and this is an integral part of his teaching - that mathematical
contents is independent of logic; mathematics can never be founded by logic alone.[Hilbert, loc.
cit.]
I am quite an adversary of Old Kant, who, in my eyes has done much harm and mischief to
philosophy, even to mankind. [G. Cantor to B. Russell, Sept. 19, 1911]
So Kant's achievements are quite controversially judged by Cantor and Hilbert. Frege's and
Dedekind's achievements are controversially judged even by Hilbert alone:
Brouwer is not, as Weyl thinks, the revolution, but the attempted repetition of a putsch with old
means, that in the past, undertaken much more dashingly, nevertheless completely failed, and
now is condemned to fail because the state power is well armed by Frege, Dedekind, and
Cantor. [D. Hilbert: "Die Neubegründung der Mathematik. Erste Mitt.", Abhandl. d. Math.
Seminars d. Univ. Hamburg, Bd. 1 (1922), S. 157-177]
... the contents of mathematics is independent of logic and therefore never ever can be founded
by logic alone. That's why the efforts of Frege and Dedekind were condemned to fail. [D. Hilbert:
"Über das Unendliche" (1925)]
The well-armed state power? After three years already on its last legs?
For original German texts and links to sources see "Das Kalenderblatt 090813".
http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf
§ 037 Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting
the belief in the existential character of the totality of all natural numbers, and hence the principle
of excluded middle in the form "Either there is a number of the given property g, or all numbers
have the property Ÿg" is without foundation. [...] The sequence of numbers which grows beyond
any stage already reached by passing to the next number, is a manifold of possibilities open
towards infinity; it remains forever in the status of creation, but is not a closed realm of things
existing in themselves. That we blindly converted one into the other is the true source of our
difficulties, including the antinomies - a source of more fundamental nature than Russell's
vicious circle principle indicated ["No totality can contain members defined in terms of itself"].
Brouwer opened our eyes and made us see how far classical mathematics, nourished by a
belief in the "absolute" that transcends all human possibilities of realization, goes beyond such
statements as can claim real meaning and truth founded on evidence. According to this view
and reading of history, classical logic was abstracted from the mathematics of finite sets and
their subsets. (The word finite is here to be taken in the precise sense that the members of such
set are explicitly exhibited one by one.) Forgetful of this limited origin, one afterwards mistook
that logic for something above and prior to all mathematics, and finally applied it, without
justification, to the mathematics of infinite sets. This is the Fall and Original sin of set theory
even if no paradoxes result from it. Not that contradictions showed up is surprising, but that they
showed up at such a late stage of the game!
[Hermann Weyl: "Mathematics and logic: A brief survey serving as a preface to a review of The
Philosophy of Bertrand Russell", American Mathematical Monthly 53 (1946) 2-13]
[Komaravolu Chandrasekharan: "Hermann Weyl, Gesammelte Abhandlungen, Vol. IV", Springer
(1968) p. 275f]
http://books.google.de/books?id=lNPriL6kG7AC&printsec=frontcover#v=onepage&q&f=false
§ 038 One of the most often heard arguments in favour of transfinite set theory is the
completeness requirement of — and real functions. It is not true.
Well, then tell me, Herr Professor Doktor Mueckenheim, how do you solve the equation
ih(∑u/∑t) = H(u) [JR, Matheology § 022, sci.logic, June 13, 2012]
In general most real numbers lack names, and we cannot effectively distinguish them. [AS,
Matheology § 022, sci.logic, June 13, 2012]
Numbers are free creations of human mind. They serve as a means to easen and to sharpen the
perception of the differences of things.
Zahlen sind freie Schöpfungen des menschlichen Geistes, sie dienen als ein Mittel, um die
Verschiedenheit der Dinge leichter und schärfer aufzufassen."
[Richard Dedekind: "Was sind und was sollen die Zahlen?" 1887, 8. Aufl. Vieweg, Braunschweig
1960, p. III]
But if not even the numbers can be distinguished, what are they good for? Real numbers that
cannot be distinguished will not complete mathematics, they will not make the real axis
continuous, they cannot guarantee that every polynomial has its zeros.
All real numbers that ever can appear in mathematical calculations have finite names (at least
the definition of the problem, like: "find the fourth root of 16") and belong to a countable set.
Therefore uncountably many unreal real numbers are good for nothing.
I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory
as a philosophy of our subject [...] platonism is the medieval metaphysics of mathematics; surely
we can do better. [S. Feferman: "Infinity in Mathematics: Is Cantor Necessary?"]
Feferman shows in his article, "Why a little bit goes a long way. Logical foundations of
scientifically applicable mathematics" on the basis of a number of case studies that the
mathematics currently required for scientific applications can all be carried out in an axiomatic
system whose basic justification does not require the actual infinite.
http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT#416,62,Folie 62
"The actual infinite is not required for the mathematics of the physical world." [S. Feferman: "In
the light of logic", Oxford Univ. Press (1998) p. 30]
http://books.google.de/books?id=AadVrcnschMC&pg
Though Gödel has been identified as the leading defender of set-theoretical platonism,
surprisingly even he at one point regarded it as unacceptable.
In his concluding chapters, Feferman uses tools from the special part of logic called proof
theory to explain how the vast part if not all of scientifically applicable mathematics can be
justified on the basis of purely arithmetical principles. At least to that extent, the question raised
in two of the essays of the volume, "Is Cantor Necessary?", is answered with a resounding "no".
[S. Feferman, loc. cit, description from the jacket flap]
http://math.stanford.edu/~feferman/book98.html
§ 039 Cantor devised set theory for application to reality. In a letter to Hilbert he wrote about
his plan of a paper on set theory and its applications:
The third part contains the applications of set theory to the natural sciences: physics, chemistry,
mineralogy, botany, zoology, anthropology, biology, physiology, medicine etc. It is what
Englishmen call "natural philosophy". In addition we have the so called "humanities", which, in
my opinion, have to be called natural sciences too, because also the "mind" belongs to nature.
Der dritte Theil bringt die Anwendungen der Mengenlehre auf die Naturwissenschaften:
Physik, Chemie, Mineralogie, Botanik, Zoologie, Anthropologie, Biologie, Physiologie, Medizin
etc. Ist also das, was die Engländer "Natural philosophy" nennen. Dazu kommen aber auch
Anwendungen auf die sogenannten "Geisteswissenschaften", die meines Erachtens als
Naturwissenschaften aufzufassen sind; denn auch der "Geist" gehört mit zur Natur.
[G. Cantor to D. Hilbert, letter of September 20, 1912]
In a letter to Mittag-Leffler Cantor explained his impetus for devising set theory:
Further I am busy with scrutinizing the applications of set theory to the physiology of organisms.
[...] I have been occupied for 14 years with these ideas of a closer exploration of the basic
nature of all organic; they are the true reason why I have undertaken the painstaking and hardly
rewarding business of investigating point sets, and all the time never lost sight of it, not for a
moment.
Further I am interested, purely theoretically, in the nature of the states and what belongs to
them, because I have my opinions on that topic which later may become formulated
mathematically; perhaps you will be taken aback, but that striking impression will disappear,
when you consider that also the state in some sense represents an organic being.
Ausserdem bin ich mit Untersuchungen über Anwendungen der Mengenlehre auf die
Naturlehre der Organismen beschäftiget [...] Mit diesen Ideen einer genaueren Ergründung des
Wesens alles Organischen beschäftige ich mich schon seit 14 Jahren, sie bilden die eigentliche
Veranlassung, weshalb ich das mühsame und wenig Dank verheissende Geschäft der
Untersuchung von Punctmengen unternommen und in diesem Zeitraum keinen Augenblick aus
den Augen verloren habe.
Ausserdem interessirt mich rein theoretisch das Wesen des Staates und was dazu gehört, weil
ich auch darüber meine Gesichtspuncte habe, die zu mathematischer Formulierung später
führen dürften; das Auffallende, was Sie darin vielleicht finden, verschwindet, wenn sie erwägen,
dass auch der Staat ein organisches Wesen gewissermaassen repräsentirt.
[G. Cantor to G. Mittag-Leffler, Sept. 22, 1884]
By applied set theory I understand what usually is called natural science or cosmology. To this
realm belong completely all natural sciences, those concerning the anorganic as well as the
organic world.
Unter angewandter Mengenlehre verstehe ich Dasjenige, was man Naturlehre oder
Kosmologie zu nennen pflegt, wozu also die sämmtlichen sogenannten Naturwissenschaften
gehören, sowohl die auf die anorganische, wie auch auf die organische Welt sich beziehenden.
[Ivor Grattan-Guinness: "An unpublished paper by Georg Cantor: Principien einer Theorie der
Ordnungstypen. Erste Mittheilung.", Acta Mathematica 124 (1970) 65-107]
I have held the following hypothesis for years: The cardinality of the body-matter is what I call, in
my investigations, the first cardinality, the cardinality of the ether-matter, on the other hand, is
the second.
... habe ich mir schon vor Jahren die Hypothese gebildet, daß die Mächtigkeit der
Körpermaterie diejenige ist, welche ich in meinen Untersuchungen die erste Mächtigkeit nenne,
daß dagegen die Mächtigkeit der Äthermaterie die zweite ist.
[G. Cantor: "Ueber verschiedene Theoreme aus der Theorie der Punktmengen in einem n-fach
ausgedehnten stetigen Raume Gn. Zweite Mitteilung.", Acta Mathematica Vol. 7, p. 105-124
(1885)]
Today we know that there is absolutely no application of infinite set theory to reality, simply
because there is no actual infinity in reality.
§ 040
Does the infinitely small exist in reality?
Quarks are the smallest elementary particles presently known. Down to 10-19 m there is no
structure detectable. Many physicists including the late W. Heisenberg are convinced that there
is no deeper structure of matter. On the other hand, the experience with molecules, atoms, and
elementary particles suggests that these physicists may be in error and that matter may be
further divisible. However, it is not divisible in infinity. There is a clear-cut limit.
Lengths which are too small to be handled by material meter sticks can be measured in terms
of wavelengths λ of electromagnetic waves, for instance.
λ = c/ν (c = 3ÿ108 m/s)
The frequency ν is given by the energy E of the photon
ν = E/h (h = 6,6ÿ10-34 Js)
and a photon cannot contain more than all the energy of the universe
E = mÿc2
which has a mass of about m = 5ÿ1055 g. This yields the complete energy E = 5ÿ1069 J. So the
unsurpassable minimal length is 4ÿ10-95 m.
Does the infinitely large exist in reality?
Modern cosmology teaches us that the universe has a beginning and is finite. But even if we
do not trust in this wisdom, we know that theory of relativity is as correct as human knowledge
can be. According to relativity theory, the accessible part of the universe is a sphere of
50ÿ109 LY radius containing a volume of 1080 m3. (This sphere is growing with time but will
remain finite forever.) "Warp" propulsion, "worm hole" traffic, and other science fiction (and
scientific fiction) does not work without time reversal. Therefore it will remain impossible to leave
(and to know more than) this finite sphere. Modern quantum mechanics has taught us that
entities which are non-measurable in principle, do not exist. Therefore, also an upper bound
(which is certainly not the supremum) of 10365 for the number of elementary spatial cells in the
universe can be calculated from the minimal length estimated above.
[W. Mückenheim: "The infinite in sciences and arts", Proc. 2nd Intern. Symp. of Mathematics
and its Connections to the Arts and Sciences (MACAS 2), B. Sriraman, C. Michelsen, A.
Beckmann, V. Freiman (eds.), Centre for Science and Mathematics Education, University of
Southern Denmark, Odense 2008, p. 265 - 272]
http://arxiv.org/abs/0709.4102
§ 041 Aristotle is the first to distinguish potential infinity and actual infinity. He bans actual
infinity from philosophy and mathematics. The idea of the infinity of God, created in Hellenism,
amalgamates - not later than in the works of Thomas Aquinatus - with the Aristotelian postulate
of the pure actuality of God. This yields the Christian perception of God's pure actuality. During
the renaissance, in particular with Bruno, the actual infinity is carried over from God to the world.
The finite world models of present science show clearly, how the superiority of the idea of actual
infinity has ceased, caused by the classical (modern) physics. In contrast it appears
disconcerting that G. Cantor explicitly established the actual infinity in mathematics during the
end of the last century. In the intellectual framework of our century - in particular when
considering existential philosophy - the actual infinity appears just like an anachronism.
Aristoteles unterscheidet als erster das Potentiell-Unendliche vom Aktual-Unendlichen - und
verbannt das Aktual-Unendliche aus der Philosophie und Mathematik. Der Gedanke der
Unendlichkeit Gottes, der aus dem Hellenismus stammt, verbindet sich - spätestens bei Thomas
- mit der von Aristoteles postulierten reinen Aktualität Gottes. So entsteht die christliche
Auffassung Gottes als aktualer Unendlichkeit. In der Renaissance, besonders bei Bruno,
überträgt sich die aktuale Unendlichkeit von Gott auf die Welt. Die endlichen Weltmodelle der
gegenwärtigen Naturwissenschaft zeigen deutlich, wie diese Herrschaft des Gedankens einer
aktualen Unendlichkeit mit der klassischen (neuzeitlichen) Physik zu Ende gegangen ist.
Befremdlich wirkt dem gegenüber die Einbeziehung des Aktual-Unendlichen in die Mathematik,
die explizit erst gegen Ende des vorigen Jahrhunderts mit G. Cantor begann. Im geistigen
Gesamtbilde unseres Jahrhunderts - insbesondere bei Berücksichtigung des existenzialistischen
Philosophierens - wirkt das Aktual-Unendliche geradezu anachronistisch.
[Paul Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) 3-11]
http://www.sgipt.org/wisms/geswis/mathe/ulorenze.htm#Das Aktual-Unendliche in der
Mathematik
§ 042 Recent history demonstrates that anyone foolhardy enough to describe his own work as
"rigorous" is headed for a fall. Therefore, we shall claim only that we do not knowingly give
erroneous arguments. We are conscious also of writing for a large and varied audience, for most
of whom clarity of meaning is more important than "rigor" in the narrow mathematical sense.
There are two more, even stronger reasons for placing our primary emphasis on logic and
clarity. Firstly, no argument is stronger than the premises that go into it, and as Harold Jeffreys
noted, those who lay the greatest stress on mathematical rigor are just the ones who, lacking a
sure sense of the real world, tie their arguments to unrealistic premises and thus destroy their
relevance.
Jeffreys likened this to trying to strengthen a building by anchoring steel beams into plaster.
An argument which makes it clear intuitively why a result is correct, is actually more trustworthy
and more likely of a permanent place in science, than is one that makes a great overt show of
mathematical rigor unaccompanied by understanding. Secondly, we have to recognize that there
are no really trustworthy standards of rigor in a mathematics that has embraced the theory of
infinite sets. Morris Kline (1980, p. 351) came close to the Jeffreys simile: "Should one design a
bridge using theory involving infinite sets or the axiom of choice? Might not the bridge collapse?"
The only real rigor we have today is in the operations of elementary arithmetic on finite sets of
finite integers, and our own bridge will be safest from collapse if we keep this in mind. [...]
Finally, some readers should be warned not to look for hidden subtleties of meaning which are
not present. [...] There are no linguistic tricks and there is no "meta-language" gobbledygook;
only plain English. We think that this will convey our message clearly enough to anyone who
seriously wants to understand it. In any event, we feel sure that no further clarity would be
achieved by taking the first few steps down that infinite regress that starts with: "What do you
mean by 'exists'?"
[E. T. Jaynes: "Probability Theory: The Logic of Science", (Fragmentary Edition of March 1996)]
http://www-biba.inrialpes.fr/Jaynes/cpreambl.pdf
http://de.wikipedia.org/wiki/Edwin_Thompson_Jaynes
§ 043 Pure mathematics and science are finally being reunited and, mercifully, the Bourbaki
plague is dying out. [Murray Gell-Mann: "Nature Conformable to Herself", Bulletin of the Santa
Fe Institute, 7 (1992) 7-10]
§ 044 Mathematics is a part of physics. Physics is an experimental science, a part of natural
science. Mathematics is the part of physics where experiments are cheap.
The Jacobi identity (which forces the heights of a triangle to cross at one point) is an
experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball).
But it can be discovered with less expense.
In the middle of the twentieth century it was attempted to divide physics and mathematics. The
consequences turned out to be catastrophic. Whole generations of mathematicians grew up
without knowing half of their science and, of course, in total ignorance of any other sciences.
They first began teaching their ugly scholastic pseudo-mathematics to their students, then to
schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under
the Sun).
Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for
application in any other science, the result was the universal hate towards mathematicians - both
on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of
the users.
The ugly building, built by undereducated mathematicians who were exhausted by their
inferiority complex and who were unable to make themselves familiar with physics [...]
predominated in the teaching of mathematics for decades. Having originated in France, this
pervertedness quickly spread to teaching of foundations of mathematics, first to university
students, then to school pupils of all lines (first in France, then in other countries, including
Russia).
To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is
commutative". He did not know what the sum was equal to and could not even understand what
he was asked about!
Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a
square, but that still has to be proved".
Judging by my teaching experience in France, the university students' idea of mathematics
(even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for
these obviously intelligent but deformed kids) is as poor as that of this pupil.
For example, these students have never seen a paraboloid and a question on the form of the
surface given by the equation xy = z2 puts the mathematicians studying at ENS into a stupor.
Drawing a curve given by parametric equations (like x = t3 - 3t, y = t4 - 2t2) on a plane is a totally
impossible problem for students (and, probably, even for most French professors of
mathematics).
[V.I. Arnold: "On teaching mathematics" (1997), Mathematics in Palais de Découverte in Paris
on 7 March 1997, Translated by A.V. Goryunov]
http://pauli.uni-muenster.de/~munsteg/arnold.html
§ 045 This question brings to the fore something that is fundamental and pervasive: that what
we are doing is finding ways for people to understand and think about mathematics.
The measure of our success is whether what we do enables people to understand and think
more clearly and effectively about mathematics.
We have a facility for thinking about processes or sequences of actions that can often be used
to good effect in mathematical reasoning. One way to think of a function is as an action, a
process, that takes the domain to the range. This is particularly valuable when composing
functions. Another use of this facility is in remembering proofs: people often remember a proof
as a process consisting of several steps. In topology, the notion of a homotopy is most often
thought of as a process taking time. Mathematically, time is no different from one more spatial
dimension, but since humans interact with it in a quite different way, it is psychologically very
different.
On the most fundamental level, the foundations of mathematics are much shakier than the
mathematics that we do. Most mathematicians adhere to foundational principles that are known
to be polite fictions. For example, it is a theorem that there does not exist any way to ever
actually construct or even define a well-ordering of the real numbers. There is considerable
evidence (but no proof) that we can get away with these polite fictions without being caught out,
but that doesn’t make them right. Set theorists construct many alternate and mutually
contradictory “mathematical universes” such that if one is consistent, the others are too. This
leaves very little confidence that one or the other is the right choice or the natural choice.
[Bill Thurston: "On proof and progress in mathematics", Bull. of the American Math. Soc. 30, 2,
(1994) 161-177]
http://arxiv.org/PS_cache/math/pdf/9404/9404236v1.pdf
http://www.hs-augsburg.de/~mueckenh/GU/GU11c
§ 046 If a non-terminating decimal is to be handled or arranged in sequence like a thing it is
sufficient to know how to handle and arrange a finite decimal of n digits, the number n being
subject to no restriction as to magnitude. The theorem would now demand that it is impossible to
set up any scheme for arranging all possible decimal fractions of n digits in a definite order, n
being subject to no restriction as to magnitude. But such a theorem is obviously false, for there
are 10n possible decimals of n digits [...] What is done in the actual diagonal Verfahren when
translated into this technique is this: it is shown that given a proposed array and any number n,
no matter how large, it is then possible to set up a decimal the first n digits of which are different
from the first n digits of any decimal to be found in the first n places of the proposed array. But
this is clearly not what is required.
The ordinary diagonal Verfahren I believe to involve a patent confusion of the program and
object aspects of the decimal fraction, which must be apparent to any who imagines himself
actually carrying out the operations demanded in the proof. In fact, I find it difficult to understand
how such a situation should have been capable of persisting in mathematics. Doubtless the
confusion is bound up with the notion of existence; the decimal fractions are supposed to "exist"
whether they can be actually produced and exhibited or not. But from the operational point of
view all such notions of "existence" must be judged to be obscured with a thick metaphysical
haze, and to be absolutely meaningless from the point of view of those restricted operations
which can be allowed in mathematical inquiry.
This repudiation of the conventional proof by the diagonal Verfahren of the non-denumerability
of the non-terminating decimals will be found to be very similar in spirit, although not in detail, to
the argument in Bentley's book [Linguistic Analysis of Mathematics]. It may be worth while to
record that the argument above was reached by me independently of Bentley [...] One can
obviously say that all the rules for writing down nonterminating decimals formulatable by the
entire human race up to any epoch in the future must be denumerable [...] I do not know what it
means to talk of numbers existing independent of the rules by what they are determined;
operationally there is nothing corresponding to the concept. If it means anything to talk about the
existence of numbers, then there must be operations for determining whether alleged numbers
exist or not, and in testing the existence of a number how shall it be identified except by means
of the rules?
[P.W. Bridgman: "A physicist's second reaction to Mengenlehre", Scripta Mathematica, Vol. II,
1934]
§ 047 Occasionally logicians inquire as to whether the current "Axioms" need to be changed
further, or augmented. The more fundamental question - whether mathematics requires any
Axioms - is not up for discussion. That would be like trying to get the high priests on the island of
Okineyab to consider not whether the Divine Ompah's Holy Phoenix has twelve or thirteen
colours in her tail (a fascinating question on which entire tomes have been written), but rather
whether the Divine Ompah exists at all. Ask that question, and icy stares are what you have to
expect, then it's off to the dungeons, mate, for a bit of retraining.
Mathematics does not require "Axioms". The job of a pure mathematician is not to build some
elaborate castle in the sky, and to proclaim that it stands up on the strength of some arbitrarily
chosen assumptions. The job is to investigate the mathematical reality of the world in which we
live. For this, no assumptions are necessary. Careful observation is necessary, clear definitions
are necessary, and correct use of language and logic are necessary. But at no point does one
need to start invoking the existence of objects or procedures that we cannot see, specify, or
implement.
The difficulty with the current reliance on "Axioms" arises from a grammatical confusion [...]
People use the term "Axiom" when often they really mean definition. Thus the "axioms" of group
theory are in fact just definitions. We say exactly what we mean by a group, that's all.
[..] Euclid may have called certain of his initial statements Axioms, but he had something else
in mind. Euclid had a lot of geometrical facts which he wanted to organize as best as he could
into a logical framework. Many decisions had to be made as to a convenient order of
presentation. He rightfully decided that simpler and more basic facts should appear before
complicated and difficult ones. So he contrived to organize things in a linear way, with most
Propositions following from previous ones by logical reasoning alone, with the exception of
certain initial statements that were taken to be self-evident. To Euclid, an Axiom was a fact that
was sufficiently obvious to not require a proof. This is a quite different meaning to the use of the
term today. Those formalists who claim that they are following in Euclid's illustrious footsteps by
casting mathematics as a game played with symbols which are not given meaning are
misrepresenting the situation.
[...] And yes, all right, the Continuum hypothesis doesn't really need to be true or false, but is
allowed to hover in some no-man's land, falling one way or the other depending on what you
believe. Cohen's proof of the independence of the Continuum hypothesis from the "Axioms"
should have been the long overdue wake-up call. In ordinary mathematics, statements are either
true, false, or they don't make sense. If you have an elaborate theory of "hierarchies upon
hierarchies of infinite sets", in which you cannot even in principle decide whether there is
anything between the first and second "infinity" on your list, then it's time to admit that you are no
longer doing mathematics.
Whenever discussions about the foundations of mathematics arise, we pay lip service to the
"Axioms" of Zermelo-Fraenkel, but do we ever use them? Hardly ever. With the notable
exception of the "Axiom of Choice", I bet that fewer than 5% of mathematicians have ever
employed even one of these "Axioms" explicitly in their published work. The average
mathematician probably can't even remember the "Axioms". I think I am typical - in two weeks
time I'll have retired them to their usual spot in some distant ballpark of my memory, mostly
beyond recall.
[...] Do you really think you need to have all the natural numbers together in a set to define the
function on natural numbers? Of course not - the rule itself, together with the specification of the
kinds of objects it inputs and outputs is enough. As computer scientists already know.
[N J Wildberger: "Set Theory: Should You Believe?"]
http://web.maths.unsw.edu.au/~norman/views2.htm
§ 048 The author [A. A. Fraenkel] is well known for his research in set theory as well as his
published textbooks in this subject. He has previously written the book "Einleitung in die
Mengenlehre" which appeared in three editions. The last edition which was published in 1928
was reprinted in New York in 1946. Whereas "Einleitung in die Mengenlehre" contained an
exposition of classical set theory as well as a survey of modern theoretical research in the
foundations of mathematics, Fraenkel has now decided to write the present book as an account
of the classical theory only. The modern aspects of foundation theory will be discussed in
another book under the title "Foundations of Set Theory" which is due to appear about 1955.
Presumably, the reason for this division of the contents of "Einleitung in die Mengenlehre" into
two different books is that the subject matter has grown too large. The reviewer, however, is not
enthusiastic about this division since such a textbook as the present one will be read primarily by
students and they might form the impression that classical set theory is securely founded just as
other parts of mathematics, e.g. arithmetic. Such an impression would, however, be misleading.
If it were not so, we could omit the entire modern foundational research without real loss to
mathematics. To the reviewer it seems unfortunate that classical set theory is developed in a
separate book so that all scruples - or almost all of them - are reserved for the second volume.
This might have the effect that most readers of this present volume will probably not become
acquainted with the criticisms at all. It is true that some hints to such scruples are given, but
most students might not think that they are important. On the other hand, it must be conceded
that the lack of knowledge of the results of foundational research will not mean much to
mathematicians who are not especially interested in the logical development of mathematics.
[Th. Skolem: "Review of: A. A. Fraenkel : Abstract Set Theory. Amsterdam & Groningen, NorthHolland Publishing Company, 1953. XII + 479 pp." Mathematica Skandinavica 1 (1953) 313.]
http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=179577
§ 049 How can a matheologian be distinguished from a mathematician? A witches' ordeal.
Take a circle (it need not be glowing) and ask the examinee to mark infinitely many intervals by
infinitely many endpoints. Shuffle the endpoints such that they slide along the circle in a
completely arbitrary way. Ask again how many endpoints are there and how many intervals
these endpoints define. The answer of a mathematician will be "infinitely many" in both cases.
The answer of a matheologian will be split. Yes, there are infinitely many endpoints, but between
them there are uncountably many (that is more than infinitely many) intervals, mainly degenerate
intervals, so called singletons: Between two endpoints there is only one point. However, the
endpoints limiting the singleton, cannot be identified and therefore cannot increase the number
of endpoints such that it is the same as the number of singletons. If you ask what makes the
singletons singletons, you will be told that there are infinitely many end- and other points, far
remote from the singleton, that stabilize the situation like the signs of the zodiac stabilize the
human fate and let all of us act as we do. Under this aspect even we could boast to be
uncountably many.
Well that's the matheologians' ordeal. If you, a reader of sci.math, meet a matheologian, try to
escape before he has caught you and turned your brain upside down.
§ 050 Already during Cantor’s life time, the reception of his ideas was more like that of new
trends in the art, such as impressionism or atonality, than that of new scientific theories. It was
highly emotionally charged and ranged from total dismissal (Kronecker’s “corrupter of youth”) to
highest praise (Hilbert’s defense of “Cantor’s Paradise”). (Notice however the commonly
overlooked nuances of both statements which subtly undermine their ardor: Kronecker implicitly
likens Cantor to Socrates, whereas Hilbert with faint mockery hints at Cantor’s conviction that
Set Theory is inspired by God.)
[Yuri I. Manin: Georg Cantor and his heritage (2002)]
http://aps.arxiv.org/PS_cache/math/pdf/0209/0209244v1.pdf
§ 051 Herren Geheimrat Hilbert und Prof. Dr. Cantor, I'd like to be Excused from your
"Paradise": It is a Paradise of Fools, and besides feels more like Hell
David Hilbert famously said: "No one shall expel us from the paradise that Cantor has created
for us."
Don't worry, dear David and dear Georg, I am not trying to kick you out. But, it won't be quite
as much fun, since you won't have the pleasure of my company. I am leaving on my own
volition.
For many years I was sitting on the fence. I knew that it was a paradise of fools, but so what?
We humans are silly creatures, and it does not harm anyone if we make believe that ¡0, ¡1, etc.
have independent existence. Granted, some of the greatest minds, like Gödel, were fanatical
platonists and believed that infinite sets existed independently of us. But if one uses the namedropping rhetorics, then one would have to accept the veracity of Astrology and Alchemy, on the
grounds that Newton and Kepler endorsed them. An equally great set theorist, Paul Cohen,
knew that it was only a game with axioms. In other words, Cohen is a sincere formalist, while
Hilbert was just using formalism as a rhetoric sword against intuitionism, and deep in his heart
he genuinely believed that Paradise was real.
My mind was made up about a month ago, during a wonderful talk (in the Integers 2005
conference in honor of Ron Graham's 70th birthday) by MIT (undergrad!) Jacob Fox (whom I am
sure you would have a chance to hear about in years to come), that meta-proved that the
answer to an extremely concrete question about coloring the points in the plane, has two
completely different answers (I think it was 3 and 4) depending on the axiom system for Set
Theory one uses. What is the right answer?, 3 or 4? Neither, of course! The question was
meaningless to begin with, since it talked about the infinite plane, and infinite is just as fictional
(in fact, much more so) than white unicorns. Many times, it works out, and one gets seemingly
reasonable answers, but Jacob Fox's example shows that these are flukes.
It is true that the Hilbert-Cantor Paradise was a practical necessity for many years, since
humans did not have computers to help them, hence lots of combinatorics was out of reach, and
so they had to cheat and use abstract nonsense, that Paul Gordan rightly criticized as theology.
But, hooray!, now we have computers and combinatorics has advanced so much. There are lots
of challenging finitary problems that are just as much fun (and to my eyes, much more fun!) to
keep us busy.
Now, don't worry all you infinitarians out there! You are welcome to stay in your Paradise of
fools. Also, lots of what you do is interesting, because if you cut-the-semantics-nonsense, then
you have beautiful combinatorial structures, like John Conway's surreal numbers that can
"handle" "infinite" ordinals (and much more beyond). But as Conway showed so well (literally!) it
is "only" a (finite!) game.
While you are welcome to stay in your Cantorian Paradise, you may want to consider
switching to my kind of Paradise, that of finite combinatorics. No offense, but most of the
infinitarian lore is sooo boring and the Bourbakian abstract nonsense leaves you with such a
bitter taste that it feels more like Hell.
But, if you decide to stick with Cantor and Hilbert, I will still talk to you. After all, eating meat is
even more ridiculous than believing in the (actual) infinity, yet I still talk to carnivores, (and even
am married to one).
[Doron Zeilberger: "Opinion 68" (2005)]
http://www.math.rutgers.edu/~zeilberg/Opinion68.html
§ 052 Most of the debate on the internet about Cantor's Theory is junk. The topic is a crank
magnet. Most of the people who participate in the debate, have no deep understanding of the
issues. However, hidden within all the noise, there does seem to be some signal. While the pure
mathematicians almost unanimously accept Cantor's Theory (with the exception of a small group
of constructivists), there are lots of intelligent people who believe it to be an absurdity.
Typically, these people are non-experts in pure mathematics, but they are people who have
found mathematics to be of great practical value in science and technology, and who like to view
mathematics itself as a science.
These "anti-Cantorians" see an underlying reality to mathematics, namely, computation. They
tend to accept the idea that the computer can be thought of as a microscope into the world of
computation, and mathematics is the science which studies the phenomena observed through
that microscope. They claim that that paradigm encompasses all of the mathematics which has
the potential to be applied to the task of understanding phenomena in the real world (e.g. in
science and engineering).
Cantor's Theory, if taken seriously, would lead us to believe that while the collection of all
objects in the world of computation is a countable set, and while the collection of all identifiable
abstractions derived from the world of computation is a countable set, there nevertheless "exist"
uncountable sets, implying (again, according to Cantor's logic) the "existence" of a super-infinite
fantasy world having no connection to the underlying reality of mathematics. The anti-Cantorians
see such a belief as an absurdity (in the sense of being disconnected from reality, rather than
merely counter-intuitive).
The mathematicians claim that they can "prove" the existence of uncountable sets, and hence
there's nothing to be debated. But that merely calls into question the nature of "proof". Certainly
infinite sets and power sets exist as abstractions. But, abstractions don't necessarily obey
exactly that same laws of logic as directly observable objects. Assuming otherwise can turn
abstractions into fantasies, and proofs into absurdities, and that's the crux of the anti-Cantorian's
argument.
The pure mathematicians tend to view mathematics as an art form. They seek to create
beautiful theories, which may happen to be connected to reality, but only by accident. Those
who apply mathematics, tend to view mathematics as a science which explores an objective
reality (the world of computation). In science, truth must have observable implications, and such
a "reality check" would reveal Cantor's Theory to be a pseudoscience; many of the formal
theorems in Cantor's Theory have no observable implications. The artists see the requirement
that mathematical statements must have observable implications as a restriction on their
intellectual freedom.
The "anti-Cantorian" view has been around ever since Cantor introduced his ideas. [...] In the
contemporary mainstream mathematical literature, there is almost no debate over the validity of
Cantor's Theory. [...] It was the advent of the internet which revealed just how prevalent the antiCantorian view still is; there seems to be a never-ending heated debate in the Usenet
newsgroups sci.math and sci.logic over the validity of Cantor's Theory. Typically, the anti-
Cantorians accuse the pure mathematicians of living in a dream world, and the mathematicians
respond by accusing the anti-Cantorians of being imbeciles, idiots and crackpots.
It is plausible that in the future, mathematics will be split into two disciplines - scientific
mathematics (i.e. the science of phenomena observable in the world of computation), and
philosophical mathematics, wherein Cantor's Theory is merely one of many possible formal
"theories" of the infinite.
[David Petry, sci.math, sci.logic, 20 Juli 2005]
http://groups.google.com/group/sci.logic/msg/02ee220b035488f9?dmode=source
§ 053 It seems reasonable that Plato were platonic in Plato's times, but is certainly surprising
the persistence of that primitive way of thinking in the community of contemporary
mathematicians [...]
But for those of us who believe in the organic nature of our brains and in its abilities of
perceiving and knowing modelled through more than 3600 millions years of organic evolution,
platonism has no longer sense. And neither self-reference nor the actual infinity may survive
away from the platonic scenario. On the other hand, it seems convenient to recall the long and
conflictive history of both notions (would them have been so conflictive if they were consistent?);
and above all their absolute uselessness in order to know the natural world. Physics and even
mathematics could go without both notions. Experimental sciences as chemistry, biology and
geology have never been related to them. The potential infinity probably suffices. Even the
number of distinguishable sites in the universe is finite. Finite and discrete: not only matter and
energy are discrete entities, space and time could also be of a discrete - quantum - nature as is
being suggested from some areas of contemporary physics as superstring theory, loop quantum
gravity, euclidean quantum gravity, quantum computation, or black holes thermodynamics.
[Antonio Leon Sanchez: "Extending Cantor's paradox" (2012)]
http://arxiv.org/abs/0809.2135
§ 054
They squabbled over lots of wine
'bout Darwin's ideas that might
humilitate mankind and take the shine
of human honour and pride!
They drank some mugs and steins so vast,
they stumbled and swayed through the doors.
They grunted audibly and crawled home at last
through dirt and mud on all fours.
(After Wilhelm Busch, Kritik des Herzens, 1874 (the year when Cantor started set theory))
http://de.wikisource.org/wiki/Sie_stritten_sich_beim_Wein_herum
Well, Husserl is, as I know with absolute certainty and as is clear from his local lectures about
proofs of God and against Darwinism, a theist and therefore is better suited as a teacher of
catholic students than the favourite candidates of Prof. Riehl.
[Cantor to canon Woker, Nov. 30, 1895]
Indeed, I think that the faculties, connect by an untearable relationship, cannot be indifferent
towards the question, whether the professor of the philosophical faculty is theist or atheist,
whether he acts pro Darwinism or contra Darwinism.
I did not want to initiate a direct action of the theological faculty in favour of Husserl, but I
meant, supported by Father Schwermer's information about the influence of the theological
faculty on the government, that a private, inofficial connivance of the candidate, so warmly
recommended by myself, would not be impossible.
[Cantor to canon Woker, Dec. 15, 1895]
As you requested I send some information concerning the candidates that presently are under
consideration by your philosophical faculty:
1. Siebeck in Gießen (1842)
2. Avenarius in Zürich (1843)
3. Eucken in Jena (1846)
4. Natorp in Marburg (1854)
5. Spitzer in Graz (1854)
6. Gross in Gießen (1861)
7. Busse in Marburg (1862)
As much as I would like to recommend the standpoint of Tolerari posse with respect to my young
friend Husserl, I have to express my greatest reservations towards those seven names. [...] Ad.
5. Has initially written a book about Darwinism, and nothing after that. Is probably a Jew and
radically liberal in every respect.
[Cantor to Heiner, Jan. 11, 1896]]
{{In order to reassure the reader's sense of justice I can say that Cantor's intrigues everytime
were condemned to fail. German appointment committees act properly according to laws and
conscience, as I can say from my own experience.
But Darwinism was not the only spiritual enemy that Cantor felt vocated to fight:}}
Now, the question rises whether it will be possible to reach the final aim, namely the total
destruction of the living-principle of freemasonry in all its facets. This is the purpose why I have
investigated and studied this dragon right through the centre of its black-blooded heart. I believe
that on this way I have been lead and furthered by the mercy of God.
[Cantor to Hermite, Febr. 11, 1896]
Original German texts can be found in: Das Kalenderblatt 090622
http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf
§ 055 Gödel takes the paradoxes very seriously; they reveal to him "the amazing fact that our
logical intuitions are self-contradictory." This attitude toward the paradoxes is of course at
complete variance with the view of Brouwer who blames the paradoxes not on some
transcendental logical intuition which deceives us, but on a gross error inadvertently committed
in the passage from finite to infinite sets. I confess that in this respect I remain steadfastly on the
side of Brouwer.
[H. Weyl: "Philosophy of Mathematics and Natural Science", Princeton, 1949]
[Komaravolu Chandrasekharan: "Hermann Weyl, Gesammelte Abhandlungen, Vol. IV", Springer
(1968) p. 602]
[Hermann Weyl: "Philosophie der Mathematik und Naturwissenschaft", Oldenbourg, 8. Aufl.
(2009)]
http://www.oldenbourgverlag.de/search/apachesolr_search/?_suche%25255Bmode%25255D=einfach&sv%25255Bolb
_vt%25255D=Weyl
§ 056 The mathematician Frege demolished the more traditional attempts to explain and
establish number and mathematical certainty. (To his satisfaction, at any rate). Frege went on to
try to give natural numbers and arithmetic a sound rational basis. To him, this meant in part
giving them a basis in a scheme of calculation. Frege faced many obstacles. Brouwer, and
Wittgenstein in his later period, proclaimed that mathematics cannot be founded by logic, or by
another layer of calculations. Frege's work was ignored in Germany; then, when his system was
finally published, it was immediately wrecked by Russell's paradox. In spite of this, Frege's
program was ultimately upheld by the professional majority (against Brouwer, for example). The
mathematical logic elaborated by Frege, and his cohorts Boole, Cantor, Peano, Skolem,
Herbrand, Hilbert, etc., became the prevailing view, or contextualization, of elementary
arithmetic.
In no way is that a trivial remark or outcome. Calling themselves new hens, the foxes seized
control of the henhouse. Arithmetic had been reductionistically re-founded in a way which
nullified its traditional consistency and uniqueness.
By no means did the discovery that Frege's system was inconsistent kill it. Instead, it was
repaired and adopted: even though the problem of repair was obdurate, and vitiated the tenet of
uniqueness of the natural numbers which crystallized Frege's original goal. (The introduction of
type theory and the Axiom of Reducibility.) Thus, the majority was prepared to resort to
"scandalously artificial" devices to prop up a system which in its straightforward formulation was
inconsistent. - Because the majority loved the new shell game. It must also be said that a
significant but weak minority opposed this course, e.g. Brouwer, Heyting, Weyl, Wittgenstein.
The lesson is that when an initially unpopular theory became popular, then prima facie
inconsistency did not kill it. It was upheld by casuistry, as it were - even though it continued to be
opposed by a minority.
[Henry Flynt: Is mathematics a scientific discipline? (1994)]
http://www.henryflynt.org/studies_sci/mathsci.html
§ 057 I have seen some ultrafinitists go so far as to challenge the existence of 2100 as a natural
number, in the sense of there being a series of "points" of that length. There is the obvious "draw
the line" objection, asking where in 21, 22, 23, ... , 2100 do we stop having "Platonistic reality"?
Here this ... is totally innocent, in that it can be easily be replaced by 100 items (names)
separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin
during a lecture of his. He asked me to be more specific. I then proceeded to start with 21 and
asked him whether this is "real" or something to that effect. He virtually immediately said yes.
Then I asked about 22, and he again said yes, but with a perceptible delay. Then 23, and yes,
but with more delay. This continued for a couple of more times, till it was obvious how he was
handling this objection. Sure, he was prepared to always answer yes, but he was going to take
2100 times as long to answer yes to 2100 then he would to answering 21. There is no way that I
could get very far with this.
[Harvey M. Friedman: "Philosophical Problems in Logic"]
http://www.math.ohio-state.edu/~friedman/manuscripts.html
http://www.math.ohio-state.edu/~friedman/pdf/Princeton532.pdf
§ 058 The only possible conclusion [given the Löwenheim-Skolem Theorem] seems to be that
notions such as countablility and uncountability are inherently relative. [...] Our description of
P(ω) as uncountable, even though correct, must be understood relative to our own current point
of view. From another point of view this very set may be countable. But I want to argue that such
relativism, compelling though it is, is subject to the by now familiar predicament that any
statement of it, if it is to be intelligible at all, will have to be understood within a framework that
casts it as a straighforward error.
It is this which I take to be Skolem's paradox. The crux of the matter is this. If there is an
implicit relativization in our claim that P(ω) is uncountable (the claim which is established by
Cantor's argument), then it ought to be possible to make it explicit (just as it is possible to make
explicit any relativization in the claim that a physical object is moving). But it is possible to do so
this only insofar as it is possible to construe our discourse about sets as discourse about a
particular collection of objects, the collection to which such claims must be relativized. And this
in turn is not possible unless we endorse the fundamental error that there is a set which contains
all the sets we intend to talk about. When it is claimed that P(ω) is not unconditionally
uncountable, we have no way of understanding this except as the demonstrably false claim that
it is not uncountable at all. Admittedly, there are interpretations of the language of set theory
under which all the "right" sentences come out true [1]
All that depends on context and interpretation (the correct, of course):
"There's - no - con - tra - dic - tion!"
Sometimes I ask myself: is it only accidentally, or is it due to my personality, that this battle shout
again and again reminds me of Orwell's cattle shout:
"Four - legs - good - two - legs - bad!"
[2] (Or was it the other way round? I don't remember exactly. I read the book before 1984
already.) and those who are shouting there?
[1] A. W. Moore: "Set Theory, Skolem's Paradox and the Tractatus", Analysis 45 (1985) 13-20.
http://analysis.oxfordjournals.org/cgi/pdf_extract/45/1/13
http://www.logicmuseum.com/cantor/skolem_moore.htm
[2] George Orwell: "Animal Farm"
http://en.wikipedia.org/wiki/Animal_Farm
§ 059 The physical limits to computation have been under active scrutiny over the past decade
or two, as theoretical investigations of the possible impact of quantum mechanical processes on
computing have begun to make contact with realizable experimental configurations. We
demonstrate here that the observed acceleration of the Universe can produce a universal limit
on the total amount of information that can be stored and processed in the future, putting an
ultimate limit on future technology for any civilization, including a time-limit on Moore's Law. The
limits we derive are stringent, and include the possibilities that the computing performed is either
distributed or local. A careful consideration of the effect of horizons on information processing is
necessary for this analysis, which suggests that the total amount of information that can be
processed by any observer is significantly less than the Hawking-Bekenstein entropy associated
with the existence of an event horizon in an accelerating universe.
[Lawrence M. Krauss, Glenn D. Starkman: "Universal Limits on Computatio" (2004)]
http://aps.arxiv.org/abs/astro-ph/0404510
Mathematicians work with abacuses
https://www.google.de/search?q=abacuses&hl=de&prmd=imvns&tbm=isch&tbo=u&source=univ
&sa=X&ei=USP1T9O0Ac7otQalkr3UBQ&sqi=2&ved=0CHIQsAQ&biw=855&bih=582
or brains, or computers (ABC). Everything beyond that scope - finished infinities, undefinable
definitions, and possibility-proofs of actions that provably are impossible to perform - is
matheology. Mathematics is what mathematicians do - not what they cannot do.
Enlightenment was a desire for human affairs to be guided by rationality rather than by faith,
superstition, or revelation; a belief in the power of human reason to change society and liberate
the individual from the restraints of custom or arbitrary authority; all backed up by a world view
increasingly validated by science rather than by religion or tradition.
[Dorinda Outram: "The Enlightenment", Cambridge University Press, Cambridge (1995)]
It is deeply deplorable that 200 years after that glorious victory of human spirit, the intellectual
counter-revolution is still alive.
§ 060 The cardinal contradiction is simply this: Cantor has a proof that there is no greatest
cardinal, and yet there are properties (such as "x = x") which belong to all entities. Hence the
cardinal number of entities having a property must be the greatest of cardinal numbers. Hence a
contradiction [1, p. 31]
An existent class is a class having at least one member. [1, p. 47] {{Surely you are joking Mr.
Russell? The class without any member is not among the existent classes?}}
Whether it is possible to rescue more of Cantor's work must probably remain doubtful until the
fundamental logical notions employed are more thoroughly understood. And whether, in
particular, Zermelo's axiom {{of choice}} is true or false {{I am shocked! An axiom could be true
or false in your age, Mr. Russell? Mathematicians in fact tried to find truth and meaning in
mathematics?}} is a question which, while more fundamental matters are in doubt, is very likely
to remain unanswered. The complete solution of our difficulties, we may surmise, is more likely
to come from clearer notions in logic than from the technical advance of mathematics; but until
the solution is found we cannot be sure how much of mathematics it will leave intact. [1, p 53]
{{O had Mr. Cantor never decided to become a mathematician!}}
Note added February 5th, 1906. - From further investigation I now feel hardly any doubt that the
no-classes theory affords the complete solution of all the difficulties stated in the first section of
this paper. [1, p 53] {{The classless society has been favoured all over the world at those times.
In fact, not much has remained: Cuba and North-Korea. But the no-classes theory is attractive
until this very day, because:}} The objections to the theory are [...] that a great part of Cantor's
theory of the transfinite, including much that it is hard to doubt, is, so far as can be seen, invalid
if there are no classes or relations. [1, p. 45] {{Is there anything that it is hard to doubt in Cantor's
theory?}}
The solution of the difficulties which formerly surrounded the mathematical infinite is probably
the greatest achievement of which our age has to boast. [2]
This is an instance of the amazing power of desire in blinding even very able men to fallacies
which would otherwise be obvious at once. [3]
[1] Bertrand Russell: "On some difficulties in the theory of transfinite numbers and order types",
Proc. London Math. Soc. (2) 4 (1906) 29-53, Received November 24th, 1905. - Read December
14, 1905
http://books.google.de/books/about/On_Some_Difficulties_in_the_Theory_of_Tr.html?id=vczfGw
AACAAJ&redir_esc=y
[2] Bertrand Russell: “The Study of Mathematics”. New Quarterly, Nov. 1907, 29-44. Reprinted in
B. Russell: "Philosophical Essays", Longmans, Green, London" (1910) also B. Russell:
"Mysticism and Logic and Other Essays" Longmans, Green & Co, London (1918). Reprinted by
Unwin, Paperbacks (1986)
[3] Bertrand Russell: "What I believe" from "Why I am not a Christian" (1957) p. 42. Bertrand
Russell: "Why I Am Not A Christian and Other Essays on Religion and Related Subjects", (Paul
Edwards, ed.), London: George Allen & Unwin (1957)
http://www.torrentz.com/4cfdc196f49eb34ea283163acfd1f79dffd667d1
http://en.wikipedia.org/wiki/Bertrand_Russell
061 Hilbert's Hotel is not a paradox, it is a very bad logical mistake, from the first paragraph. It
is based on the same terrible mistake that underlies all transfinite math. The mistake is believing
that the word "transfinite" can mean something. What it means in practice is really "transinfinite".
Mathematicians believe that something can exist beyond infinity.
If you accept the addition of 1 to infinity, then it means that you don't understand infinity to
begin with. All the math that takes place in the transinfinite is quite simply false. Notice that I do
not say it is physically baseless, or mystical, or avant garde, or any other half-way adjective. It is
false. It is wrong. It is a horrible, terrible mistake, one that is very difficult to understand. It is
further proof that Modern math and physics have followed the same path as Modern art and
music and architecture. It can only be explained as a cultural pathology, one where selfproclaimed intellectuals exhibit the most transparent symptoms of rational negligence. They are
outlandishly irrational, and do not care that they are. They are proud to be irrational. They
believe - due to a misreading of Nietzsche perhaps - that irrationality is a cohort of creativity. Or
it is a stand-in, a substitute. A paradox therefore becomes a distinction. A badge of courage. A
brave acceptance of Nature's refusal to make sense (as Feynman might have put it.) If we
somehow survive this cultural pathology, the future will look upon our time in horror and
wonderment. How did we ever reach such fantastic levels of intellectual fakery and denial,
especially in a century steeped in the warnings of Freud to beware of just this illness?
[Miles Mathis: "Introductory Remarks on Cantor"]
http://milesmathis.com/cant.html
§ 062 Matheologians are accustomed to ignore the difference between potential and actual
infinity. Usually they even deny that there is any difference. In this paragraph I will make another
attempt to show this difference.
Potential infinity is a never ending, never completed sequence or chain of steps like the
following, constructed of initial segments of the ordered set of positive even numbers.
|{2}| = 1 < 2
|{2, 4}| = 2 < 4
|{2, 4, 6}| = 3 < 4, 6
|{2, 4, 6, 8}| = 4 < 6, 8
|{2, 4, 6, 8, 10}| = 5 < 6, 8, 10
|{2, 4, 6, 8, 10, 12}| = 6 < 8, 10, 12
...
This sequence is not more than all its terms. It cannot surpass all its terms. It is incapable of
reaching a cardinality larger than every positive even number, the failure getting clearer and
clearer with every term at every step.
The existence of alephs, however, requires actual infinity, completed infinity, finished infinity. But
then we have to swallow results like the following:
Consider an urn and infinitely many actions performed within one hour (the first one needs 1/2
hour, the second one 1/4 hour, the third one 1/8 hour and so on).
Fill in 1, 2, remove 1
Fill in 3, 4, remove 2
Fill in 5, 6, remove 3
continue.
After one hour the urn is empty, because for every number the time of removal can be
determined. So for the set X of numbers residing in the urn we obtain
LimtimeØ1h |X| ∫ |LimtimeØ1h X|
This prevents any application of set theory to reality although it was Cantor's outspoken aim to
apply set theory to reality (cp., e.g., his letter to Mittag-Leffler of Sept. 22, 1884). And it prevents
any application to mathematics too, because in mathematics the limit of the sequence
21
2.1
432.1
43.21
6543.21
654.321
...
is not 0 but ¶.
Or assume the existence of all rational numbers of [0, 1], enumerated, for instance, according to
Cantor's method, starting with 0. If they all exist, then all permutations should exist, because
each number is in finite distance from the first number 0, enumerated by 1. But then also the
permutation with all rational numbers enumerated and sorted according to their magnitude
should be obtainable within aleph_0 steps and, therefore, should exist, shouldn't it? This
prevents any application of set theory to mathematics either.
Therefore only potential infinity can prevail. This is the true reason why matheologians deny to
recognize the difference between potential and actual infinity. While most of the followers
certainly honestly claim that they never thought about that problem, I cannot believe that the
leading matheologians always have observed the same ignorance.
§ 063 {{It is impossible to order or to well-order items that cannot be identified. Zermelo's
corresponding proof is incorrect - just as is his edition of Cantor's letters:}}
[...] the final section of the correspondence with Cantor starts only in July 1899. This was the
part from which extracts were published in the edition of Cantor's papers by Zermelo using the
transcriptions made by Cavaillès. [...]
The standard of editing of the extracts is bad. [...]
The collection begins with a letter from Cantor of 28 July 1899. lt is the most famous of them
all [...] lt is often cited in the literature on the foundations of mathematics, and was translated into
English in [J. van Heijenoort: "From Frege to Gödel ..." (1967, Cambridge, Mass.) 113-117.]
There does not exist a letter in this form.
[I. Grattan-Guinness: "The Rediscovery of the Cantor-Dedekind Correspondence", Jahresbericht
DMV 76 (1974) p. 126f]
§ 064 Pure mathematics will remain more reliable than most other forms of knowledge, but its
claim to a unique status will no longer be sustainable.
For centuries mathematics has been seen as the one area of human endeavor in which it is
possible to discover irrefutable, timeless truths. Indeed, theorems proved by Euclid are just as
true today as they were when first written down more than 2000 years ago. That the sun will rise
tomorrow is less certain than that two plus two will remain equal to four.
However, the 20th century witnessed at least three crises that shook the foundations on which
the certainty of mathematics seemed to rest. [Goedel, Four-Color Theorem, Classification of
Finite Simple Groups. ...]
A problem that can be formulated in a few sentences has a solution more than ten thousand
pages long. The proof has never been written down in its entirety, may never be written down,
and as presently envisaged would not be comprehensible to any single individual. The result is
important and has been used in a wide variety of other problems in group theory, but it might not
be correct.
These three crises could be hinting that the currently dominant Platonic conception of
mathematics is inadequate. As Davies remarks:
{{These}} crises may simply be the analogy of realizing that human beings will never be able to
construct buildings a thousand kilometres high and that imagining what such buildings might
"really" be like is simply indulging in fantasies.
We are witnessing a profound and irreversible change in mathematics, Davies argues, which will
affect decisively its character:
{{Mathematics}} will be seen as the creation of finite human beings, liable to error in the same
way as all other activities in which we indulge. Just as in engineering, mathematicians will have
to declare their degree of confidence that certain results are reliable, rather than being able to
declare flatly that the proofs are correct.
Davies's article "Whither Mathematics?" (PDF, 448 KB) is available at Mathematics: The Loss of
Certainty.
[Science Blog (2005)]
http://scienceblog.com/community/older/2005/10/200509095.shtml
§ 065 The induction principle is this: if a property holds for 0, and if whenever it holds for a
number n it also holds for n + 1, then the property holds for all numbers. For example, let θ(n) be
the property that there exists a number m such that 2ÿm = nÿ(n + 1). Then θ(0) (let m = 0).
Suppose 2ÿm = nÿ(n + 1). Then 2ÿ(m + n + 1) = (n + 1)ÿ((n + 1) + 1), and thus if θ(n) then θ(n + 1).
The induction principle allows us to conclude θ(n) for all numbers n. As a second example, let
π(n) be the property that there exists a non-zero number m that is divisible by all numbers from 1
to n. Then π(0) (let m = 1). Suppose m is a non-zero number that is divisible by all numbers from
1 to n. Then mÿ(n + 1) is a non-zero number that is divisible by all numbers from 1 to n + 1, and
thus if π(n) then π(n + 1). The induction principle would allow us to conclude π(n) for all numbers
n.
The reason for mistrusting the induction principle is that it involves an impredicative concept of
number. lt is not correct to argue that induction only involves the numbers from 0 to n; the
property of n being established may be a formula with bound variables that are thought of as
ranging over all numbers. That is, the induction principle assumes that the natural number
system is given. A number is conceived to be an object satisfying every inductive formula; for a
particular inductive formula, therefore, the bound variables are conceived to range over objects
satisfying every inductive formula, including the one in question.
In the first example, at least one can say in advance how big is the number m whose existence
is asserted by θ(n): it is no bigger than nÿ(n + 1). This induction is bounded, and one can hope
that a predicative treatment of numbers can be constructed that yields the result θ(n). In the
second example, the number m whose existence is asserted by π(n) cannot be bounded in
terms of the data of the problem.
lt appears to be universally taken for granted by mathematicians, whatever their views on
foundational questions may be, that the impredicativity inherent in the induction principle is
harmless - that there is a concept of number given in advance of all mathematical constructions,
that discourse within the domain of numbers is meaningful. But numbers are symbolic
constructions; a construction does not exist until it is made; when something new is made, it is
something new and not a selection from a pre-existing collection. There is no map of the world
because the world is coming into being.
[Edward Nelson: "Predicative arithmetic", Princeton University Press, Princeton (1986)]
http://www.math.princeton.edu/~nelson/books/pa.pdf
http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT#417,66,Folie 66
§ 066 [...] according to our view it is meaningless to talk about the set of all points of the
continuum.
Numbers facilitate counting, measuring, and calculating.
The numbers belong to the realm of thinking, the continuum belongs to the realm of visualizing. I
repeat, for me the continuum is not identical with the set —.
[...] the logical short circuit results from transforming the infinitely increasing number of digits of
the potential infinite into an actual infinite and identifying ◊2 with the never ending decimal
representation 1.4142 ...
[Detlef Laugwitz: "Zahlen und Kontinuum", BI, Zürich (1986)]
Or the short circuit results from identifying 2-1 + 2-2 + ... + 2-k + ... with the natural number 1.
Limnض 1/n = 0 means the variable number 1/n will never be 0.
We'll have always have 1/n > 0. In modern language: " n: 1/n > 0.
Similarly,
Limnض 2-1 + 2-2 + ... + 2-n = 1
does not mean anything else than
2-1 + 2-2 + ... + 2-n = 1 - 2-n
and 2-n is never 0.
No last natural number L can be defined. And even if it could, 2-L = 0 would certainly not be
satisfied. Therefore
2-1 + 2-2 + ... + 2-n + ... = 1
is only a short cut. Every interpretation as a sum of actually infinitely many terms would be just
the short circuit that Laugwitz criticizes.
If you disregard the very simplest cases, there is in all of mathematics not a single infinite series
whose sum has been rigorously determined.
[Niels Henrik Abel quoted in G. F. Simmons: "Calculus Gems", New York (1992)]
With the exception of the geometric series, there does not exist in all of mathematics a single
infinite series whose sum has been determined rigorously.
[Niels Henrik Abel quoted in E. Maor: "To infinity and beyond", Birkhäuser, Boston (1986)]
I don't know which of the two quotes is more to the point. But that does not matter. In its
important clause they are sufficiently concurrent: "... there is in all of mathematics not a single
infinite series whose sum has been rigorously determined" because the finished infinite is not
part of mathematics and unfinished series haven't fixed sums.
§ 067 Cantor’s Problem
What is born on Sacred Cows is as much religion as is any theology.
This paper offers a contrary conclusion to Cantor’s argument, together with implications to the
theory of computation. [...] We first dispose of misconceptions arising of terms. The word
"infinite" is a fabrication of mental ether - a vacuous expression lacking mathematical precision with many mystical connotations. [...] We define a set to be computationally-countable if once
initiated, producing a monotonic sequence of integers, the process of counting - whatever that
might be - at some future time is assured to cease [...]. We define a set to be computationallyuncountable where once initiated, exclusive association being constant, the process of counting
does not cease. [...] We will use the term “denumerable” where we mean to infer the more
conventional concept of “countable” meaning an injection or bijection with the set of integers. [...]
The principal set [...] S of all strings composed of elements from the set containing the symbols 0
(zero) and 1 (one). How many elements are in the set S? [...] we can then repeat the
concatenation process ad infinitum. Independent of whatever the cardinality of S was presumed
to be when we began, this process will "count" our discovery of additional elements without
ceasing. Such would seem to reasonably assure the set S is computationally-uncountable;
regardless any initial presumptions as to its cardinality. [...] S is denumerable by definition. [...]
Assume some set exists for which no injection into S exists. Let that set be represented by the
symbol Q. We first partition all plausible sets into two disjoint subsets: Q1 and Q2. Let Q1 be
those sets for which representation of elements is derived from a computationally-countable set
of symbols. Let Q2 be those sets for which representation of elements is derived from a
computationally-uncountable set of symbols. Let Q’ be some set in Q1 defined above with
representation of elements of Q’ derived of strings of symbols in some computationallycountable set P’. The symbols in P’ can be placed into a 1 - 1 correspondence with some set of
integers in Ÿ, thus in 1 - 1 correspondence with some set of elements of S. Then by simple
substitution rewriting of their representation all elements of Q’ are mapped to elements of S. The
consequence of said mapping is at least an injection from Q’ to a set contained in S. It follows
then that for any set Q’ that |Q’| is not greater than |S|. Let Q’’ be some set in Q2 defined above
with representation of elements of Q’’ derived of strings of symbols in some computationallyuncountable set P’’. By applying Cantor’s Diagonalization method, having established for any
subset of P’’ an injective assignment from S one can obtain yet another element in S to assign
the next yet unassigned element in P’’, using the assignments thus far made to assure the
mapping is a proper injection (i.e. 1 - 1). The injection of the set of symbols into S, while not
computable in total, arguably exists. It then follows of the same principles as held for Q’ above
that S contains at least one set into which an injection from Q’’ is formed through simple
substitution rewriting of the symbols of each string.
It can therefore be concluded that there exists no set Q having cardinality greater than the
cardinality of S, as any set Q can be shown to be either in Q1 or Q2 as defined above, and thus
“contained” by correspondence with a set contained in S through an injection into S. The later is
a result of considered importance to the conclusions that Cantor’s Problem does admit. Now to
the essential question of whether — has an injection into any set contained in S. Since — is
represented by strings over the set P’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, ‐, .}, a computationallycountable (finite) set, it can be concluded that S contains at least one set into which an injection
from — exists because — is in Q1 as later was defined above. Such a conclusion contradicts
Cantor's conclusion as to the relative cardinality of — and Ÿ; given that it has been shown that Ÿ
has a bijection to S by constructive proof and — has at least an injection into S. If we assume
that — exists in a bijection relation to S, such grants at best |—| = |Ÿ|. [...] set C = {c | c {m, w}*}
[...] is contained in Q1, as defined above. By direct substitution of symbols m Ø 0 and w Ø 1, any
string in C can be rewritten to a string in S. Given that there does not exist in C two strings c1, c2
such that c1 = c2, the rewriting of strings in C produces by definition a bijection with S. This
results in a contradiction for two reasons. The bijection with S is such that C is denumerable by
the same means with which S is placed into bijection relation with Ÿ. While at the same time, S
is not denumerable if the Diagonalization argument is valid. The latter then stands in
contradiction of the bijection between Ÿ and S.
[Charles Sauerbier (2009)]
http://arxiv.org/ftp/arxiv/papers/0912/0912.0228.pdf
§ 068 To most mathematicians, the title of this article will, I suppose, appear a bit strange: it is
so obvious that 265536 is a natural number that there would seem to be no rational basis for
questioning it. Yet there have been objections to the claim that all such exponential expressions
name a natural number, two of the best known being due to Paul Bernays and Edward Nelson.
Bernays, in "On Platonism in Mathematics", rhetorically questions whether 67257729 can be
represented by an "Arabic numeral" (he does not, however, press the discussion). By contrast,
Nelson, in "Predicative Arithmetic", develops a large body of theory which he then advances to
support his belief that 265536 is not a natural number or that, more generally, exponentiation is
not a total function. [...] For while it does not limit the use of induction it does imply that the effect
of induction is context-dependent. It also implies that when the objects of discussion are
linguistic entities (and in this paper the position advocated is that "natural numbers" or better
"natural number notations" are linguistic entities) then that collection of entities may vary as a
result of discussion about them. A consequence of this is that the "natural numbers" of today are
not the same as the "natural numbers" of yesterday. Although the possibility of such denotational
shifts remains inconceivable to most mathematicians, it seems to be more compatible with the
history both of the cultural growth (and of growth in individuals) of the number concept than is
the traditional, Platonic picture of an unchanging, timeless, and notation-independent sequence
of numbers.
[David Isles: "What evidence is there that 265536 is a natural number?", Notre Dame Journal of
Formal Logic, 33,4 (1992) 485-480.]
http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.ndjfl/1093
634481&page=record
§ 069 Summary: A study of the philosophical and historical foundations of infinity highlights the
problematic development of infinity. Aristotle distinguished between potential and actual infinity,
but rejected the latter. lndeed, the interpretation of actual infinity leads to contradictions as seen
in the paradoxes of Zeno. lt is difficult for a human being to understand actual infinity {{not only
for a human being but for every intelligent being that can understand the intended meaning of
actual infinity at all}}. Our logical schemes are adapted to finite objects and events. {{Praises
be!}} Research shows that students focus primarily on infinity as a dynamic or neverending
process {{they may have read Fraenkel and Gödel, cp. KB 090923}}. Individuals may have
contradictory intuitive thoughts at different times without being aware of cognitive conflict. The
intuitive thoughts of students about both the actual (at once) infinite and potential (successive)
infinity are very complex. The problematic nature of actual infinity and the contradictory intuitive
cognition should be the starting point in the teaching of the concept infinity.
Dubinsky (2005b:261) wys ook daarop dat studente glo dat die vergelyking 0,99999... = 1 vals
is. {{Taken literally, they are right. Since the actual infinite does not exist, 0.99999... is at best by
definition to be interpreted as 1 - as a limit without explicitly written limit symbol.}} Studente dink
dat (1) 0,99999... 'n bietjie kleiner is as 1, die naaste wat jy daaraan kan kom sonder om dit
werklik te bereik; en (2) die verskil tussen die twee (09999... en 1) oneindig klein is {{potentially,
yes}} - Tall noem dit die intreding van die ifinitesimale (1978:6); of (3) dat 0,9999... die laaste
getal voor 1 is. [p. 76]
This thesis contains beautiful fractals and several grahics by Escher.
[Rinette Mathlener: "Die problematiek van die begrip oneindigheid", Dissertation, Magister
Educationis, Universiteit van Suid-Afrika(2008)]
http://uir.unisa.ac.za/handle/10500/1927
http://uir.unisa.ac.za/bitstream/handle/10500/1927/dissertation.pdf?sequence=1
§ 070 The iterative conception can only help resolve the paradoxes if we view it as clarifying
the intuitive concept of a collection, not as introducing a new, distinct “set” concept. But this
clarification is elusive, as can be seen by looking at some of the versions of the conception
which have appeared in the literature:
Sets are "formed", "constructed", or "collected" from their elements in a succession of stages ...
([18], p. 506)
According to the iterative conception, sets are created stage-by-stage, using as their elements
only those which have been created at earlier stages. ([19], p. 183)
In the metaphor of the iterative conception, the steps that build up sets are “operations” of
“gathering together” sets to form “new” sets. ([21], p. 637)
Thus a set is formed by selecting certain objects ... we want to consider a set as an object and
thus to allow it to be a member of another set ... When we are forming a set z by choosing its
members, we do not yet have the object z, and hence cannot use it as a member of z. ([22], p.
322-323)
It should be apparent from this selection that the nature of the set-forming operation is extremely
unclear. There seems to be no general agreement even as to whether this is an actual operation
which could in any sense be carried out, or instead some kind of impenetrable metaphor. The
problem is apparent in Boolos’s remark that “a rough statement of the idea ... contains such
expressions as ‘stage’, ‘is formed at’, ‘earlier than’, ‘keep on going’, which must be exorcised
from any formal theory of sets. From the rough description it sounds as if sets were continually
being created, which is not the case” ([4], p. 491). Yet Boolos does not follow his rough
statement with a more informative informal description that avoids the objectionable phrases,
and it seems doubtful that he could. Without these expressions there is no informal description.
This difficulty is connected to the ontological problem, about which none of the authors cited
above has anything meaningful to say: if we have no idea what sets are supposed to
be,obviously there is little we can say about how they are supposed to be formed. Yet the idea
that sets are in some sense “formed” from elements which enjoy some kind of “prior” existence
is crucial to the iterative conception’s ability to evade the classical paradoxes. The point is
supposed to be that the set of all sets, or the set of all ordinals, or Russell’s set, are illegitimate
on the iterative account precisely because they cannot be “formed”. So it seems fair to say that
the iterative conception successfully deals with the paradoxes only to the extent that it presents
a clear picture of the operation of set formation, which is to say, not at all.
[2] P. Benecerraf and H. Putnam, eds., Philosophy of Mathematics: Selected Readings (second
edition), 1983.
[4] G. Boolos, The iterative conception of set, in [2], 486-502.
[9] K. Gödel, What is Cantor’s continuum problem?, in [2], 470-485.
[18] C. Parsons, What is the iterative conception of set?, in [2], 503-529.
[19] M. D. Potter, Iterative set theory, Philosophical Quarterly 43 (1993), 178-193.
[20] W. V. Quine, Immanence and validity, Dialectica 45 (1991), 219-230.
[21] M. F. Sharlow, Proper classes via the iterative conception of set, J. Symbolic Logic 52
(1987), 636-650.
[22] J. R. Shoenfield, Axioms of set theory, in Handbook of Mathematical Logic, J. Barwise, ed.,
1977, 321-344.
[Nik Weaver: "Is set theory indispensable?" (2009)]
http://arxiv.org/pdf/0905.1680.pdf
§ 071 The Hausdorff Sphere Paradox [...] (here X, Y, Z are disjoint sets which nearly cover the
sphere, and X is congruent to Y, in the sense that a rotation of the sphere makes X coincide with
Y, and likewise Y is congruent to Z. But what is extraordinary is the claim that X is also
congruent to the union of Y and Z, even though Y ∫ Z). We are, like Poincaré and Weyl, puzzled
by how mathematicians can accept and publish such results; why do they not see in this a
blatant contradiction which invalidates the reasoning they are using?
Nevertheless, L. J. Savage (1962) accepted this antinomy as literal fact and, applying it to
probability theory, said that someone may be so rash as to blurt out that he considers congruent
sets on the sphere equally probable; but the Hausdorff result shows that his beliefs cannot
actually have that property. The present writer, pondering this, has been forced to the opposite
conclusion: my belief in the existence of a state of knowledge which considers congruent sets on
a sphere equally probable, is vastly stronger than my belief in the soundness of the reasoning
which led to the Hausdorff result.
Presumably, the Hausdorff sphere paradox and the Russell Barber paradox have similar
explanations; one is trying to define weird sets with self-contradictory properties, so of course,
from that mess it will be possible to deduce any absurd proposition we please. [...]
Russell's theory of types can dispose of a few paradoxes, but far from all of them. Even with
the best of good will on both sides, it would require at least another generation to bring about the
reconciliation of pure mathematics and science. For now, it is the responsibility of those who
specialize in infinite set theory to put their own house in order before trying to export their
product to other fields. Until this is accomplished, those of us who work in probability theory or
any other area of applied mathematics have a right to demand that this disease, for which we
are not responsible, be quarantined and kept out of our field. In this view, too, we are not alone;
and indeed have the support of many non-Bourbakist mathematicians.
[E. T. Jaynes: "Probability Theory: The Logic of Science", (Fragmentary Edition of March 1996)]
http://www-biba.inrialpes.fr/Jaynes/cappb8.pdf
§ 072 No matter how much the content of mathematics exploits paradox, mathematicians
express dedication to policing their doctrine against inconsistency. Mathematicians do not
welcome those who attempt inconsistency proofs of favored theories.
What was profound was that results were contextualized so that they ceased to be inconsistency
threats. The Löwenheim-Skolem paradox; Skolem's w-inconsistent model of Peano arithmetic
(also the conjunction of Gödel's completeness and incompleteness theorems); w-inconsistency
of Quine's "New Foundations" set theory; independence of the Axiom of Choice; etc. But
mathematics had always proceeded like this: e.g. Dedekind had taken Galileo's paradox as the
definition of infinity.
The alternative would be that the content of academic mathematics eventuates from sophistry
and majority preference. As to the latter, Paul Lorenzen says: You will become famous if you
please famous people - and all famous mathematicians like axiomatic set theory.
I will refer to such considerations as professional discipline (or even coercion). If this is the
situation, then the profession will protect itself from heresies and criticism not by superior
reasoning {{no doubt!}}, but by additional professional discipline in conjunction with additional
casuistry.
At the beginning of rational mathematics stands the result concerning the incommensurability of
the side and diagonal of a square. This severe embarrassment, which can be conceived as an
elementary refutation of mathematics, was instead co-opted as a program for the development
of mathematics. (With legitimacy of proof by contradiction as a crucial consideration allowing
this.) And yet the status of irrational numbers and the continuum of reals has been questionable
to this very day.
Already a precedent for this approach has been provided by historical studies of the Axiom of
Choice (with the Banach-Tarski paradox) by Gregory Moore and Stan Wagon. From their
excavations, we learn that Émile Borel published a book, Les Paradoxes de l'Infini (3rd ed.,
Paris, 1946), which on p. 210 said that the Banach-Tarski paradox amounts to an inconsistency
proof of the Axiom of Choice.
I do not take a compliant view of mathematics. I consider it as a historically given doctrinal
institution; placing in suspension any claim that there is an ideal perfect mathematics. Given
Hennix's combined historical and rational scrutiny of the development of mathematical doctrine, I
will propose that the main factor in the establishment of "truth" in mathematics is professional
procedure and discipline.
There already are many results which are inconsistency proofs in effect. Only professional
discipline forestalls the obvious interpretation of these proofs as inconsistency proofs. Quasiinconsistency proofs are neutralized or co-opted by negotiation and the addition of layers of
interpretation.
The most celebrated results of the twentieth century (probably earlier centuries as well) came
from skirting paradox while claiming not to land in it. A paradoxical positing is made deliberately,
and then is deflected so that, as interpreted, it is not a contradiction.
Truth is negotiated on the basis of manipulation of import by distorting interpretations.
Interpretation takes the form of discarding traditional intentions concerning mathematical
structure: the privileged position of Euclidian geometry; the invariance of dimension; the
association between integer and magnitude; uniqueness of the natural number series; etc.
From time to time, results are discovered which patently embarrass the conventional wisdom, or
controvert popular tenets. [The Gödel theorems.] Then follows a political manipulation, to distort
the unwanted result by interpretation so that it is seen to "enhance" the popular tenet rather than
to controvert it.
Even if my sense of the situation is right, the appearance of such a professionally compelling
proof would be a more a matter of packaging and selling than anything else. [...] The biggest
hurdle such an attempted proof faces is professional discipline. Whether inconsistency proofs
are recognized to have occurred is subject to entirely "political" manipulation.
So, even though, for example, the Hausdorff-Banach-Tarski paradox has been called the most
paradoxical result of the twentieth century, classical mathematicians have to convince
themselves that it is natural, because it is a consequence of the Axiom of Choice, which
classical mathematicians are determined to uphold, because the Axiom of Choice is required for
important theorems which classical mathematicians regard as intuitively natural.
[Henry Flynt: "Is mathematics a scientific discipline?" (1994)]
http://www.henryflynt.org/studies_sci/mathsci.html
§ 073 Consistency and Madness
Go to a mental hospital and I'll bet you will meet people there who call themselves Napoleon, or
Jesus Christ, or Elvis Presley, or Albert Einstein. For the sake of simplicity, let us take the man
with the Einstein complex. Now suppose that you take such a man apart and you decide to talk
with him, in order to convince him that Albert Einstein is dead and buried. And that his real name
is Johnson. And that he is just the man around the corner. No genius at all. Do you think you are
going to be successful?
You talk to him for more than one hour, trying to convince him that he should give up his
picture of the world. At last, you ask him if he has understood your arguments. I'll bet his answer
will be like the following: "Yes, of course I understand ! Because Albert Einstein, he is a genius.
He is so clever that he can understand, of course, any of your arguments. That's why. And
what's more, I am Albert Einstein."
Now replace "Einstein" by any mainstream mathematician and you're done. Nobody can deny
that people in a mental hospital have a consistent picture of the world. Now the good news of
consistency is that, once you are on the right track, you will remain on the right track. The axiom
system for Euclidian Geometry is a good example of this manner of being consistent.
But here comes the bad news.
Once you are on the wrong track, you will always be on the wrong track.
There is no way to tell a mathematician that the axioms of the Zermelo Fraenkel / axiom of
Choice (ZFC) system are "not good". Because then he will defend himself as our would-be
Einstein character did. All of his arguments will be consistent with the system he believes in. He
wants to remain in his vicious circle, safe and well. Calling everybody else a "dude" and a
"crank" and a "zealot". And there is no way out. There is no cure for his madness. Because
mathematics is what mathematicians do.
[Han de Bruijn: Natural philosophy]
http://www.alternatievewiskunde.nl/QED/natural.htm#oo
§ 074 Dedekind tried to describe an infinite class by saying that it is a class which is similar to a
proper subclass of itself. [...] I am to investigate in a particular case whether a class is finite or
not, whether a certain row of trees, say, is finite or infinite. So, in accordance with the definition, I
take a subclass of the row of trees and investigate whether it is similar (i.e. can be co-ordinated
one-to-one) to the whole class! (Here already the whole thing has become laughable.) It hasn’t
any meaning; for, if I take a "finite class" as a subclass, the attempt to co-ordinate it with the
whole class must eo ipso fail: and if I make the attempt with an infinite class - but already that is
a piece of nonsense, for if it is infinite, I cannot make an attempt to co-ordinate it. [...] An infinite
class is not a class which contains more members than a finite one, in the ordinary sense of the
word "more". If we say that an infinite number is greater than a finite one, that doesn't make the
two comparable, because in that statement the word "greater" hasn’t the same meaning as it
has say in the proposition 5 > 4!
The form of expression "m = 2n correlates a class with one of its proper subclasses" uses a
misleading analogy to clothe a trivial sense in a paradoxical form. (And instead of being
ashamed of this paradoxical form as something ridiculous, people plume themselves on a victory
over all prejudices of the understanding). It is exactly as if one changed the rules of chess and
said it had been shown that chess could also be played quite differently.
When "all apples" are spoken of, it isn’t, so to speak, any concern of logic how many apples
there are. With numbers it is different; logic is responsible for each and every one of them.
Mathematics consists entirely of calculations.
In mathematics description and object are equivalent. {{Therefore numbers that cannot be
described cannot exist.}} "The fifth number of the number series has these properties" says the
same as "5 has these properties". The properties of a house do not follow from its position in a
row of houses; but the properties of a number are the properties of a position.
[L. Wittgenstein: "Philosophical Grammar", Basil Blackwell, Oxford (1969), quoted from E.D.
Buckner: "THE LOGIC MUSEUM" (2005), unfortunately no longer on the web.]
http://www.amazon.de/Philosophical-Grammar-LudwigWittgenstein/dp/0631123504/ref=sr_1_1?ie=UTF8&qid=1286647502&sr=1-1
There is no path to infinity, not even an endless one. [§ 123]
It isn't just impossible "for us men" to run through the natural numbers one by one; it's
impossible, it means nothing. […] you can’t talk about all numbers, because there's no such
thing as all numbers. [§ 124]
An "infinitely complicated law" means no law at all. [§ 125]
There's no such thing as "all numbers" simply because there are infinitely many. [§ 126]
Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It
correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of
classes, of which one is correlated with the other, but which are never related as class and
subclass. Neither is this infinite process itself in some sense or other such a pair of classes.
In the superstition that m = 2n correlates a class with its subclass, we merely have yet another
case of ambiguous grammar. [§ 141]
Generality in mathematics is a direction, an arrow pointing along the series generated by an
operation. And you can even say that the arrow points to infinity; but does that mean that there is
something - infinity - at which it points, as at a thing? Construed in that way, it must of course
lead to endless nonsense. [§ 142]
If I were to say "If we were acquainted with an infinite extension, then it would be all right to talk
of an actual infinite", that would really be like saying, "If there were a sense of abracadabra then
it would be all right to talk about abracadabraic sense perception". [§ 144]
Set theory is wrong because it apparently presupposes a symbolism which doesn't exist instead
of one that does exist (is alone possible). It builds on a fictitious symbolism, therefore on
nonsense. [§ 174]
[L. Wittgenstein: "Philosophical Remarks", quoted from E.D. Buckner: "THE LOGIC MUSEUM"
(2005), unfortunately no longer on the web]
§ 075 Several examples are used to illustrate how we deal cavalierly with infinities and
unphysical systems in physics.
[...] At this point, we can ask what all this {{Cantor's infinites}} implies for physics and physical
systems. There are significant implications for physical systems if we accept the major premise
that rather than being merely potential, real infinities do exist. We shall now cite some examples
to illustrate some of these implications.
The case of an infinitely long line charge distribution is a simple example. In this case both the
length and the electric charge are of cardinality C for the continuum and consequently the ratio λ
representing the charge density is finite. The case of the thermodynamic limit invoked in
statistical mechanics is a fascinating counter-example. If we treat them as real infinities, then the
ratio N/V involves the number of atoms which is countable and thus of cardinality ¡0 whereas
the volume of the system is of cardinality C x C x C = C. For this ratio to be finite we must
require that C must equal a finite number times ¡0. This last statement is however, not true for
real infinities. Hence it follows that the finiteness of the number density N/V would not have been
possible if N and V were real infinities. Indeed the finiteness of the number density follows from
the fact that each of these quantities is finite to begin with, as was discussed earlier.
[...] We conclude by making a few general observations.
Thermodynamic systems are devoid of infinities and are inherently finite. N is countable and
large and V is measurable and macroscopically large but all physical parameters are finite and
measurable and finite, including the number density. There is a class of physical systems
containing infinities but which can be re-examined by using methods which have successfully
prevailed in thermodynamics and statistical mechanics, with a view to resolve the problem
infinities in these systems.
[...] Finally there may be physical systems containing real infinities which cannot be
transformed away. These systems may perhaps be understood only by a re-examination based
on Cantor’s transfinite mathematics. {{May or may not.}} In this context, it is useful to remember
that Cantor’s theory is well grounded in physical reality {{where can we find that reality?}}: it is
based on arithmetic and set theory. {{That holds also for astrology, numerology, and
Traumdeutung. But an argument to join one of these sects cannot be obtained from this paper.}}
{{Footnote 9}} An interesting point in the history of mathematics is the continuum hypothesis:
that there is no cardinality bigger than ¡0 and smaller than C. This notion is neither proved nor
disproved. {{Meanwhile it has been proved: there is no cardinality bigger than ¡0. See, for
instance, § 062}}
[P. Narayana Swamy: "Infinities in Physics and Transfinite numbers in Mathematics" (1999)]
http://arxiv.org/abs/math-ph/9909033
§ 076 Mathematics, we hold, deals with multiple models of the world. It is not subsumed in any
one big model or by any one grand system of axioms.
The idea that set theory is relative is not new; it was clearly stated for axiomatic set theory by
Skolem in 1922 [9]. We intend simply to draw some of the philosophic consequences of that
relativity. For the Platonist, there is a real world of sets, existing forever, described only
approximately by the Zermelo-Fraenkel axioms or by their modifications. It may be that some
final insight will give a definite axiom system, but the sets themselves are the underlying
mathematical reality.
In our view, such a Platonic world is speculative. It cannot be clearly explained as a matter of
fact (ontologically) or as an object of human knowledge (epistemologically). Moreover, such
ideal worlds rapidly become too elaborate; they must display not only the sets but all the other
separate structures which mathematicians have described or will discover. The real nature of
these structures does not lie in their often artificial construction from set theory, but in their
relation to simple mathematical ideas or to basic human activities.
Hence, we hold that mathematics is not the study of intangible Platonic worlds, but of tangible
formal systems which have arisen from real human activities.
Many students of set theory do not follow what I have called the "Grand Set Theoretic
Foundation" but instead follow Cantor to emphasize the intuitive notion of a set as a collection
which is a real object in its own right. For them set theory is not subsumed by the ZermeloFraenkel axiom system or by any other first-order formal system. It may be studied formally by
other means; using infinitary languages or second-order logic Such Cantorian sets are just as
real as numbers. Indeed, one might say that number theory is formalized only in part by Peano's
arithmetic in just the way set theory is formalized, but only in part, by Zermelo-Fraenkel. (There
can be true properties of whole numbers not demonstrated in Peano arithmetic.)
From this point of view, set theory is just another branch of mathematics {{if at all}}. If in this
view set theory is not taken to be the foundation of mathematics {{relieving}}, it can be
assimilated with our proposal that mathematics consists of formal disciplines derived from a
variety of human activities. {{Human activities? Is omega-tasking a human activity? Well, in
some sense. At least believing in is - like worshipping of the devil (with a lot of formal rules).}}
The various earlier philosophies of mathematics [...] each arose out of the dominant aspects of
mathematics as then understood. For example, Platonism arose in Greece and applied to
mathematics there because it fitted Greek geometry; it has been popular among mathematicians
recently because it fitted well with the view that mathematics derives from axioms for sets.
Logicism arose together with the discovery and formalization of mathematical logic. Intuitionism
was the child of emphasis on numbers as the starting point of mathematics and on intuition as a
basis of topology. Formalism arose with the development of axiomatic methods and the
discovery that proof theory might give consistency proofs for abstract mathematics. Empiricism
sprang from the 19th-century view of mathematics as almost coterminal with theoretical physics;
it was much influenced by Kant's dichotomy between analytic and synthetic.
Now we search for a philosophy of mathematics better attuned to the present state of the
subject.
2. Paul C. Eklof, Whitehead's problem is undecidable, this Monthly, 83 (1976) 775-787.
9. Th. Skolem, Einige Bemerkungen zur axiomatische Begründung der Mengenlehre. Fifth
Congress of Scandinavian Mathematicians, 1922, Helsingfors, 1923, pp. 2 17-232.
[Saunders Mac Lane: "Mathematical models: A sketch for the philosophy of mathematics", The
American Mathematical Monthly, Vol. 88, No. 7 (1981) 462-472.]
http://home.dei.polimi.it/schiaffo/TFIS/philofmaths.pdf
The answer is: MatheRealism.
mhtml:http://www.hs-augsburg.de/~mueckenh/MR.mht
http://arxiv.org/abs/math/0505649
§ 077 Well known is the story of Tristram Shandy
[Laurence Sterne: "The Life and Opinions of Tristram Shandy" (1759-1767)]
http://en.wikipedia.org/wiki/The_Life_and_Opinions_of_Tristram_Shandy,_Gentleman
who undertakes to write his biography, in fact so pedantically, that he needs for each of the first
days of his life a full year. Of course he will never get ready if continuing that way. But if he
would live infinitely long (for instance a countable infinity of years), then his biography would get
"ready", because every day in his life, how late ever, finally would get its description.
[Adolf A. Fraenkel: "Einleitung in die Mengenlehre" 3rd ed., Springer, Berlin (1928) p. 24]
Original German Text:
http://www.hs-augsburg.de/~mueckenh/GU/GU12c.PPT#395,21,Folie 21
... for instance the story of Tristram Shandy who writes his autobiography so pedantically that
the description of each day takes him a year. If he is mortal he can never terminate; but if he
lived forever then no part of his biography would remain unwritten, for to each day of his life a
year devoted to that day's description would correspond.
[A.A. Fraenkel, A. Levy: "Abstract Set Theory", North Holland, Amsterdam (1976) p. 30]
§ 078 Digital Philosophy (DP) is a new way of thinking about the fundamental workings of
processes in nature. DP is an atomic theory carried to a logical extreme where all quantities in
nature are finite and discrete. This means that, theoretically, any quantity can be represented
exactly by an integer. Further, DP implies that nature harbors no infinities, infinitesimals,
continuities, or locally determined random variables. This paper explores Digital Philosophy by
examining the consequences of these premises. [...] Digital Philosophy makes sense with regard
to any system if the following assumptions are true:
All the fundamental quantities that represent the state information of the system are ultimately
discrete. In principle, an integer can always be an exact representation of every such quantity.
For example, there is always an integral number of neutrons in a particular atom. Therefore,
configurations of bits, like the binary digits in a computer, can correspond exactly to the most
microscopic representation of that kind of state information.
In principle, the temporal evolution of the state information (numbers and kinds of particles) of
such a system can be exactly modeled by a digital informational process similar to what goes on
in a computer. Such models are straightforward in the case where we are keeping track only of
the numbers and kinds of particles. For example, if an oracle announces that a neutron decayed
into a proton, an electron, and a neutrino, it’s easy to see how a computer could exactly keep
track of the changes to the numbers and kinds of particles in the system. Subtract 1 from the
number of neutrons, and add 1 to each of the numbers of protons, electrons, and neutrinos.
The possibility that DP may apply to various fields of science {{except transfinite set theory}}
motivates this study.
http://www.digitalphilosophy.org/
http://en.wikipedia.org/wiki/Edward_Fredkin
§ 079 A second negative effect of transfinite theory is that by positioning transfinite theory as
the basis arithmetic and logic, transfinite theorists have accomplished the unintended goal of
basing mathematics on mystical concepts: namely, the completed infinity. Such a result is highly
attractive to the mystical mind. Again, classical logicians would abhor the idea of basing
reasoning on mystical concepts, but the mystical mind adores it... gaining further acceptance for
transfinite theory.
This leads to one of the weirdest twists in modern mathematics: The intuitionalist school, led
by LEJ Brouwer actually demanded a higher degree of logical rigor than the Logicists school that
was happy with a mathematics founded on the slippery slope of transfinite theory, completed
infinities and paradoxes. Of course, we have seen the same twists in politics. The "people's"
party is generally a pseudonym for a dictatorship. Tax reform almost always leads to higher
taxes. Intuitionism in mathematics does not mean that people rely on their "intuition." Intuitionist
school demands that all axioms be based on comprehendible ideas. The logistical schools holds
that the axioms don't really matter, so long as symbolic logic used is consistent. The ideas
expressed by the symbols can be insane. [...]
In my opinion, the fact that schools no longer teach logic is the worst effect of transfinite
theory.
[Kevin Delaney: "Curing the Disease"]
http://descmath.com/diag/cure.html
§ 080 Borel's thesis is that the overwhelming majority of numbers will always remain
inaccessible to the human race as we know it, in the sense that it will never be possible to define
these numbers effectively in such a manner that any two mathematicians will be certain that they
are speaking about one and the same entity.
If an enumerable set, such as that of the natural numbers, is considered instead of the unit
interval, then the assignment of equal probabilities to the elements of this set reduces to zero the
global probability of any number of accessible integers, which, according to Borel, is absurd
because it precludes the possibility of ever getting one of these numbers, so that every choice
leads to an inaccessible number. {{Think of a number ... The choice is very restricted.}}
Borel discusses the familiar decomposition of the circumference of a circle into an enumerable
number of mutually exclusive, congruent sets of points. He asserts that it is not possible to
attribute equal probabilities to these sets without running into contradictions, and that it is
therefore necessary to attribute unequal probabilities to them. "But then we contradict the
Euclidean principle of equality, according to which two superposable figures are equal." As the
construction of these sets "requires the use of Zermelo's axiom, our conclusion is that it is
necessary to choose between Zermelo's axiom and Euclid's axiom according to which two
superposable figures are equal, that is to say, identical from all points of view, and that, in
particular, equal probabilities correspond to them. The simultaneous application of the two
axioms leads, in fact, to a contradiction." (Borel, needless to say, chooses "Euclid's axiom.")
{{He was just a real mathematician.}} This argument is open to objections. First, it is not
inconceivable that a nonmeasurable set can be constructed without the intervention of Zermelo's
axiom. Secondly, there is another way in which two superposable figures may be "identical from
all points of view," without having equal probabilities correspond to them, and that is, by having
no probabilities correspond to them {{why then not starting off with a completely meaningless
mathematics?}}. Euclid's axiom cannot be interpreted as stating that congruent figures - if the
"figures" in question are not the elementary Euclidean ones {{either they are "figures" hence
"elementary Euclidian", or no "figures" at all}} - have probabilities and that these probabilities are
equal. [...] Complex numbers were once considered meaningless, whereas today some
mathematicians consider Zermelo's axiom and its consequences meaningless. {{In fact, one can
discern by this question mathematicians from matheologians.}} Borel's notion of accessibility,
although of heuristic significance, seems too subjective, temporal, and, by precluding intrinsically
the possibility of delimiting the realm of the accessible {{admitedly, for delimiting paradoxes like
that of Hausdorff-Banach-Tarski are well suited}}, vague, according to his own standards, to
"define in a precise manner a science of the accessible and of the real." {{Quite that he did contrary to Zermelo.}}
[F. Bagemihl: "Review: É. Borel: 'Les nombres inaccessibles', Gauthier- Villars, Paris (1952)",
Bull. Amer. Math. Soc. 59,4 (1953) 406-409]
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/118
3518025
§ 081 Skolem's conclusion was that the notions of set theory are relative to the universe of sets
under consideration (Skolem 1923, p. 224): The axiomatic founding of set theory leads to a
relativity of set theoretic notions, and this relativity is inseparably connected with every
systematic axiomatization.
Later Skolem even strengthened his opinion by what can be regarded the essential motto of socalled "Skolem relativism" (Skolem 1929, p. 48): There is no possibility of introducing something
absolutely uncountable, but by a pure dogma.
Because of this relativistic attitude he avoided traditional set theory and became negligent of
problems concerned with semantical notions.
Fraenkel was not sure about the correctness of the proof of the Löwenheim-Skolem theorem.
von Neumann instantly recognized the importance of the result but he reacted with scepticism
about the possibility of overcoming the weakness of axiomatizations they reveal (1925, p. 240).
[Heinz-Dieter Ebbinghaus: "Ernst Zermelo, An Approach to His Life and Work", Springer (2007)
199f]
§ 082 Cantor offered several proofs that there is no one-one correlation between the real
numbers and the natural numbers, but only the presumption that there are infinite numbers can
turn whatever impossibility there is here into a seeming demonstration that the number of the
real numbers is greater than the number of natural numbers.
Cantor defined real numbers in terms of Cauchy sequences of rationals, specifically in terms of
equivalence classes of such sequences (Suppes [26], pp 161, 181). But he merely presumed
there were such equivalence classes, not realising that Russell's Paradox, amongst other things,
required such a presumption to be justified in any particular case. Now it was on the basis of this
assumption that Cantor was able to prove that the real number system, unlike the rational
number system, was complete, i.e. that every Cauchy sequence of real numbers, unlike every
Cauchy sequence of rational numbers, has a limit ([26], p 185). And that leads to the
representation of every real number as not just the would-be limit of a sequence of finite
decimals, but also a limit which is actually reached, so that a real number is identical with a
certain infinite decimal ([26], pp 189, 191). The fact that there cannot be such completed decimal
expansions therefore shows that the sets Cantor used to define the reals do not exist: the
Cauchy sequences of rationals equivalent to a given one do not form a set (c.f. [27], p 92). Now
in addition to the above, well-known "proofs" of the non-denumerability of certain sets, in terms
of unending decimals, and in terms of the subsets of the natural numbers, Cantor also gave a
proof of the non-denumerability of the reals which rested solely on the completeness of the real
number system (Dauben [8], p 51, Grattan-Guinness [10], pp 185-6, see also § 6.2). But the
Platonically real limit he there presumed to exist is now shown not to exist, which means we
remain compelled to see the infinite as an undifferentiated lack of number.
[8] Dauben, J.W., Georg Cantor, Princeton U.P., Princeton, 1990.
[10] Grattan-Guinness, I., From the Calculus to Set Theory 1630-1910, Duckworth, London,
1980.
[26] Suppes, P., Axiomatic Set Theory, Van Nostrand, Princeton N.J., 1960.
[27] Tiles, M., The Philosophy of Set Theory, Blackwell, Oxford, 1989.
[Hartley Slater: "The Uniform Solution of the Paradoxes" (2004)]
§ 083 As a foundation for mathematics, then, set theory is far less firm than what is founded
upon it; for common sense in set theory is discredited by the paradoxes. Clearly we must not
look to the set-theoretic foundation of mathematics as a way of allaying misgivings regarding the
soundness of classical mathematics. Such misgivings are scarce anyway, once such offenses
against reason as the infinitesimal have been set right. [...]
For the one thing we insist on, as we sort through the various possible plans for passable set
theories, is that our set theory be such as to reproduce, in the eventual superstructure, the
accepted laws of classical mathematics. This requirement is even useful as a partial guide when
in devising a set theory we have to choose among intuitively undecidable alternatives. We may
look upon set theory, or its notation, as just a conveniently restricted vocabulary in which to
formulate a general axiom system for classical mathematics - let the sets fall where they may.
{{In any case mathematics has priority. If set theory is unable to reproduce the simple and
unambiguous fact that the Binary Tree cannot contain more recognizable paths than nodes, then
set theory is unsuitable as a foundation of recognizable mathematics describing recognizable
reality.}}
[Willard V. O. Quine: "The ways of paradox and other essays", Harvard University Press (1966)
p. 31f]
http://books.google.de/books?id=YReOv31gdVIC&source=gbs_navlinks_s
§ 084 When discussing the validity of the Axiom of Choice, the most common argument for not
taking it as gospel is the Banach-Tarski paradox. Yet, this never particularly bothered me. The
argument against the Axiom of Choice which really hit a chord I first heard at the Olivetti Club,
our graduate colloquium. It’s an extension of a basic logic puzzle, so let’s review that one first.
100 prisoners are placed in a line, facing forward so they can see everyone in front of them in
line. The warden will place either a black or white hat on each prisoner’s head, and then starting
from the back of the line, he will ask each prisoner what the color of his own hat is (ie, he first
asks the person who can see all other prisoners). Any prisoner who is correct may go free.
Every prisoner can hear everyone else’s guesses and whether or not they were right. If all the
prisoners can agree on a strategy beforehand, what is the best strategy?
[...] the first guy counts the total number of white hats. If it is odd, he says “white”, and if it is
even, he says “black”. Then the guy in front of him can count the number of white hats he can
see, and if differs from the parity the first guy counted, he knows his hat is white. But now the
next guy knows the parity of white hats the first guy saw, and whether or not the second guy had
a white hat, so he can compare it to the white hats he sees, and find out if his own hat is white.
This argument repeats, and so everyone except the first guy guesses correctly.
Its interesting to notice that a larger number of hat colors poses no problem here. For any set
of hat colors, the prisoners can pick an abelian group structure on. Then, the first prisoner
guesses the "sum" of all the hat colors he can see. The next guy can then subtract the sum of
the hat colors he sees from the hat color the first guy said to find his own hat color. Again, this
argument repeats, and so everyone except the first guy gets out. For the case of black and
white, the previous argument used black = 0 (mod 2) and white = 1 (mod 2).
This is all well and good, but it doesn’t seem to help the countably infinite prisoners in the
second puzzle. Since they can’t hear anyone else’s guess, they can’t set up a similar system for
passing on information. So what can they do?
First, instead of thinking of hat colors, they just turn white into 1 and black into 0 (like above).
Then, a possible scenario of hats on their heads is an infinite sequence of 1’s and 0’s. Call two
such sequences ‘equivalent’ if they are equal after a finite number of entries. This is an
equivalence relation, and so we can talk about equivalence classes of sequences.
Next, the prisoners invoke the Axiom of Choice to pick an element in each equivalence class,
which they all agree on and memorize. Now, when they are put in line and get a hat, they will be
able to see all but a finite part of the sequence, and so they can all tell what equivalence class
they are in. Their strategy is then to guess as if they were in the pre-chosen element in that
equivalence class.
How well does this work? Well, the sequence they are actually in and the representative
element they picked with the axiom of choice must be equivalent, so they are the same after a
finite number of entries. Therefore, after a finite number of incorrect guesses, each prisoner will
miraculously guess his hat color correctly!
This solution is also pretty stable, in that most attempts to make the puzzle harder don’t break
it. The warden can know their plan and even know their precise choice of representative
sequences. If so, he can make sure any arbitrarily large finite number of them are wrong, but he
can’t get an infinite number of them. Also, the number of hat colors can be arbitrarily big; the
same solution works identically.
This last point is pretty trippy. In the two color case, its very reasonable for any prisoner to
guess his hat color correctly, and also for arbitrarily large numbers of them to get it right in a row.
Effectively, at no finite point in the guessing do the results of the optimal strategy appear to differ
from random guessing. However, if there are uncountably many hat colors, then the probability
of any prisoner randomly guessing his hat color is 0. One can reasonably expect no prisoners to
be correct for random guessing, so when eventually that first prisoner guesses correctly, the
warden should be rightly shocked (though not as shocked as he will be when all but a finite
number of prisoners guess correctly).
I find this solution deeply troubling to the intuitive correctness of the axiom of choice. Sure, this
is based primarily on my intuition for finite things and a naive hope that they should extend to
infinities. I think particularly troubling is in the uncountably many colors case, where any given
prisoner has no chance to guess his hat color correctly, and yet almost all prisoners are correct.
[Greg Muller: "The Axiom of Choice is Wrong", September 13, 2007]
http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/
After a talk by Mike O’Connor and an older publication here:
http://mathforum.org/wagon/fall05/p1035.html
§ 085 People have asked me, "How can you, a nominalist, do work in set theory and logic,
which are theories about things you do not believe in?" ... I believe there is value in fairy tales
and in the study of fairy tales.
[A. Burdman-Feferman, S. Feferman: "Alfred Tarski - Life and Logic", Cambridge Univ. Press
(2004) p. 52]
http://books.google.de/books?id=wqktlxHo9wkC&printsec=frontcover&dq=%22+life+and+logic%
22&hl=de&ei=YzQwTO7iCJ2SOK6o3MUB&sa=X&oi=book_result&ct=result&resnum=1&ved=0
CCwQ6AEwAA#v=onepage&q&f=false
§ 086 The reaction to Skolem's results was split. For example, Fraenkel was not sure about the
correctness of the proof of the Löwenheim-Skolem theorem, and he seems to have had
difficulties in analysing the role of logic with sufficient rigour to understand Skolem's paradox [...]
A "War" Against Skolem
What about Zermelo? When faced with the existence of countable models of first-order set
theory, his first reaction was not the natural one, namely to check Skolem's proof and evaluate
the result - it was immediate rejection {{there nogthing has changed}}. Apparently, the motivation
of ensuring "the valuable parts of set theory" which had led his axiomatizations from 1908 and
from the Grenzzahlen paper had not only meant allowing the deduction of important set-theoretic
facts, but had included the goal of describing adequately the set-theoretic universe with its
variety of infinite cardinalities. Now it was clear that Skolem's system, like perhaps his own,
failed in this respect. Moreover, Skolem's method together with the epistemological
consequences Skolem had drawn from his results, could mean a real danger for mathematics
{{not at all! - quite the opposite}} like that caused by the intuitionists: In his Warsaw notes W3 he
had clearly stated that "true mathematics is indispensably based on the assumption of infinite
domains," among these domains, for instance, the uncountable continuum of the real numbers.
Hence, following Skolem, "already the problem of the power of the continuum loses its true
meaning".
Henceforth Zermelo's foundational work centred around the aim of overcoming Skolem's
relativism and providing a framework in which to treat set theory and mathematics adequately.
Baer speaks of a real war Zermelo had started, wishing him "Heil und Sieg und fette Beute", at
the same time pleading for peace [...]
However, peace was not to come. [...] a vivid impression of Zermelo's uncomprising
engagement, at the same time also revealing his
worries:
It is well known that inconsistent premises can prove anything one wants; however, even the
strangest consequences that Skolem and others have drawn from their basic assumption, for
instance the relativity of the notion of subset or equicardinality, still seem to be insufficient to
raise doubts about a doctrine that, for various people, already won the power of a dogma that is
beyond all criticism. [...] His remedy consisted of infinitary languages {{save nonsense by
nonsense}}. [...] Skolem had considered such a possibility, too, but had discarded it because of a
vicious circle (Skolem 1923, p. 224):
In order to get something absolutely uncountable either the axioms themselves would have to
be present in an absolutely uncountably infinite number or one would have to have an axiom
which could provide an absolutely uncountable set of first-order sentences. However, in all
cases this leads to a circular introduction of higher infinities, that means, on an axiomatic basis
higher infinities exist only in a relative sense.
[Heinz-Dieter Ebbinghaus, Volker Peckhaus: "Ernst Zermelo: an approach to his life and work",
Springer (2007) p. 199 ff]
http://www.springer.com/math/history+of+mathematics/book/978-3-540-495512?cm_mmc=Google-_-Book%20Search-_-Springer-_-0
§ 087 The Banach-Tarski Gyroscope is an intricate mechanism believed to have been
constructed using the Axiom of Choice. On each complete rotation counterclockwise, the
Banach-Tarski Gyroscope doubles in volume while maintaining its shape and density; on
rotating clockwise, the volume is halved. When first discovered, fortunately in the midst of
interstellar space, the Banach-Tarski Gyroscope was tragically mistaken for an ordinary desk
ornament. Subsequently it required a significant portion of the available energy of the
contemporary galactic civilization to reverse the rotation before nearby star systems were
endangered; fortunately, the Banach-Tarski Gyroscope still obeys lightspeed limitations on
rotation rates, and cannot grow rapidly once expanding past planetary size. After the subsequent
investigation, the Banach-Tarski Gyroscope was spun clockwise and left spinning.
http://news.ycombinator.com/item?id=411727
§ 088 Consider a sequence of indexed natural numbers
11
2 1, 3 2
4 1, 5 2, 6 3
...
The sequence of the indices
1
1, 2
1, 2, 3
...
has limit Ù. But the sequence of natural numbers
1
2, 3
4, 5, 6
...
has limit «, because Ù will be exhausted.
What is the limit of the indexed numbers?
The indexed numbers may be carried by the balls of a supertask which runs as follows: In the
nth step fill into a vase n balls, take off the n - 1 balls that were inside before.
§ 089 In a 1924 paper Tarski proved that seven well-known propositions in cardinal arithmetic
whose proofs use the axiom of choice are actually equivalent to the axiom. In the same year he
published the first systematic development of a theory of finite sets, based on Zermelo's axioms,
but with the negation of the axiom of infinity and no axiom of choice. Dedekind and Hausdorff
had envisaged such a project, but Tarski was the first to realize it completely. {{Set theory
without finished infinity? That's like amalgamating set theory with chocolate - brown too, but
delicious.}}
[...] The eminent French mathematician Denjoy later incorporated substantial portions of the
paper into one of his books [...]. Still in the same year, Tarski published with Stefan Banach a
paper [1] that quickly became famous. Its main theorem asserts that any two bounded sets with
interior points are equivalent by finite decomposition. For example, a sphere can be
decomposed into a finite number of pieces that can be reassembled, using rigid motions
(translations, rotations, and reflections), into two spheres, each of which is congruent to the
original one. More dramatically, a sphere the size of a pea can be decomposed into a finite
number of pieces that can be reassembled to make a sphere as big as the sun. The proof
makes essential use of the axiom of choice. The fact that the axiom has such paradoxical
consequences was seen by some as evidence that it should not be accepted. {{Émile Borel
published a book, Les Paradoxes de l'Infini, which on p. 210 said that the Banach-Tarski
paradox amounts to an inconsistency proof of the Axiom of Choice.}}
Between 1923 and 1926, Tarski discovered that a number of implications in cardinal arithmetic
that had traditionally been proved using the axiom of choice could in fact be proved without it (at
the price of using a more complicated argument). He announced these and many other results in
a 1926 paper that was jointly written with Adolf Lindenbaum. There one also finds, for instance,
the theorem that the generalized continuum hypothesis implies the axiom of choice. A total of
146 theorems are listed in the paper, all of them without proof. {{Why not? If the theorems were
conving.}} We sense that in Lindenbaum, Tarski had found a kindred spirit: the results came so
fast that they didn't have time to write them up properly. Sierpinski spent some of the difficult
years during World War II, when Warsaw University was closed, working out the proofs of the
theorems in this paper. Tarski seems to have had a passion for set theory and, in particular, for
cardinal arithmetic during this period. A proof of the continuum hypothesis once came to him in a
dream, and the "proof" was so good it took him two weeks to find the mistake. {{Unbelievable. It
is so simple: There is no continuum.}}
[1] Stefan Banach and Alfred Tarski, "Sur la decomposition des ensembles de points en parties
respectivement congruentes," Fundamenta Mathematicae 6, 1924, 244-277.
[S. Givant: "Unifying Threads in Alfred Tarski's Work", The Mathematical Intelligencer 21,1
(1999) 47-58]
§ 090 The complete set of rational numbers exists and is countable. Let
q1, q2, q3, q4, q5, ...
be an enumeration.
Consider the first two elements
|q1, q2| q3, q4, q5, ...
and order them by magnitude:
|q1', q2'| q3, q4, q5, ...
Consider the first three elements
|q1, q2, q3| q4, q5, ...
and order them by magnitude.
|q1'', q2'', q3''| q4, q5, ...
Continue such that in the nth step the first n elements are ordered by magnitude.
In the limit all rational numbers have been ordered by magnitude (*) - if limits of nonconverging sequences have any meaning. Otherwise, the enumeration of any non-converging
sequence is meaningless too. Compare the limit of the sequence (an) with an = (1, 2, 3, ..., n).
(*) The set theoretical limit of the sequences of sets ordered by magnitude can be calculated as
follows. (Apostrophes are left out, because although q' may differ from q'', every apostrophized q
will remain in the set for ever. By construction every ordered set contains at most one q out of
order.)
Let Qk = (q1, q2, q3, ... , qk) be the kth initial segment of the set:
LimSup (Qk) = …n=1...¶ »k=n...¶ Qk
= …n=1...¶ ((q1, ... , qn) » (q1, ... , qn, qn+1) U (q1, ... , qn, qn+1, qn+2) U ...)
= …n=1...¶ (q1, q2, q3, ... )
= (q1, q2, q3, ... )
LimInf (Qk) = »n=1...¶ …k=n...¶ Qk
= »n=1...¶ ((q1, ... , qn) … (q1, ... , qn, qn+1) … (q1, ... , qn, qn+1, qn+2) … ...)
= (q1) » (q1, q2) » (q1, q2, q3) » ... » (q1, q2, q3, ... )
= (q1, q2, q3, ... )
§ 091 No set-theoretically definable well-ordering of the continuum can be proved to exist from
the Zermelo-Fraenkel axioms together with the axiom of choice and the generalized continuum
hypothesis.
[S. Feferman: "Some applications of the notions of forcing and generic Sets", Talk at the
International Symposium on the Theory of Models, Berkeley (1963)]
It is well known, that no well-ordering of the reals can be accomplished by mortal humans. But
usually matheology can at least prove that things, that cannot be done, can be done. Now even
this proof fails! What a pity!
On the other hand, well-ordering of the reals must be done. At all costs! Otherwise the
hierarchy of infinities would break down and research on inaccessible cardinals would appear
like nonsense. So let us pray to the Gods of matheology that they do what no mortal human can
do: Well-ordering the real numbers. Perhaps they can even provide a list of all real numbers?
But that must be kept secret! Because otherwise the research on inaccessible cardinals ...
§ 092 My second best proof contradicting set theory
1) Define a sequence of points pn in the unit interval: pn = 1/n.
These points define intervals
Ak = [1/n, 1/(n+1)] for odd n
and
Bj = [1/n, 1/(n+1)] for even n.
The intervals of sort A === and B --- are alternating. If the points are denoted by n, we have
something like the following configuration.
...7--6==5--4====3---------2===========1
Theorem. If two neighbouring points pn and pn+1 are exchanged, the number of intervals
remains the same.
...7--6==5--3====4---------2===========1
The intervals remain alternating. In particular, the number of intervals cannot increase.
2) Define a set of intervals Im in the unit interval such that interval Im has length
|Im| = 10-m and covers the rational number qm of a suitable enumeration of all rational numbers of
the unit interval.
Then the union of all Im has measure § 1/9. The remaining part of the unit interval has
measure ¥ 8/9 and is split into uncountably many singletons.
A sketch of the intervals Im ~~~ is given here:
...a b~~~~~~~~~c~~~~~~~~~d~~~~~~e f~~~g~~~h i~~~j k~~~l
We cannot exclude intervals within intervals like c~~~d within b~~~e or, alternatively,
overlapping intervals like b~~~d and c~~~~e and also adjacent intervals like f~~~g and g~~~h.
3) Let the endpoints pn of the configuration described in (1) move in an arbitrary way, say
powered by little ants or by the Gods of matheology. Then it cannot be excluded that the pn and
the endpoints of the Im of (2) will coincide (no particular order is required).
...a b~~~~~~~~~c~~~~~~~~~d~~~~~~e f~~~g~~~h i~~~j k~~~l
...3=7--------------11=========5--------12=4---2===9-8==10-6==1
As our theorem shows, there will be not more than ¡0 intervals in the end position. This includes
the set of Im and the set of intervals in the complement. In case that intervals fall into intervals,
the complete number can be reduced. In no case it can grow.
Therefore the assertion of uncountably many degenerate intervals (so called singletons - but
there cannot exist irrational singletons without rational numbers separating them) in the
complement has been contradicted.
§ 093 The difficulty we are confronted with is that ZFC makes a claim we find implausible. To
say we can't criticize ZFC since ZFC is our theory of sets is obviously to beg the question
whether we ought to adopt it despite claims about cardinality that we might regard as exorbitant.
[George Boolos: "Must We Believe in Set Theory?"]
Cancer robbed our community of an outstanding philosopher. One is tempted to say
“philosopher and logician,” but as Richard Cartwright remarked in his eulogy for George Boolos,
“he would have not been altogether happy with the description: accurate, no doubt, but faintly
redundant - a little like describing someone as ‘mathematician and algebraist.’”
[...] George Boolos made significant contributions in every area of logic in which he worked. [...]
Gödel, in contrast to Boolos, argued that the axioms of choice and replacement do follow from
the iterative conception. In article 7, an introduction to a posthumously published lecture by
Gödel, Boolos takes issue with Gödel’s Platonistic claim that the axioms of ZFC (Zermelo
Frankel set theory with Choice) “force themselves upon us as true.” Even if the axioms articulate
a natural and compelling conception of set, they need not correspond to anything objectively
real.
Article 2 contains Boolos’ defense of Fraenkel’s, in contrast to Zermelo’s, position that first-order
but not second-order logic is applicable to set theory. Boolos criticizes the view of Charles
Parsons (and D. A. Martin) that it makes sense to use second-order quantifiers when first-order
quantifiers range over entities that do not form a set. Boolos’ answer to the title of article 8, “Must
We Believe in Set Theory?” is "no": the phenomenological argument (due to Gödel) does not
imply that the axioms of set theory correspond to something real, and the indispensability
argument (due to Carnap) that mathematics is required by our best physical theory, is dismissed
as “rubbish.”
[Gary Mar: "Book Review: Logic, Logic and Logic, George Boolos. Harvard University Press,
1998. ix + 443 pages. Hardcover $45, paperback $22.95. ISBN 0-674-53767-X", Essays in
Philosophy, Vol. 1 No. 2, June 2000]
http://commons.pacificu.edu/cgi/viewcontent.cgi?article=1013&context=eip
§ 094 Nicole d'Oresme (1323 - 1382) proved the harmonic series to be divergent. Alas he
needs ¡0 sums of the form
(1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + ... + 1/16) + ...
with in total 2¡0 unit fractions. If there are less than 2¡0 natural numbers, then there are also less
than 2¡0 unit fractions and the proof of Nicole d'Oresme will fail.
But 2¡0 elements are countable, as is easily proved by the nodes of the Binary Tree:
0
1, 2
3, 4, 5, 6
7, ...
The bijection proves 2¡0 = ¡0. There is but one problem: The "limit-level" of the Binary Tree does
not contain any numbered node. The natural numbers are all exhausted before. (For a definition
of limit by liminf and limsup cp. § 090.) That means, the paths of the Binary tree, i.e. the real
numbers of the unit interval, cannot differ "in the limit". Can they be distinguished at all?
§ 095 Dr. D.F.M. Strauss, is professor of philosophy at the University of the Free State in
Bloemfontein, South Africa.
In Chapter II Strauss addresses various philosophical problems in mathematics. Mathematics is
concerned with number and space, the first two modalities. The prime issue in mathematics is
how to treat infinity. Strauss discusses three main foundational crises in the history of
mathematics: (1) the discovery of irrationals, (2) infinitesimal calculus, and (3) modern set
theory. All three involve the relation between potential and actual infinity. Much attention is
devoted to the conflict between Cantor's treatment of actually infinite sets and the intuitionists'
rejection of actual infinity.
The Dutch mathematician L.E.J. Brouwer (1882-1966), an ardent promoter of intuitionism,
lived in Amsterdam at the same time as Dooyeweerd and had some influence on Dooyeweerd.
Dooyeweerd acknowledged only the potential infinite; he found the idea of the actual infinite
unacceptable. {{Obviously there are many people sharing this opinion. Unfortunately modern
censorship cares that these people rarely get a chance to write in mathematical journals.
Compare the situation with Hilbert, who fired Brouwer from the board of the Mathematische
Annalen, starting what Einstein called the war of the frogs and the mice. For this
Machtergreifung in 1928 Hilbert had less authorization than Hitler had for his in 1933.}} Strauss,
however, argues that Dooyeweerdian philosophy actually provides grounds for both types of
infinity. Strauss distinguishes between the successive infinite and the at once infinite. The
successive infinite is associated with numbers and determines every denumerable, endless
succession of numbers (e.g., the integers or rational numbers). The at once infinite, on the other
hand, is associated with the continuous extension of space. The latter represents a higher order
of infinity; it cannot be reduced to a successive infinity since space cannot be reduced to number
{{that's why uncountable sets are an absurdity.}}
Review: D.F.M. Strauss "Paradigms in Mathematics, Physics and Biology: Their Philosophical
Roots", Tekskor Bk, Danhof, South Africa, (2001, revised 2004), 177 pp.
§ 096 It is true, I believe, that the meaning of an expression is a matter of how it is used. But
this does not mean that meaning can be reduced to use. Rather, meaning permeates use. We
manifest our understanding of expressions in our use of them. We communicate with others by
shared access to one another's use of them. There is nothing here to suggest that we should
always be able to "pin down" what an expression means. [...] How then do people manage to
grasp such meaning?
Well, how do they? They observe expressions being used. They try to see the point of the use.
They try to use the expressions in the same way, under the guidance of promptings, corrections,
and encouragement from others.
Yes, but if there is nothing in how an expression is used which they can have access to before
understanding it and which actually serves to determine the expression's (full infinite) meaning,
how does any of this help? Will they not be confronted by something which strikes them as
being, at best, radically inconclusive and, at worst, so much incomprehensible babble?
Initially perhaps. But they eventually come to understand. It is true that this can seem quite
mysterious. What we have to do, however, is to see it as perfectly natural. People just do have
shared interests, and a shared sense of what is significant and of how things relate to one
another. (These are partly innate and partly inculcated.) As a result, people are able to
understand one another. They are able to see what other people are up to. They are able to
grasp what expressions mean. In the mathematical case, there is no reason why being
subjected to (some of) the truths of a formal theory - seeing how these truths are proved and the
kinds of justifications that are proffered for them - should not give someone a sense of how to
carry on, even though not all the truths have been, or could be, captured.
I am leaning heavily at the moment on some of Wittgenstein's later work on meaning and
language use. [...] Wittgenstein was regarded by many as one of the chief architects, along with
Russell, of what became known as logical atomism. [...] "What we cannot speak about," he
wrote in conclusion, "we must pass over in silence."
However - and this is the twist - what we cannot speak about, or what cannot be said, can,
Wittgenstein maintained, be shown: the nonsense in the Tractatus had arisen from an attempt to
put genuine insights into words. This distinction between what can be said and what can be
shown - the saying/showing distinction - was a linchpin of the whole book. No feature of the
world as a whole could properly be conveyed in words. The framework in which all the facts
were held together was not itself a fact. Features of the world as a whole, its overall shape and
form, were a matter not of its being how it was but of its being however it was. They were a
matter not of what could be said but of what was involved in saying anything at all. They were
what could be shown.
[A.W. Moore: “The Infinite”, 2nd ed., Routledge, New York (2005), p. 183-188]
This should be heeded by addicts of the formal-definitions obsession, in particular BourbakiDieudonné and others of that ilk.
§ 097 According to the status quo, the continuum is properly modelled by the "real numbers".
What is a real number? Let's start with an easier question: What is a rational number? Here
comes set theory to our aid. It is, according to some accounts, nothing but an equivalence class
of ordered pairs of integers. Thus when my six year old daughter uses the fraction what she is
really doing is using the "equivalence class" [...] Sequences generated by algorithms can be
specified by those algorithms, but what possibly could it mean to discuss a "sequence" which is
not generated by such a finite rule? Such an object would contain an "infinite amount" of
information, and there are no concrete examples of such things in the known universe. This is
metaphysics masquerading as mathematics.
[N J Wildberger: "Set Theory: Should You Believe?"]
http://web.maths.unsw.edu.au/~norman/views2.htm
§ 098 The critical analysis which has given physics its new confidence [Einstein's "operational"
technique] has, up to the present, been almost exclusively confined to an examination of the
nature of the physical concepts which the physicist uses. But since mathematics is coming to
play an increasingly important role in the new physics, it is evident that a critical examination of
the nature of the fundamental concepts of mathematics is a task of the immediate future for the
physicist. It was therefore not without a certain amount of dismay that I suddenly became aware
that in the mathematics of the present day there are doubts, uncertainties, and differences of
opinion on fundamental questions which are at least not unlike the bewilderment of physics
when confronted with the new phenomena of the quantum domain. My awakening to a
consciousness of the situation in mathematics I owe to that extraordinary well written little book
by E.T. Bell, "The Queen of the Sciences". Within a few weeks of my reading this book A. F.
Bentley's "Linguistic Analysis of Mathematics appeared. This I skimmed hastily, gathering from it
a most vivid impression of the chaotic state of affairs in the "fundamental" fields of mathematics
[...]
These two expositions made it evident that Mengenlehre was that branch of mathematics in
which perhaps there were the most serious differences of opinion and in which fundamental
questions were most to the fore. [...]
The reactions to this second acquaintance with Mengenlehre were as different as can well be
imagined from those of my first naive contact, when I suppose I got the usual kick out of feeling
that I was playing with the infinite.
[...] the meaning of a term or concept is contained in those operations which are performed in
making application of the term or concept to relevant situations. I can only report that as a matter
of personal analysis I find this operational aspect at the bottom of all meaning, but my inference
is that other persons go through similar processes [...] we may perhaps say that self-consistency
is in some way intimately connected with real things. [...] The accepted method of proving that
some system of postulates does not conceal some contradiction is to exhibit some "real",
"existing" system which satisfies the postulates. Nothing further in the way of proof or analysis is
felt to be necessary; the feeling that actually existing things are not self-contradictory is so
elemental as almost to constitute a definition of what we mean by self-consistent. Now when we
are concerned with "things" we are evidently concerned with some form of experience, so that
we may make an even broader statement and say that experience is not self-contradictory. [...] it
is once obvious that the operational technique automatically secures to mathematics the sine
qua non of self-consistency, for operations actually carried out, whether physical or mental, are a
special form of experience, so that any mathematical concept or argument analyzed into actual
operations must have the self-consistency of all experience. [...]
"point" has no meaning unless it is defined, and this involves the specification of some sort of
procedure. "All the points of a line" as a purely intuitional concept apart from the rules by which
points are determined, can have no operational meaning, and accordingly is to be held for
mathematics as an entirely meaningless concept. [...] "All the points of a line" means no more
than "All the rules for determining points on a line" {{which are known to form a countable set}}.
[...] In other words, we have no more reason to describe the points on a line as nondenumerable than the non-terminating decimals. The repudiation of the diagonal Verfahren for
the decimals at the same time removes all reason for thinking the points on a line nondenumerable.
In fact, a consistent application of the operational criterion of meaning appears to demand the
complete discard of the notion of infinities of different orders. We never have "actual" infinites [...]
but only rules for operation which are not self-terminating. How can there be different sorts of
non-self-terminatingness? At any stage in the process the rule either permits us to go on and
take the next step or it does not [...] and that is all there is to it. [...] Mengenlehre is similarly
supposed to have established the existence of transcendentals by showing that all algebraic
numbers are denumerable. This proof I would reject, holding that the mere act of assigning
operational meaning to the transcendentals of itself ensures that they are denumerable. As a
matter of fact, only a few transcendentals have been established. Mengenlehre is powerless to
show whether any given number, such as e or π, is transcendental or not, and the detailed
analysis necessary in any given case for establishing transcendence is not avoided by
Mengenlehre. From the operational view a transcendental is determined by a program of
procedure of some sort; Mengenlehre has nothing to add to the situation. And this, as far as my
elementary reading goes, exhausts the contributions which Mengenlehre has made in other
fields.
[P.W. Bridgman: "A physicist's second reaction to Mengenlehre", Scripta Mathematica, Vol. II,
1934]
We should repeat this remarkable sentence: "Mengenlehre has nothing to add to the situation.
And this, as far as my elementary reading goes, exhausts the contributions which Mengenlehre
has made in other fields." But we can emphasize this remarkable sentence, also from a less
elementary point of view. Walter Felscher
http://de.wikipedia.org/wiki/Felscher
autor of a text book on set theory in three volumes writes: "Concerning the application of
transfinite numbers in other mathematical disciplines {{outside of mathematics nobody claims
any use of set theory (for the always present exceptions of the rule compare Kalenderblatt
120422 to 120429)}}, the great expectations originally put on set theory have been fulfilled only
in few special cases ..."
[Walter Felscher: "Naive Mengen und abstrakte Zahlen III", Bibl. Inst., Mannheim (1979) p. 25]
http://books.google.de/books/about/Naive_Mengen_und_abstrakte_Zahlen.html?id=s0c6AAAA
MAAJ&redir_esc=y
§ 099 Alexander Zenkin: OPEN LETTER
TO: The Bulletin of Symbolic Logic
CC: The International Mathematical Union
Dear Professor Blass,
As you certainly guessed, the question is not only about a publication of my comments
"Whether the Lord Exists in G. Cantor's Transfinite 'Paradise'" to the scandally-known, quasi"pedagogical" W. Hodges' paper "An Editor Recalls Some Hopeless Papers" [...] in your BSLjournal. The question is about much more important problems. I believe that BSL-papers like the
W. Hodges' one are a dangerous phenomenon from scientific, educational, and social points of
view. There are the following reasons to state that.
1. The main conclusion of the W. Hodges' paper that "there is nothing wrong with Cantor's
argument" is wrong fundamentally and therefore, having been proclaimed in the BSL, such a
conclusion disorients a wide scientific community (pedagogical, mathematical, logical,
philosophical, cognitive psychological, etc. ones) as to one of the most important problems of the
humankind culture as a whole - the problem on the veritable nature of Infinity.
2. The high symbolic logic level of the BSL-publications is a recognized standard of a metamathematical thinking and an attractive pattern for young generations of meta-mathematicians
and symbolic logicians. However a one-sided publishing BSL-policy (not to publish points of view
differing from the traditional set theoretical opinion) deprives the young generation of the
democratic right to make independently its own scientific choice between two historical,
contradictory points of view as to the true nature of Infinity: i.e., between the today traditional
Cantor's and all modern axiomatic set theory's opinion, on the one hand, and the opinion of
Aristotle, Leibniz, Kant, Gauss, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer,
Wittgenstein, Weil, Luzin, Quine, and today - Sol. Feferman, Ja. Peregrin, V. Turchin, P.
Vopenka, etc. [...]
3. One of the most important reasons for my flat objection against the W. Hodges' and similar
meta-mathematical papers is their deforming influence on mathematical education and their
dangerous social consequences as a whole. As far back as the middle of 50s of the XX century,
the outstanding American mathematician, John von Neumann [...] warned: "Too much
formalization and symbolization in the theory of mathematics is dangerous for the healthy
development of the science of mathematics".
In the beginning of the 60s, a large group (about 75) of outstanding mathematicians of
America and Canada (including Richard Bellman of Rand Corporation, Richard Courant of New
York University, Н.О. Pollak of Bell Telephone Laboratories, George Polya of Stanford
University, Andre Weil of Institute for Advanced Study, and others) tried to attract attention of
mathematical community to the same problem - to the danger to provoke a stable disgust of
pupils, students and their parents (who, by the way, are today's Presidents, Government-men,
Congressmen, Government ministers, etc.) to mathematics by means of a premature, excessive,
deterrent, and simply thoughtless formalization of mathematical education.
In their known Memorandum "ON ТНЕ MATHEMATICS CURRICULUM OF THE HIGH
SCHOOL" (American Mathematical Monthly, 1962, March, 189-193) they, in particular, wrote.
"It would [...] bе а tragedy if the curriculum reform [...] should be misdirected and the golden
opportunity wasted. There are, unfortunately, factors and forces in the current scene which may
lead us astray. [...] premature formalization may lead to sterility; premature introduction of
abstractions meets resistance especially from critical minds who, before accepting an
abstraction, wish to know why it is relevant and how it could be used. In its cultural significance
as well as in its practical use, mathematics is linked to the other sciences and the other sciences
are linked to mathematics, which is their language and their essential instrument. Mathematics
separated from the other sciences loses one of its most important sources of interest and
motivation. [...] We wish especially that the new curricula should reflect more the connection
between mathematics and science and carefully heed the distinction between matters logically
prior and matters which should have priority in teaching. Only in this way can we hope that the
basic values of mathematics, its real meaning, purpose, and usefulness will be made accessible
to all students [...]"
In conclusion, they again accentuate and expressed their "concern about а trend to excessive
emphasis on abstraction in the teaching of mathematics"
As the posterior history showed, this very serious, very anxious, and high professional warning
of outstanding mathematicians of the middle of the XX c. as to the danger of the "excessive
formalization and symbolization of mathematical education" was not heard.
Today the situation is further aggravated. The Vice-President of the International Mathematical
Union, Academician of the Russian Academy of Sciences, outstanding mathematician and
mathematical educator, professor Vladimir I. Arnold of Steklov Mathematical Institute (Moscow)
in his numerous papers of the last decade again tries hard to attract attention of mathematicians
and educational community to the catastrophic situation in modern mathematics and
mathematical education. The main reason is the same one – a (today already) global superformalization or, using his term, “bourbakization” [...] of the modern mathematics as a whole
(see, e.g., V.I. Arnold, "International Mathematical Congress in Berlin." [...])
“Our brain, - writes Arnold, - has two halves: one <the left-hemisphere> is responsible for the
multiplication of polynomials and languages <i.e., for the abstract, formal, rational thinking>, and
the other half <the right-hemisphere> is responsible for orientation of figures in space and all the
things important in real life <i.e., for the intuitive, visual, creative thinking>. Mathematics is
geometry when you have to use both halves. In the middle of the XX Century, a [...] mafia of
“left-hemispheric” mathematicians could exclude geometry from mathematical education (firstly
in France, and then in other countries), replaced all informal aspects of this discipline by a
training in a formal manipulation with abstract “notions” <i.e., with empty names, terms, symbols,
etc.>. All geometry, and consequently all connection of mathematics with the real world and with
other sciences was excluded from mathematical education. Such the “abstract” description of
mathematics is unfit neither for education, nor for any practical applications.
…Compelling miserable schoolboys/girls to learn such <formalized mathematics>, “lefthemispherical criminals” created a modern distinctly negative attitude of society and
governments to mathematics.
…The aversion to mathematics which government ministers, exposed to such the experience of
such the education in school, have is a healthy and valid reaction. Unfortunately, this their
disgust spreads on all mathematics without exclusions, and that can kill it as a whole”.
... these “left-hemispherical invalids” were able to cultivate whole generations of mathematicians
that don’t understand any other approach to mathematics and are able only to teach next
generations by the same way.
… It is awful to think what kind of pressure the Bourbakists put on (evidently nonsilly) students to
reduce them to formal machines! This kind of formalized education is completely useless for any
practical problem and even dangerous, leading to Chernobyl-type accidents. [...]
… Modern formalized (bourbakized) education in mathematics is an exact antithesis for teaching
the critical thinking and the true scientific foundations of mathematics. Such the mathematical
education is dangerous for a humankind as a whole”.
The “clinical” picture of the “bourbakism” drawn by Prof. Arnold verily can be called a mental
Acquired Immunodeficiency Syndrome of brain, i.e., shortly a MENTAL-AIDS. As an experience
testifies, the MENTAL-AIDS is a very infectious illness which affects especially easy an
unprotected kid’s brain, unfortunately, without any perspective to get well: as far back as XVIII c.
the great English philosopher G. Berkeley said that "a human-being mind, immersed in high
level abstractions from a young age, loses an adequate perception of the real world to its adult
age".
I shouldn't be surprised if many parents of modern schoolboys/girls and students would like to
bring an action against modern Cantorians and their official "scientific" communities and journals
because of their deliberate cultivation and propagation of such the dreadful infectious social
disease as that MENTAL-AIDS.
However that may be, today, in the very beginning of the XXI Century, we have a much more
painful diagnosis concerning prospects of modern mathematics and mathematical education.
I state and can prove that the main historical source of this dangerous social illness is just the
modern cantorianism with its pure abstract, ambitious transfinite constructions with an empty
ontology, based upon the only Cantor's theorem on the uncountability of real numbers. [...]
I am sure that there is a lot of judicious mathematicians and simply provident parents of future
mathematicians of genius who would not like that their children became "formal machines" used
to execute criminal, terroristic, anti-human "deductive" plans. I hope to have their active support.
Nobody, including the BSL-team, will save the Cantor's transfinite "paradise".
Sincerely yours,
Alexander Zenkin [...]
P.S.1. I have attentively read the enclosed BSL-reviewer's report [...] and regret too that the
report reviewer distorted the sense of the Comments-1 deliberately, high professionally and
fundamentally, and misled you and the mathematical and symbolic logic community as to the
important problems touched upon in the Comments-1. [...]
P.S.2. [...] my system VISAD (for VISual Anaysis of Data), based on the Cognitive Computer
Graphics (CCG) conception, has fulfilled a comparative analysis of CCG-images of the W.
Hodges' paper text and the anonymous BSL-reviewer's report text and has established that the
both authors are the same face <are as like as two peas>. - It is quite interesting result from the
professional scientific ethics point of view, is not so?
http://www.ccas.ru/alexzen/papers/OPEN_LETTER-2_to_the_BSL.doc
§ 100 The pure mathematician can do what he pleases, but the applied mathematician must be
at least partially sane.
[J. Willard Gibbs, quoted in Morris Kline: "Mathematics: The Loss of Certainty", Oxford University
Press (1980) p. 285]
http://books.google.de/books?id=RNwnUL33epsC&pg=PA285&lpg=PA285&dq=%22The+pure+
mathematician+can+do+what+he+pleases%22&source=bl&ots=P84nY_76Mg&sig=6qOnuCW2
RzyzU8vBgoqPIpR8AkA&hl=de&sa=X&ei=zCIQULroN8X4QTFkIGwCQ&sqi=2&ved=0CFMQ6AEwAw#v=onepage&q=%22The%20pure%20mathematici
an%20can%20do%20what%20he%20pleases%22&f=false
§ 101 The first virtue of a matheologian is the ability to anaesthetize himself (or herself) against
the contradictions arising from matheology.
§ 102 Our contemporary orthodoxy: to show that there are so-and-sos is to prove "so-and-sos
exist" from the axioms of set theory.
[Penelope Maddy: "Mathematical Existence", Bull. Symbolic Logic 11 (2005) 351]
http://www.jstor.org/pss/1578738
Note: contemporary means not timeless.
§ 103 Today I received by mail an offprint of "Statements concerning the theory of the
transfinite" with a handwritten dedication: "H. A. Schwarz in memory of our old friendship
dedicated by the author." After having had the opportunity to go through it leisurely, I cannot
conceal that it appears to me as a pathological aberration. What on earth have the Fathers of
the Church to do with the irrational numbers? I really hope my fear might not come true, that our
patient has left the straight and narrow like the poor Zöllner* who never found the way back to
scientific business. The more I think over these cases the more I am forced to get aware of the
similarity of symptoms. Might we manage to lead the poor young man back to serious work!
Otherwise it will come to a bad end with him. [H.A. Schwarz to C. Weierstraß, Oct. 17, 1887]
*) Johann Karl Friedrich Zöllner (1834-1882) was a professor of astrophysics who later got
involved in depth in philosophical studies and after all became an adherent of spiritism.
German original text from Herbert Meschkowski: "Georg Cantor: Leben, Werk und Wirkung",
2nd ed., Bibl. Inst., Mannheim (1981) pp. 266-267 can be found in: Das Kalenderblatt 091019
http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf
§ 104 I cannot consider the set of positive integers as given, for the concept of the actual
infinite strikes me as insufficiently natural to consider it by itself. (Nikolai Nikolaevich Luzin to
Kazimierz Kuratowski)
[Naum Yakovlevich Vilenkin: "In search of infinity", Birkhäuser, Boston (1995) p. 126]
http://books.google.de/books?id=cU3HQFek7L0C&printsec=frontcover&source=gbs_v2_summa
ry_r&cad=0#v=onepage&q=&f=false
{{Obvious explanation for matheologians:}} For Luzin this was torture. His progress in
mathematics at the Gymnasium became worse and worse, so that his father was obliged to
engage a tutor ...
[J.J. O'Connor, E.F. Robertson: Luzin Biography, Mac Tutor]
http://www-history.mcs.st-and.ac.uk/Biographies/Luzin.html
§ 105 What if I were to travel back in time and kill my past self? If my past self died, then there
would be no I to travel back in time, so I wouldn't kill my past self after all. So then the time-trip
would take place, and I would kill my past self. And so on. I was also disturbed by the fact that if
the future is already there, then there is some sense in which our free will is an illusion.
Gödel seemed to believe that not only is the future already there, but worse, that it is, in
principle, possible to predict completely the action of some given person. {{There he believed in
completeness?}}
I objected that if there were a completely accurate theory predicting my actions, then I could
prove the theory false-by learning the theory and then doing the opposite of what it predicted.
According to my notes, Gödel's response went as follows: "It should be possible to form a
complete theory of human behavior, i.e., to predict from the hereditary and environmental givens
what a person will do. However, if a mischievous person learns of this theory, he can act in a
way so as to negate it. Hence I conclude that such a theory exists, but that no mischievous
person will learn of it. In the same way, time-travel is possible, but no person will ever manage to
kill his past self." Gödel laughed his laugh then, and concluded, "The a priori is greatly
neglected. Logic is very powerful."
Apropos of the free will question, on another occasion he said: "There is no contradiction
between free will and knowing in advance precisely what one will do. If one knows oneself
completely then this is the situation. One does not deliberately do the opposite of what one
wants." {{There exists free will, but nobody uses it to decide alternatively. - No contradiction.}}
[...] Gödel's credo, "I do objective mathematics." By this, Gödel meant that mathematical
entities exist independently of the activities of mathematicians, in much the same way that the
stars would be there even if there were no astronomers to look at them. For Gödel,
mathematics, even the mathematics of the infinite, was an essentially empirical science. [...]
Cantor's Continuum Problem is undecidable on the basis our present-day theories of
mathematics. For the formalists this means that the continuum question has no definite answer.
But for Platonist like Gödel, this means only that we have not yet "looked" at the continuum hard
enough to see what the answer is. [...] the same possibilities of thought are open to everyone, so
that we can take the world of possible forms as objective and absolute. Possibility is observerindependent, and therefore real, because it is not subject to our will. [...] anyone who takes the
trouble to learn some mathematics can "see" the set of natural numbers for himself. So, Gödel
reasoned, it must be that the set of natural numbers has an independent existence, an existence
as a certain abstract possibility of thought. I asked him how best to perceive pure abstract
possibility. He said three things. i) First one must close off the other senses, for instance, by
lying down in a quiet place. It is not enough, however, to perform this negative action, one must
actively seek with the mind. ii) It is a mistake to let everyday reality condition possibility, and only
to imagine the combinings and permutations of physical objects - the mind is capable of directly
perceiving infinite sets. iii) The ultimate goal of such thought, and of all philosophy, is the
perception of the Absolute. Gödel rounded off these comments with a remark on Plato: "When
Plautus could fully perceive the Good, his philosophy ended."
Gödel shared with Einstein a certain mystical turn of thought. {{Gödel may have been a great
mythologican. Einstein's known legacy does not support Rucker's claims - on the contrary:
http://www.hillmanweb.com/reason/inspiration/einstein.html
}}
The central teaching of mysticism is this: Reality is One. The practice of mysticism consists in
finding ways to experience this higher unity directly. The One has variously been called the
Good, God, the Cosmos, the Mind, the Void, or (perhaps most neutrally) the Absolute. No door
in the labyrinthine castle of science opens directly onto the Absolute. But if one understands the
maze well enough, it is possible to jump out of the system and experience the Absolute for
oneself. {{Selfexperience of the Absolute is the essential element of every religion.}}
[...] I asked Gödel if he believed there is a Single Mind behind all the various appearances and
activities of the world.
He replied that, yes, the Mind is the thing that is structured, but that the Mind exists
independently of its individual properties.
I then asked if he believed that the Mind is everywhere, as opposed to being localized in the
brains of people.
Gödel replied, "Of course. This is the basic mystic teaching."
We talked a little set theory, and then I asked him my last question: "What causes the illusion
of the passage of time?"
Gödel spoke not directly to this question. [...] He went on to relate the getting rid of belief in the
passage of time to the struggle to experience the One Mind of mysticism. Finally he said this:
"The illusion of the passage of time arises because we think of occupying different realities. In
fact, we occupy only different givens. There is only one reality."
I wanted to visit Gödel again, but he told me that he was too ill. In the middle of January 1978,
I dreamed I was at his bedside.
There was a chess board on the covers in front of him. Gödel reached his hand out and
knocked the board over, tipping the men onto the floor. The chessboard expanded to an infinite
mathematical plane. And then that, too, vanished. There was a brief play of symbols, and then
emptiness - an emptiness flooded with even white light {{confirming the intercerebral existence
form of the mind and thereby the possible existence of a mindscape capable of absorbing
uncountable numbers of numbers and ideas. (The rocks of the Moon were there before the lunar
module landed; and all the possible thoughts are already out there in the Mindscape. An idea is
already there in the Mindscape whether or not someone is thinking it. (Rucker))}}
The next day I learned that Kurt Gödel was dead {{In order to make the most of this mysterious
text: finally imitate (or imagine - according to your skills) the call of the screech owl.}}
[Rudy Rucker: "Infinity and the Mind", Princeton University Press, Princeton (2005) pp. 36, 168ff]
§ 106 Views to the effect that Platonism is correct but only for certain relatively "concrete"
mathematical "objects". Other mathematical "objects" are man made, and are not part of an
external reality.
Under such a view, what is to be made of the part of mathematics that lies outside the scope
of Platonism? An obvious response is to reject it as utterly meaningless.
[Harvey M. Friedman: "Philosophical problems in logic" (2002) p. 9]
http://www.math.osu.edu/~friedman.8/pdf/Princeton532.pdf
§ 107 Cantor had a different set of numinous feelings about the infinite. He was not only a great
mathematician, but a very religious man and by some standards a mystic. Yet his mysticism was
supported by his mathematics, which to him was at least as strong an argument for the
mathematics as for the mysticism. Apart from claiming divine inspiration for his work, we don't
know exactly what spiritual views he linked to his mathematics, but his theorems give support to
the following. Measured in meters, we are tiny specks compared to the universe at large. But
measured in dimensionless points, we are as large as the universe: a proper subset, but one
with the same cardinality as the whole. Similarly, measured in meters, we may be off in a corner
of the universe. But measured in points, the distance is equally great in all directions, whether
universe is finite or infinite; that puts us in the center, wherever we are. Measured in days, our
lives are insignificant hiccups in the expanse of past and future time. But measured in points of
time, our lives are as long as universe is old. We are as small as we seem, but simultaneously,
by a most reasonable measure, co-extensive with the totality of being in both space and time.
{{These few properties already should allow the average rational scientist (as opposed to the
mystic matheologian) to recognize how "reasonable" this measure is.}}
[Peter Suber: "Infinite Reflections", St. John's Review, XLIV, 2 (1998)]
http://www.earlham.edu/~peters/writing/infinity.htm#sublimity
§ 108 The main part of the paper is devoted to show that the real numbers are denumerable.
The explicit denumerable sequence that contains all real numbers will be given. The general
element that generates the sequence will be written as well as the first few elements of that
sequence. That there is one-to-one correspondence between the real numbers and the
elements of the explicitly written sequence will be proven by the three independent proofs. [...] It
is also proven that the Cantor’s 1873 proof of non denumerability is not correct since it
implicates non denumerability of rational numbers. In addition it is proven that the numbers
generated by the diagonal procedure in Cantor’s 1891 proof are not different from the numbers
in the assumed denumerable set.
[S. Vlahovica, B. Vlahovic: "Countability of the Real Numbers", arXiv:math/0403169 (2004)]
http://arxiv.org/abs/math/0403169
§ 109 Cantor’s diagonal method has been the proof for the uncountable infinite set of real
numbers to be “larger” than the countable infinite set of natural numbers. In the following work
Cantor’s method is refuted and it is proven that the cardinality of real numbers is the same as
the cardinality of natural numbers. Contrary to Cantor’s method, this proof is constructive and
estimates directly the cardinality of the real numbers and compares it to the natural numbers by
constructing an injection to the prime numbers. The work is written in German and shall be
translated into English. Until then the German version is the only source for the proof.
[J. Grami, A. Grami: "Die reellen Zahlen sind abzählbar" (2009)]
www.real-numbers.de
§ 110 Belief in the existence of the infinite comes mainly from five considerations:
(1) From the nature of time - for it is infinite.
(2) From the division of magnitudes - for the mathematicians also use the notion of the infinite.
Further, how can the infinite be itself any thing, unless both number and magnitude, of which it is
an essential attribute, exist in that way? If they are not substances, a fortiori the infinite is not. It
is plain, too, that the infinite cannot be an actual thing and a substance and principle.
This discussion, however, involves the more general question whether the infinite can be
present in mathematical objects and things which are intelligible and do not have extension, as
well as among sensible objects. Our inquiry (as physicists) is limited to its special subject-matter,
the objects of sense, and we have to ask whether there is or is not among them a body which is
infinite in the direction of increase.
We may begin with a dialectical argument and show as follows that there is no such thing. If
'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible
or sensible. Nor can number taken in abstraction be infinite, for number or that which has
number is numerable. If then the numerable can be numbered, it would also be possible to go
through the infinite.
It is plain from these arguments that there is no body which is actually infinite. But on the other
hand to suppose that the infinite does not exist in any way leads obviously to many impossible
consequences: there will be a beginning and an end of time, a magnitude will not be divisible
into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither
alternative seems possible, an arbiter must be called in; and clearly there is a sense in which the
infinite exists and another in which it does not. We must keep in mind that the word 'is' means
either what potentially is or what fully is. Further, a thing is infinite either by addition or by
division.
Now, as we have seen, magnitude is not actually infinite. But by division it is infinite. (There is
no difficulty in refuting the theory of indivisible lines.) The alternative then remains that the
infinite has a potential existence.
The infinite exhibits itself in different ways-in time, in the generations of man, and in the division
of magnitudes. For generally the infinite has this mode of existence: one thing is always being
taken after another, and each thing that is taken is always finite, but always different.
But in the direction of largeness it is always possible to think of a larger number: for the number
of times a magnitude can be bisected is infinite. Hence this infinite is potential, never actual: the
number of parts that can be taken always surpasses any assigned number. But this number is
not separable from the process of bisection, and its infinity is not a permanent actuality but
consists in a process of coming to be, like time and the number of time.
With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no
infinite in the direction of increase. For the size which it can potentially be, it can also actually be.
Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned
magnitude; for if it were possible there would be something bigger than the heavens.
Our account does not rob the mathematicians of their science, by disproving the actual
existence of the infinite in the direction of increase, in the sense of the untraversable. In point of
fact they do not need the infinite and do not use it. They postulate only that the finite straight line
may be produced as far as they wish. It is possible to have divided in the same ratio as the
largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will
make no difference to them to have such an infinite instead, while its existence will be in the
sphere of real magnitudes.
It remains to dispose of the arguments which are supposed to support the view that the infinite
exists not only potentially but as a separate thing. Some have no cogency; others can be met by
fresh objections that are valid.
[Aristoteles: "Physics, Book III", Part 4 (350 v. Chr.)]
http://www.greektexts.com/library/Aristotle/Physics/eng/index.html
http://classics.mit.edu/Aristotle/physics.3.iii.html
§ 111 Let m and n be two different characters, and consider a set M of elements
E = (x1, x2, ..., xν, ...)
which depend on infinitely many coordinates x1, x2, ..., xν, ..., and where each of the coordinates
is either m or w. Let M be the totality of all elements E.
To the elements of M belong e.g. the following three:
EI = (m, m, m, m, ...),
EII = (w, w, w, w, ...),
EIII = (m, w, m, w, ...).
I maintain now that such a manifold M does not have the power of the series 1, 2, 3, ..., ν, ... .
This follows from the following proposition:
"If E1, E2, ..., Eν, ... is any simply infinite series of elements of the manifold M, then there
always exists an element E0 of M, which is not equal to any element Eν."
For proof, let there be
E1 = (a1,1, a1,2, ..., a1,ν, ...)
E2 = (a2,1, a2,2, ..., a2,ν, ...)
...
Eμ = (aμ,1, aμ,2, ..., aμ,ν, ...)
...
where the characters aμ,ν are either m or w. Then there is a series b1, b2, ..., bν, …, defined so
that bν is also equal to m or w but is different from aν,ν.
Thus, if aν,ν. = m, then bν = w, and if aν,ν. = w, then bν = m.
Then consider the element
E0 = (b1, b2, b3, ...)
of M, then one sees straight away, that the equation
E0 = Eμ
cannot be satisfied by any positive integer μ, otherwise for that μ and for all values of ν
bν = aμ,ν
and so we would in particular have
bμ = aμ,μ
which through the definition of bν is impossible. From this proposition it follows immediately that
the totality of all elements of M cannot be put into the sequence: E1, E2, ..., Eν, ... otherwise we
would have the contradiction, that a thing E0 would be both an element of M, but also not an
element of M.
[G. Cantor: "Über eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht DMV I
(1890-91) 75-78]
A proof by contradiction fails, if only one counter example can be found. Here it is:
Consider the sequence
E1 = (w, m, m, m, m, m, m, ...)
E2 = (m, w, m, m, m, m, m, ...)
E3 = (m, m, w, m, m, m, m, ...)
E4 = (m, m, m, w, m, m, m, ...)
...
This matrix formed by the aμ,ν has no line μ and no column ν with all characters aμ,ν = m. Since
such a line or column would need infinitely many predecessors, namely all lines or columns with
a finite number of m before the w, it cannot belong to the sequence. (It is the limit of the
sequence.)
Define E0 = (b1, b2, b3, ...) by bμ = m ∫ w = aμ,μ.
The first μ characters of E0 agree with the first μ characters of all Eν for all ν > μ. Since there is
no last μ and no last ν, this situation does never change. Otherwise we would have the
contradiction that a matrix has more* characters m on the diagonal than it has in any line and in
any column**.
*) i.e. a number of m before the first w that is larger than every finite number.
**) where the number of m before the first w is always finite.
§ 112 There is no evidence that Cantor himself ever considered the possibility that the
continuum hypothesis is undecidable. Obviously formal investigations were far from his mind. [p.
213]
For him mathematical theorems were theses about something being; he even was convinced
that the cardinal numbers ¡0 and ¡ were corresponding to realities in the physical world. We are
afraid, he would not have enjoyed the "solution" of his questions by the modern foundational
researcher. [p. 213] {{This sentence could be improved by another pair of quotation marks.}}
Obviously it was difficult for Cantor to express in hard mathematical language, what he
imagined. His "definitions" could appear rather questionable to a critical thinker like Kronecker.
[p. 229] {{Good intuition is certainly preferable over formalism.}}
I have no doubts concerning the truth of the transfinite that I have recognized by help of God
and have been studying in its diversity and unity for more than 20 years. Every year and nearly
every day advances me in this science. [...] From no other subjects of created nature I have
safer knowledge than of the theorems of transfinite number- and type-theory. [G. Cantor to Pater
I. Jeiler, Whitsun 1888] {{Of course this intuition must not fail.}}
Or is it advisable to completely refrain from set theory in primary school? Today we tend to
recommend that. [p. 227] {{And not only there!}}
[Quotations with page numbers from Herbert Meschkowski: "Georg Cantor: Leben, Werk und
Wirkung", 2nd ed., Bibl. Inst., Mannheim (1981)]
For German original texts see: Das Kalenderblatt 100707
http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf
§ 113 [...] gradually and unwittingly mathematicians began to introduce concepts that had little
or no direct physical meaning. [...] concepts and theories that do not have immediate physical
interpretation [...] gained acceptance. [...] The gradual rise and acceptance of the view that
mathematics should embrace arbitrary structures that need have no bearing, immediate or
ultimate, on the study of nature led to a schism that is described today as pure versus applied
mathematics. [1] "The Fall and Original sin of set theory" [2], leads to matheology that praises a
mixture of "mush of the continuum" (Brei des Kontinuums) with "space sauce" (Raumsauce) [3]
as exquisite brainfood, but nevertheless is not ashamed to call the real numbers a set. Sets
consist of distinguishable elements.
[1] M. Kline: "Mathematical Thought from Ancient to Modern Times", Oxford University Press
(1972) 1029, 1031, 1036.
http://books.google.de/books?id=Lvco6V8fBpoC&pg=PA1029&lpg=PA1029&dq=%22gradually+
and+unwittingly+mathematicians+began+to+introduce%22&source=bl&ots=BmiyX56Ts_&sig=Jl
H8x1AL88Q8DWDyoEpmMCJVdI&hl=de&sa=X&ei=Mr9uUIq0BeLl4QSIlIDQBw&sqi=2&ved=0CDMQ6AEwA
g#v=onepage&q=%22gradually%20and%20unwittingly%20mathematicians%20began%20to%2
0introduce%22&f=false
[2] H. Weyl: "Mathematics and logic", American Mathematical Monthly 53, 1946, p. 2.
[3] H. Weyl: "Über die neue Grundlagenkrise der Mathematik", Math. Zeitschrift 10 (1921)
reprinted in: "Gesammelte Abhandlungen, II", Springer, Berlin (1968) 149f.
http://books.google.de/books?id=OGnthgn0H9AC&pg=PA143&hl=de&source=gbs_toc_r&cad=4
#v=onepage&q&f=false
http://de.scribd.com/doc/49885193/Hermann-Weyl-Ueber-Die-Neue-Grundlagenkrise-DerMathematik
§ 114 The requirement that every element of a set shall be a set itself seems questionable.
Formally that may work and simplifies the formalism. But what about the application of set theory
on geometry and physics? {{That appears to be the least important problem. Set theory has no
application anyway.}} [Zermelo to Fraenkel, Jan. 20, 1924]
For German original texts see: Das Kalenderblatt 100319
http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf
§ 115 According to Russell, the structure of the infinite and the continuum were completely
revealed by Cantor and Dedekind, and the concept of an infinitesimal had been found to be
incoherent and was “banished from mathematics” through the work of Weierstrass and others
[1901, pp. 88, 90]. These themes were reiterated in Russell’s often reprinted Mathematics and
the Metaphysician [1918] and further developed in both editions of Russell’s The Principles of
Mathematics [1903; 1937], the works which perhaps more than any other helped to promulgate
these ideas among historians and philosophers of mathematics. In the two editions of the latter
work, however, the banishment of infinitesimals that Russell spoke of in 1901 was given an
apparent theoretical urgency. No longer was it simply that “nobody could discover what the
infinitely little might be,” [1901, p. 90] but rather, according to Russell, the kinds of infinitesimals
that had been of principal interest to mathematicians were shown to be either “mathematical
fictions” whose existence would imply a contradiction [1903, p. 336; 1937, p. 336] or, outright
“self-contradictory,” as in the case of an infinitesimal line segment [1903, p. 368; 1937, p. 368].
In support of these contentions Russell could cite no less an authority than Georg Cantor, the
founder of the theory of infinite sets.
Having accepted along with Russell that infinitesimals had indeed been shown to be
incoherent, and that (with the possible exception of constructivist alternatives) the nature of the
infinite and the continuum had been essentially laid bear by Cantor and Dedekind, following the
development of nonstandard analysis in 1961, a good number of historians and philosophers of
mathematics (as well as a number of mathematicians and logicians) readily embraced the now
commonplace view that is typified by the following remarks:
In the nineteenth century infinitesimals were driven out of mathematics once and for all, or so it
seemed. [P. Davis and R. Hersh 1972, p. 78]
But ...
the German logician Abraham Robinson (1918–1974), who invented what is known as nonstandard analysis, thereby eventually conferred sense on the notion of an infinitesimal greater
than 0 but less than any finite number. [Moore 1990; 2001, p. 69]
Indeed ...
nonstandard analysis ..., created by Abraham Robinson in the early 1960s, used techniques of
mathematical logic and model theory to introduce a rigorous theory of both [non-Cantorian]
infinite and infinitesimal numbers. This, in turn, required a reevaluation of the long-standing
opposition, historically, among mathematicians to infinitesimals in particular. [Dauben 1992a, pp.
113-114]
[Philip Ehrlich: "The Rise of non-Archimedean Mathematics and the Roots of a Misconception I:
The Emergence of non-Archimedean Systems of Magnitudes", Arch. Hist. Exact Sci. 60 (2006)
1-121]
And the moral of this story? Not long ago there existed in mathematics a very questionable
opinion of the past that finally has been overcome. That must not happen in matheology - the
only science standing absolutely safe.
§ 116 How can the assumption of the infinite be justified?
Could not just this seemingly so fruitful hypothsesis of the infinite have introduced straigth
contradictions into mathematics, thereby destroying the basic nature of this science that is so
proud upon its consistency?
[On the hypothesis of the infinite, Ernst Zermelo's Warsaw notes W4 (p. 171), reported in H.-D.
Ebbinghaus, V. Peckhaus: "Ernst Zermelo, An Approach to His Life and Work", Springer (2007)
p. 292.]
For German original texts see: Das Kalenderblatt 100322
http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf
§ 117 Finally, considering all the applications and recognizing the whole host of transfinite
conclusions of the most difficult and painstaking sort that are involved for instance in relativity
theory and quantum theory, and how nature precisely follows these results, the beam of the
fixed star, Mercury, and the complicated spectra here on earth and in a distance of hundred
thousands of lightyears. Should we in view of these facts hesitate only one second to apply
tertium non datur {{to infinite sets}} only because of the beautiful eyes of Kronecker and because
of some philosophers who are disguised as mathematicians and for reasons that are completely
arbitrary and not even can be formulated precisely? (David Hilbert 1931, 387f)
[Volker Peckhaus: "Becker und Zermelo"]
{{It seems that Hilbert really believed in the scientific application of transfinity.
Concerning the rumor that at a time not more than twelve scholars had understood (general)
relativity theory, we can safely say that Hilbert was not among that dozen.}} Under this aspect
Zermelo's judgement (in a letter of August 28, 1928 to Marvin Farber) about Hilbert's (and
Ackermann's) Foundations of Theoretical Logic becomes understandable: "Hilbert's just
published logic is more than miserable, and also from his long announced Foundations of
Mathematics I do no longer expect anything spectacular."
[H.-D. Ebbinghaus, V. Peckhaus: "Ernst Zermelo, An Approach to His Life and Work", Springer
(2007) 293]
Zermelo defended his point of view with clear insights and discerning arguments, but also with
polemical formulations and sometimes hurtful sharpness. The controversial attitude shining
through here has become a dominating facet of his image {{that makes him simpatico}}. Further
controversies such as those with Ludwig Boltzmann about the foundations of the kinetic theory
of heat and with Kurt Gödel and Thoralf Skolem about the finitary character of mathematical
reasoning support this view. [loc. cit. p. VII, preface]
http://books.google.de/books?id=G1nQU0GKvx8C&pg=PR7&lpg=PR7&dq=%22his+point+of+vi
ew+with+clear+insights%22&source=bl&ots=TiX2_dD9rX&sig=jZYPErW5-Fg7eJXx9dGje2PM9Q&hl=de&ei=Ei6iS5bNIKSAnQPmhMCFCg&sa=X&oi=book_result&ct=result&resnum=1&ved
=0CAYQ6AEwAA#v=onepage&q=%22his%20point%20of%20view%20with%20clear%20insight
s%22&f=false
For German original texts see: Das Kalenderblatt 100323
http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf
§ 118 Marvin Farber took "Notes on the foundations of Mathematical Logic" from discussions
with Zermelo between 24 April and 10 June 1924 (8 page typescript).
Models of axiom systems are conceived as substrates. A substrate is "a system of relations
between the elements of a domain". It is here that Zermelo clearly distances himself from
Hilbert's programme of consistency proofs. With regards to the still open question of the
consistency of arithmetic it says (ibid., 2):
A substrate is presupposed, as, for example, in the case of the series of real numbers.
Something is presupposed which transcends the perceptual realm. The mathematicians must
have the courage to do this, as Zermelo states it. In the assumption or postulation of a substrate,
the freedom of contradiction of the axioms is presupposed [...].
Zermelo differs from Hilbert on this. It is Hilbert's view that it must be proved.
The scientific estrangement from Hilbert my have led Zermelo to feel also a personal one.
{{Finally however Zermelo has won: The freedom of contradiction of the axioms is presupposed
and, as we can learn by an infinitude of striking examples, always postsupposed too.}}
[H.-D. Ebbinghaus, V. Peckhaus: "Ernst Zermelo, An Approach to His Life and Work", Springer
(2007) p. 156]
§ 119 The point of all this is that just as the finiteness of our physical bodies does not imply that
every physical object is finite, the finiteness of the number of cells in our brains does not mean
that every mental object is finite. Well ... are there any infinite minds, thoughts, ideas, or forms or
what have you in the Mindscape? [...] If infinite forms are actually out there in the Mindscape,
then maybe we can, by some strange trick of mental perspective, see some of these forms.
{{One of these strange tricks is presumably the consumption of enough alcohol.}}
[Rudy Rucker: "Infinity and the Mind", Princeton University Press, Princeton (2005) p. 38]
§ 120 f = x2 fl df/dx = 2x + dx
The infinitesimal dx disappears because it is much smaller than the finite 2x, explains Marquis
de l'Hospital in the first textbook on Calculus. But what happens at x = 0 ?, asks D. Laugwitz in
"Zahlen und Kontinuum" on p. 25.
§ 121 It is not unknown that mathematics besides the large benefit for the practical life has also
a second, not less important although less obvious benefit for the practicing and sharpening of
the intellect. The latter is what the state mainly is intending when prescribing the study of
mathematics for every kind of university study. [B. Bolzano: "Betrachtungen über einige
Gegenstände der Elementargeometrie“ (1804)]
{{Bolzano is the inventor of the word Menge (set). But since Menge in German means "a lot", he
declared:}} Also allow me to call a collection, that contains only two parts, a set. [J. Berg (Hrsg.):
"Bernard Bolzano, Einleitung zur Grössenlehre", Friedrich Frommann Verlag, Stuttgart (1975),
Bolzano-Gesamtausgabe, Reihe II Band 7, p. 152].
A Menge consisting of only one or none element Bolzano would hardly have recognized as a
Menge. When will the first sets of negative cardinality be invented? Or has it happened already?
For German original texts see: Das Kalenderblatt 091223
http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf
§ 122 The concept of infinity has been for hundreds of years one of the most fascinating and
elusive ideas to tantalize the minds of scholars and lay people alike. The theory of infinite sets
lies at the heart of much of mathematics, yet is has produced a series of paradoxes that have
led many scholars to doubt the soundness of its foundations {{not the disciples of matheology, of
course}}. [Naum Yakovlevich Vilenkin: "In search of infinity", Birkhäuser, Boston (1995), Cover]
http://books.google.de/books?id=cU3HQFek7L0C&source=gbs_navlinks_s
http://www.weltbild.de/3/16939032-1/buch/in-search-of-infinity.html
§ 123 Suppose a contradiction were to be found in the axioms of set theory. Do you seriously
believe that a bridge would fall down?
http://www-history.mcs.st-and.ac.uk/Quotations/Ramsey.html
§ 124 The third point is that under these conditions it is straightforward to show that the
procedure “Give me any numeral n you can imagine, I will give you the next one” has to break
down at a certain point. Ask any person to imagine a very large numeral, say, in decimal
presentation. Usually what we do is to form a picture, say, we see a blackboard and it is covered
with ciphers all over. But that won’t do. For once we have such a picture, it is obvious that is
communicable, hence that it is finitely expressible and hence that there is room to imagine the
next numeral and to communicate it. Thus, the alternative must be that the numeral is so large
that it cannot be imagined, thereby making it senseless to talk about the next one. I will return to
the implicit paradoxical nature of what I just wrote. What is being asked is to imagine a numeral
so huge that it cannot be imagined.
[Jean Paul Van Bendegem: "Why the largest number imaginable is still a finite number", p. 11]
(a) Labels are used merely as labels: if the world is finite, so is the set of labels, and it is
impossible to label all “objects” in the world.
(b) Labels form a structured set. In this case the labelling process can become more economical
and more efficient, but it remains the case that the set of labels stays finite.
(c) Labels form a structured set inserted in a theoretical framework.
Here two subcases can be distinguished:
(c1) There are interpretations of the theoretical framework that refer to “objects” in the world.
Obviously in this case everything remains finite again on the assumption that the world is finite.
(c2) There are no specific interpretations that refer to “objects” in the world. It is then always
possible to find finite quasi-models that are derived from the classical infinite models of the
theoretical framework. In some cases (as shown in the example above) these quasi-models can
be seen as extensions of the classical model since it is possible to keep all classically true
statements true in the quasi-model. Thus in those cases no truths are lost.
The last case also applies to all labels that can be imagined by a labelling machine, if the
requirement is that the labels should be communicable. Hence, if it is representable, it is obvious
that we can imagine something larger, as we usually represent something in an environment,
hence additional space is available. What we have to imagine, is a label such that if we try to
represent it, we should fail to do so. Hence the agreement with Priest’s description quoted at the
beginning of this paper: “so large that it has no physical or psychological significance …”. It is
paradoxical to be sure. If formulated in terms of questions, the problem becomes immediately
obvious. The question “What is the largest label or numeral that is not imaginable?”, should not
be answered by “The label so-and-so with properties such-and-uch”, because then it has been
imagined, thereby not answering the question. The answer must be: “Whatever it is, that label”.
An alternative reply would be: “The largest label is that label about which questions such as the
question posed cannot be asked”. It is that label that ceases to be that label as soon as
something is said about it. {{A property that can change - potential infinity.}} A conclusion that fits
in nicely with the argued for vagueness of the largest label.
[Jean Paul Van Bendegem: "Why the largest number imaginable is still a finite number", p. 16]
http://www.vub.ac.be/CLWF/members/jean/the%20largest%20number%20imaginable.pdf
Consider a label that has been constructed using all matter of the universe except that little heap
that is necessary to maintain your consciousness. If you try to increase the Kolmogorovcomplexity of the label, then you will have to use matter from said heap. This process must stop
somewhere. But you will no longer be able to recognize its limit.
§ 125 The leap into the beyond occurs when the sequence of numbers that is never complete
but remains open toward the infinite is made into a closed aggregate of objects existing in
themselves. Giving the numbers the status of objects becomes dangerous only when this is
done. [p. 38]
In advancing to higher and more general theories the inapplicability of the simple laws of
classical logic eventually results in an almost unbearable awkwardness. And the mathematician
watches with pain the larger part of his towering edifice which he believed to be built of concrete
blocks dissolve into mist before his eyes. [p. 54]
[Hermann Weyl: "Philosophy of Mathematics and Natural Science" (1949); reprinted with a new
introduction by Frank Wilczek, Princeton Univ. Press (2009)]
http://press.princeton.edu/titles/8960.html
http://frankwilczek.com/weyl05.pdf
§ 126
(A) Mathematics is common sense;
(B) Do not ask whether a statement is true until you know what it means;
(C) A proof is any completely convincing argument;
(D) Meaningful distinctions deserve to be preserved.
[Errett Bishop: "Schizophrenia in contemporary mathematics", Amer. Math. Soc. Colloquium
Lecture, Seventy-eighth summer meeting, University of Montana, Missoula, Montana (1973)]
Mathematical Reviews (MathSciNet): MR788163
http://www-history.mcs.st-and.ac.uk/Biographies/Bishop.html
http://en.wikipedia.org/wiki/Errett_Bishop
§ 127 The intercession is an alternative measure for infinite sets of finite numbers.
Definition: Two infinite sets, A and B, intercede (each other) if they can be put in an
intercession, i.e., if they can be ordered such that A is dense in B and B is dense in A. In other
words, between two elements of A there is at least one element of B and, vice versa, between
two elements of B there is at least one element of A.
The intercession of sets with nonempty intersection, e.g., the intercession of a set with itself,
requires distinction of identical elements. As an example an intercession of the set of positive
integers and the set of even positive integers, 1, 2, 3, ... and 2', 4', 6', ..., is given by
1, 2', 2, 4', 3, 6', ....
The intercession includes Cantor's definition of equivalent (or equipotent) sets by one-to-one
correspondence (or bijection): Two equivalent sets always intercede each other, i.e., they can
always be put in an intercession. The intercession is an equivalence relation.
All infinite sets of finite numbers (like the integers, the rationals or the reals) belong, under this
relation, to the same equivalence class.
[W. Mückenheim: "Die Mathematik des Unendlichen", Shaker, Aachen (2006) 116-117]
http://planetmath.org/encyclopedia/Intercede.html
§ 128 Robert Grosseteste (1168 - 1253), Bishop of Lincoln and teacher of Roger Bacon in
Oxford claimed: "The number of points in a segment one ell long is its true measure."
Also John Baconthorpe (? - 1346), called Doctor resolutus, brought honour on his epithet and
courageoulsly opposed the contemporary scholastic opinion "infinitum actu non datur" by stating:
"There is the actual infinite in number, time, quantity."
§ 129 Unfortunately, I was introduced to Cantor in a course of real analysis. To say that most
mathematics professors who teach Cantor's ideas are mentally challenged, is being rather kind
to them. My first impressions of Cantor were not good, but my impressions of the professors who
still teach his theories, are significantly worse. (John Gabriel)
Vox populi - vox Dei ?
§ 130 What is so bad about contradictions?
I shall address the title question, and the answer I shall give is: rnaybe nothing much. Let me
first explain how, exactly, the question is to be understood. I shall Interpret it to mean: What is
wrong with believing some contradictions? I emphasize the 'some'; the question 'What is wrong
with believing all contradictions?' is quite different, and, I am sure, has a quite different answer. It
would be irrational to believe that I am a fried egg. [...].
I think that there is nothing wrong with believing some contradictions. I believe, for example,
that it is rational (rationally possible - indeed, rationally obligatory) to believe that the liar
sentence is both true and false. [...] I have discovered, in advocating views such as this, that
audiences suppose them to be a priori unacceptable. When pressed as to why, they come up
with a number of arguments. I shall consider five of the most important, and show their lack of
substance. They can be summarized as follows:
(1) Contradictions entail everything.
(2) Contradictions cannot be true.
(3) Contradictions cannot be believed rationally.
(4) If contradictions were acceptable, people could never be rationally criticized.
(5) If contradictions were acceptable, no one could deny anything.
I am sure that there must be other possible objections, but the above are the most fundamental
that I have encountered. I shall take them in that order. What I have to say about the first
objection is the largest, because it lays the basis for all the others. [...]
The objection is that rational belief is closed under entailment, but a contradiction entails
everything. Hence, if someone believed a contradiction, they ought to believe everything, which
is too much. I certainly agree that believing everything is too much: I have already said that there
is an important difference between some and all. {{The following contradiction certainly does not
belong to the category "some:
"The absence of an object in the union of all finite initial sequences implies its absence in the
infinite set."
"The absence of an object in the union of all finite initial sequences does not imply its absence
in the infinite set."}}
[Graham Priest: "What is so bad about contradictions?", JSTOR The Journal of Philosophy, Vol.
95, No. 8 (Aug. 1998) 410-426]
http://www.jstor.org/pss/2564636
§ 131 The following correspondence has been made available to me by one of the
correspondents.
Dear NN,
[...] The key to Cantor's argument (as it is to Dedekind's definition of the reals via Dedekind cuts)
is the notion of a completed infinity - anathema to Aristotle and much later to Gauss, but the rock
on which all of our modern view of set theory - and hence mathematics {{how can a rational
thinking human make such an error?}} - is based. So in Cantor's proof, we have to take the
purported list of all positive reals in (0, 1), in 1-1 correspondence with the naturals, as completed
before we look further. There is no room to add another, nor any changes that can be made: all
the positive reals in (0, 1) are already enumerated. Now begin his diagonalization: as the
diagonal passes through the nth entry, we make the 5-6 change in 'our' decimal in accordance
with what we find in the nth place of the nth entry. Should what we have so far made occur later
in the list, as the rth entry, our diagonal line will pass though it too, and we'll make the same kind
of change in its rth decimal place. No conundrum after all, then - if you're willing to accept the
idea of a completed infinity (here in the form of a completed list) with all of its implications. The
constructivists aren't willing so to accept - but this leaves them with problems perhaps even
more serious (such as being unable to prove the intermediate value theorem) {{this situation is
certainly not improved by accepting a contradiction}}. You might conclude that all's not well in
this mathematical Denmark - perhaps we need to rethink the foundations on a basis other than
set theory, if it leads to such outrageous consequences. {{The foundations are already there.
Please do not go astray in the infinite. You will not find them there. Foundations are always on
the bottom: I + I = II , I + II = III , ... }}
§ 132 The following correspondence has been made available to me by one of the
correspondents.
The reason that I did not answer your email message earlier is that I was re-reading
Mueckenheim's papers to understand his arguments better. Somehow I do not interpret what
Mueckenheim says as you did. I think you are looking at Mueckenheim's work through the eyes
of a constructivist and Mueckenheim is not a constructivist. There is, I believe, a deeper
message in Mueckenheim's work that we are applying a different standard of interpretation or
reasoning to Cantor's proof of uncountability of reals than his proof of countability of rationals.
{{That has been clearly recognized by the writer: For the "proof of countability" the finite
representations are used, for the "proof of uncountability" the infinite representations are used,
which do not exist without finite definitions and which, therefore, cannot be counted at all.}} I
don't know how to explain it but there are some hidden assumptions about the list of reals that
Cantor based his diagonal argument on. For example, Cantor used the idea completed infinity
here, meaning he assumed all reals were in the list! But he said nothing about how the list was
constructed! In fact, if I give you a real number, you could not tell me where the number is in the
list. Accepting the assumption of completed infinity is, in my mind, equivalent to accepting God
on faith! I am suddenly feeling a connection between religion and math. {{That is not surprising
with respect to Cantor's world view, and it is not reprehensible either - but it is not related to
mathematics.}} Basically, if you can construct the list of reals, it will automatically be countable!
{{Of course. We can only construct what can exist in principle.}} So Cantor started with a
contradiction right from the beginning of his diagonal argument.
§ 133 Each real number of the interval [0, 1] can be represented by an infinite path in a given
binary tree. In Section 2 the binary tree is projected on a grid N × N and it is shown that the set
of the infinite paths corresponds one-to-one to the set N. The Theorems 2.1 and 2.2 give the first
proof and the Theorem 2.3 provides a second proof. Section 3 examines the Cantor’s proof of
1891. The Section shows that (i) if the diagonal method is correct, then any denumerable list L to
which the diagonal method was applied is incomplete (Theorem 3.1), (ii) if some complete list
exists and if the diagonal method is correct, then a complete list cannot be represented in the
form used in Cantor’s proof (Theorem 3.2) and (iii) being L incomplete nothing affirms or denies
that |N| is the cardinality of the set of real numbers of the interval [0, 1] (Theorem 3.3). Sections
2 and 3 mean that (i) there is the list, denoted by LH, such that it contains all members of N and
all members of F and to each member of F of the list corresponds one-to-one a member of N of
the list and (ii) if the diagonal method used in Cantor’s 1891 proof [1] is valid, then LH has a
different form of the form of the list L used by Cantor. In Section 4 we try to show that LH has the
same form as L.
[J. C. Ferreira: "The cardinality of the set of real numbers" (2001)]
http://arxiv.org/PS_cache/math/pdf/0108/0108119v6.pdf
§ 134 The sentence "This irrational number exists" means
1) This irrational number has a name; and
2) we can decide whether it is less than or greater than any rational number we might name.
Our idea of length is that it is continuous. And since we think of measuring length as the
distance from the origin O along the x-axis, the thought was that the values of x must reflect the
continuum of lengths by being a continuum of numbers. One way to express that is to say that,
corresponding to each endpoint P of OP, there is a real number x, the coordinate of P, which is
the measured length of OP. In other words, we must be able to measure every length.
But will that be possible? Will it be possible to name the ratio that every length will have to a
unit of measure?
No. It is impossible to name every point in a continuum – a continuum of names is an
absurdity. Names are discrete. And nameless numbers do not exist, not even potentially. There
is no arithmetical continuum. [...]
Infinite decimals?
"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did
it for half an hour a day. Why, sometimes I've believed as many as six impossible things before
breakfast." (Alice in Wonderland)
Each real number in the supposed continuum between 0 and 1, at any rate, has that form.
But an infinite decimal has no name. It is not that we will never finish naming it. We cannot
even begin. Infinite decimals, therefore, since they do not have names, are not numbers.
Just because something is written with the symbols we use for numbers -- 1, 2, 3, and so on -does not make it a number, any more than something written with the symbols we use for words
-- "obakqe" -- makes it a word.
Equivalently, infinite decimals are not numbers because with them it is impossible to solve the
four problems of arithmetic. We cannot name the sum of infinite decimals; we cannot name their
difference; we cannot name their product; and we cannot name their quotient. Infinite decimals
are not numbers.
If a student in an arithmetic class were to say, "Although I cannot name it, teacher, the sum
exists; and to know that is sufficient," then the student might deserve an A in metaphysics, but in
arithmetic she would certainly fail.
The symbol for an infinite decimal, although it is called a real number, is intended to refer to a
point on what is called the real line. But a point is a concept completely different from a number.
And to achieve their identity by postulation is both a tautology -- "To every point on the line there
corresponds a point on the line" -- and an acknowledgement of defeat.
We can try to make sense of an infinite decimal, however, as being an abbreviation for a limit.
[...] To suppose, however, that there could be algorithms for computing a continuum of real
numbers, would require a continuum of algorithms. Again that is absurd. Algorithms are
discrete. And in the absence of an algorithm, it will be impossible to place a supposedly infinite
decimal, such as .24059165378..., with respect to order relative to any rational number.
In the absence of an algorithm, .24059165378... is nothing but a sequence of made up digits
followed by three dots. It is not the symbol of a number.
In fact, the English mathematician and father of artificial intelligence Alan Turing proved the
following: To compute the decimal expansion of a real number, it is possible to create an
algorithm for only a countable number of them.
Why the obsession with a continuum of numbers? It was aggravated by the apparent demand
of coordinate geometry: For every point on the x-axis there must be a number which is its
coordinate. But it does not matter that an arithmetical continuum is a fiction. In the actual
practice of calculus, it never comes up. When we do a calculation, we name a number. That is
all anyone has ever done or ever will do, even though the theoretical explanation for what we do
might be nonsense. {{What sense would it make to maintain that obvious nonsense?}}
In short, inasmuch as measurements -- numbers that we can know and name -- are the essence
of the physical sciences, the theory of real numbers is not a theory of measurement. Together
with its associated set theory ("The set of real numbers", "The set of points on a line"), the theory
of real numbers is the most prominent current example of fantasy mathematics.
Lawrence Spector (2010)
For a newer version look here:
http://www.themathpage.com/aCalc/real.htm
§ 135 One central agent of the connection between mathematics and religion is the concept of
infinity (but it is not the only one!). From its first appearance under the name of "apeiron" with
Anaximander of Miletus (610-546 BC), to the recent work of Hugh Woodin [9], this is a
permanent theme in mathematics -- H. Weyl even wrote that mathematics is "the science of the
Infinite" [8] -- but the theme is also permanent in the philosophy of mathematics, and the word
End is not yet written. This is a fascinating story that has inspired philosophers and theologians,
poets and mathematicians. One can follow the birth of the concept, corresponding to attributes
of God (or space or time) with mathematics filling more and more space through the centuries,
until the Cantorian parthenogenesis between mathematics and religion (but still with a trace of its
origins with the theological absolute to escape the paradox of the set of all sets). [...] A second
theme running through many chapters of the book is the search for a global vision uniting
mathematics and religion. This can be found first in the school of Pythagoras, the object of "The
Pythagoreans," an interesting study by Reviel Netz. Netz suggests, through an analysis of the
mystery of the Pythagorean cult, that religion and mathematics might be able to interact,
because they share some way of "rationalizing mystery" through analogies and metaphors. The
global unity of mathematics with religion is central in Plato's work, and in his followers' such as
Plotinus and Proclus, but also much later in modern times.
[8] Hermann Weyl, The Open World (God and the Universe, Causality, Infinity), Yale, 1933,
Reprint Oxbow Press, 1989.
[9] Hugh W. Woodin, "The continuum hypothesis. I.," Notices Amer. Math. Soc.48 (2001) 567576. Part II. Notices Amer. Math. Soc. 48 (2001) 681-690.
[Mathematics and the Divine. A Historical Study edited by Teun Koetsier and Luc Bergmans,
Amsterdam, Elsevier, 2005, Hardbound, 716 pp., US $250, ISBN-$3: 978-0-444-50328-2, ISBNIO: 0-444-50328-5 Rewieved by Jean-Michel Kantor in The Mathematical Intelligencer 30, 4
(2008) 70-71]
§ 136 By Cantor was - it is well known as biggest beast alephant grown.
But aleph is a number that
turned out too large - thus ant fell dead.
Syntax under poetic licence. After Christian Morgenstern:
http://www.gedichtsuche.de/gedichtanfaenge_ueber.php?id=278.
http://en.wikipedia.org/wiki/Christian_Morgenstern
§ 137 A moment of contemplation,
of resting,
of reflection:
When have you identified a number by an infinite string of symbols for the last time?
Was there a first time?
§ 138 "God himself cannot persist without wise men" – Luther said, and with every right; but
"God himself can even less persist without unwise men" – that good Luther did not say!
[F. Nietzsche: "Die fröhliche Wissenschaft", 3. Buch, Schmeitzner, Chemnitz (1882)]
http://gutenberg.spiegel.de/buch/3245/6
{{By the way, same holds for matheology.}}
In this way an ambitious innovator always attains his goal; he becomes a famous philosopher
and the corruption of youth happens on a large scale.
[Cantor to Loofs, Feb. 24, 1900, about Nietzsche]
Corruptor of youth? Isn't Kronecker quoted with just those words? For instance by Manin:
[Yuri I. Manin:Georg Cantor and his heritage, arXiv:math/0209244v1]
http://aps.arxiv.org/PS_cache/math/pdf/0209/0209244v1.pdf
§ 139 The question of greatest urgency confronting nineteenth- and early twentieth-century
mathematicians was arguably that of the status of the infinite within mathematics. Zermelo’s
s1921, comprising five multi-part philosophical “theses”, should be understood in that spirit.
Despite its brevity, s1921 is somewhat repetitive. It seems that Zermelo had no intention of
publishing it even as part of some longer piece. Instead, s1921 likely functioned as a personal
manifesto; clearly, Zermelo sees himself as breaking new ground here {{breaking new ground? or breaking through thin ice?}}. If conceived in July 1921, in fact, s1921 would contain the
earliest intimation of the theory of systems of infinitely long propositions [...]
[R. Gregory Taylor: "Introductory note to 'Zermelo s1921 - Theses concerning the infinite in
mathematics'", Ernst Zermelo - Collected Works/Gesammelte Werke Volume I - Set Theory,
Miscellanea / Band I - Mengenlehre, Varia (Schriften der Mathematisch-naturwissenschaftlichen
Klasse der Heidelberger Akademie der Wissenschaften, 2010, Volume 21, 302-307)]
http://www.springerlink.com/content/w4301h8878h711t6/
§ 140 Ms C dies and goes to hell, or to a place that seems like hell. The devil approaches and
offers to play a game of chance. If she wins, she can go to heaven. If she loses, she will stay in
hell forever; there is no second chance to play the game. If Ms C plays today, she has a 1/2
chance of winning. Tomorrow the probability will be 2/3. Then 3/4, 4/5, 5/6, etc., with no end to
the series. Thus every passing day increases her chances of winning. At what point should she
play the game? The answer is not obvious: after any given number of days spent waiting, it will
still be possible to improve her chances by waiting yet another day. And any increase in the
probability of winning a game with infinite stakes has an infinite utility. For example, if she waits
a year, her probability of winning the game would be approximately .997268; if she waits one
more day, the probability would increase to .997275, a difference of only .000007. Yet, even
.000007 multiplied by infinity is infinite. On the other hand, it seems reasonable to suppose the
cost of delaying for a day to be finite - a day's more suffering in hell. So the infinite expected
benefit from a delay will always exceed the cost. This logic might suggest that Ms C should wait
forever, but clearly such a strategy would be self defeating: why should she stay forever in a
place in order to increase her chances of leaving it? So the question remains: what should Ms C
do?'
[E.J. Gracely: "Playing games with eternity: The devil's offer", Analysis 48.3 (1988) p. 113]
http://www.balliol.ox.ac.uk/sites/default/files/Dudman-1988-Indicative-and-Subjunctive.pdf
{{The verdict of eternal damnation - one of few practical applications of set theory.}}
§ 141 The initially warm relationship between Hilbert and Brouwer began to cool in the
twenties, when Brouwer started to campaign for his foundational views. Hilbert accepted the
challenge - he took the threat of an intuitionistic revolution seriously. Brouwer lectured
successfully at meetings of the German Mathematical Society. His series of Berlin lectures in
1927 caused a considerable stir; there was even some popular reference to a Putsch in
mathematics. [Dirk van Dalen: "The War of the Frogs and the Mice", The Mathematical
Intelligencer 12, 4 (1990) 17-31]
Brouwer came to Göttingen to deliver a talk on his ideas to the Mathematics Club.
"You say that we can't know whether in the decimal representation of π ten 9's occur in
succession," someone objected after Brouwer finished. "Maybe we can't know - but God knows!"
{{Isn't "matheology" an appropriate label?}}
To this Brouwer replied dryly, "I do not have a pipeline to God."
After a lively discussion Hilbert finally stood up.
"With your methods," he said to Brouwer, "most of the results of modern mathematics would
have to be abandoned {{compare Franz Lemmermeyer's accusation that I am at war with
modern mathematics, abandoning the results oft the last 2500 years
http://www.zentralblatt-math.org/zmath/en/search/?q=an:1204.00016&format=complete
http://www.hs-augsburg.de/~mueckenh/Kommentar/
}}, and to me the important thing is not to get fewer results but to get more results." {{Even on
the risk that the "results" are of as little value as the results that can be derived from the bodily
Assumption of Virgin Mary?}}
He sat down to the enthusiastic applause. {{Four legs good, two legs bad.}}
[Constance Reid: "Hilbert", Springer (1970) p. 184f]
http://books.google.de/books/about/Hilbert.html?id=mR4SdJGD7tEC&redir_esc=y
§ 142 Baire considered any infinite set, denumerable or not, as ‘virtual’ - an object defined by
certain conventions. Thus if one is given an infinite set, “it is false ... to consider the subsets of
this set as given”. {{Baires point of view at first glance seems unresonable and disconcerting.
How can it be that the subsets of a given set can be not given if even an axiom "proves" that the
power set is given? But Baire has received a late confirmation. It has turned out that some
elements of some sets of real numbers and some elements of the power set of the natural
numbers cannot be identified by finite definitions whereas infinite sequences without finite
definition cannot be identified at all. Thus they are not given, at least they cannot be taken.}} A
fortiori it made no sense to conceive, as Zermelo did, that the choice of an element had been
made in each subset. {{That is plainly impossible if the objects are not given.}} [...] Baire insisted
on regarding Zermelo’s choices as dependent: “One takes a distinguished element m1 from the
set M; there remains M - m1, in which one takes a distinguished element m2, etc” {{Zermelo felt
obliged to show that γ-sets with one and two elements exist. This mathematically completely
superfluous action is the psychologically important abracadabra of the magician. So Zermelo
elegantly passes by the proof that every subset of a given set exists. Had he tried it, his failure
would have become obvious immediately. But better late than never!}}
[Gregory H. Moore: "The Origins of Zermelos Axiomatization of Set Theory" (1978)]
http://www.jstor.org/pss/30226178
§ 143 This article undertakes a critical reappraisal of arguments in support of Cantor’s theory of
transfinite numbers. The following results are reported:
- Cantor’s proofs of nondenumerability are refuted by analyzing the logical inconsistencies in
implementation of the reductio method of proof and by identifying errors. Particular attention is
given to the diagonalization argument and to the interpretation of the axiom of infinity.
- Three constructive proofs have been designed that support the denumerability of the power set
of the natural numbers, P(Ù), thus implying the denumerability of the set of the real numbers —.
These results lead to a Theorem of the Continuum that supersedes Cantor’s Continuum
Hypothesis and establishes the countable nature of the real number line, suggesting that all
infinite sets are denumerable. Some immediate implications of denumerability are discussed:
- Valid proofs should not include inconceivable statements, defined as statements that can be
found to be false and always lead to contradiction. This is formalized in a Principle of
Conceivable Proof.
- Substantial simplification of the axiomatic principles of set theory can be achieved by excluding
transfinite numbers. To facilitate the comparison of sets, infinite as well as finite, the concept of
relative cardinality is introduced.
- Proofs of incompleteness that use diagonal arguments (e.g. those used in Gödel’s Theorems)
are refuted. A constructive proof, based on the denumerability of P(Ù), is presented to
demonstrate the existence of a theory of first-order arithmetic that is consistent, sound, negationcomplete, decidable and (assumed p.r. adequate) able to prove its own consistency. Such a
result reinstates Hilbert’s Programme and brings arithmetic completeness to the forefront of
mathematics.
[J. A. Perez: "Addressing mathematical inconsistency: Cantor and Gödel refuted" Arxiv (2010)]
http://arxiv.org/ftp/arxiv/papers/1002/1002.4433.pdf
§ 144 Introduction. I dedicate this essay to the two-dozen-odd people whose refutations of
Cantor’s diagonal argument (I mean the one proving that the set of real numbers and the set of
natural numbers have different cardinalities) have come to me either as referee or as editor in
the last twenty years or so. [...] Cantor’s argument is short and lucid. It has been around now for
over a hundred years. Probably every professional mathematician alive today has studied it and
found no fallacy in it.
{{Paul Bernays is no longer alive but his recognition is so much a commonplace that every living
mathematician should know it: "It is not an exaggeration to say that platonism reigns today in
mathematics. But on the other hand, we see that this tendency has been criticized in principle
since its first appearance and has given rise to many discussions. This criticism was reinforced
by the paradoxes discovered in set theory, even though these antinomies refute only extreme
platonism. It is this absolute platonism which has been shown untenable by the antinomies.
Nonetheless, if we pursue the thought that each real number is defined by an arithmetical law
the idea of the totality of real numbers is no longer indispensable." [Paul Bernays: "On Platonism
in Mathematics", (1934) p. 6f] And with extreme platonism also Cantor's diagonal argument
vanishes. Who does know this and who does, in addition, share the scepticism of Solomon
Feferman ["Infinity in Mathematics: Is Cantor Necessary?"]: "I am convinced that the platonism
which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject [...]
platonism is the medieval metaphysics of mathematics; surely we can do better." cannot share
the absolute claim articulated here. The author should know it, and if knowing it, he should say it.
But he says something quite different:}}
There is a point of culture here. Several of the authors said that they had trained as
philosophers, and I suspect that in fact most of them had. In English-speaking philosophy (and
much European philosophy too) you are taught not to take anything on trust, particularly if it
seems obvious and undeniable. You are also taught to criticise anything said by earlier
philosophers. Mathematics is not like that; one has to accept some facts as given and not up for
argument. {{Philosophers often are charged to be bad mathematicians. Kant, for instance,
according to Cantor [letter to Russell, Sept. 19, 1911], "was so bad a mathematician".}}
Nobody should be surprised when philosophers who move into another area take their habits
with them. (In the days when I taught philosophy {{... that explains a lot!}}
[Wilfrid Hodges: "An editor recalls some hopeless papers", The Bulletin of Symbolic Logic 4,1
(1998)]
§ 145 This book advocates nothing less than the elimination of the infinite from mathematics.
[...] If we accept Brouwer's view, the only sets which exist are those which are countable and
have been effectively well-ordered. The author remarks, and this is his principal point, that we
can as well go the whole way and admit the existence only of finite sets. Any statement about a
countable, effectively well-ordered set can by a circumlocution be translated into a statement
about the rule by which the elements follow each other in the well-ordering, which rule is
something finite and definite. For example, we are shown how it is possible in his scheme to
prove that every bounded monotonic sequence of irrational numbers has a limit. [...] It seems to
me that the author, besides producing an interesting book, has made a good case for the
contention that if we accept Brouwerism, we can get along theoretically without the notion of an
infinite set, whether or not that notion is meaningless, as the author maintains. However, I do not
believe that the views of Brouwer will ever find general acceptance among mathematicians. As
this is not the place for an elaborate discussion of the questions raised by the intuitionists, I
should merely like to add the following minor point to the prevailing confusion. If the continuum
hypothesis is true, it is conceivable that someone may someday discover an effective way of
well-ordering the real number continuum so that every number has only a countable number of
predecessors. If this were done it would be practically a refutation of Brouwerism. It might seem,
then, that either the intuitionists must prove that it cannot be done, or must proceed with a sort of
sword of Damocles hanging over their heads. {{There seems to prevail a fundamental
misunderstanding in the examiner's imagination. - Every real well-ordering would show the
countability of the continuum, because then every real number would be named. But there are
only countably many names. This paragraph shows that in 1931 the impossibility of a wellordering of the continuum had not yet been generally recognized by mathematicians. Probably
not much has changed yet, and students of mathematics don't even know what they are
supposed to defend (see § 002).}}
[Orrin Frink: Review of Felix Kaufmann: "Das Unendliche in der Mathematik und seine
Ausschaltung", Deuticke, Leipzig (1930), Bull. Amer. Math. Soc. 37 (1931) 149-150.]
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/118
3494611
§ 146
Men who cannot work receive a pension. That's a commandment of humanity.
Men who are dead cannot work. That's a hardship of misfortune.
Men who are dead receive a pension. That's called a syllogism.
Syllogism is a part of logic. Logic has been invented by Greeks.
So, if in Greece someone is dead ...
(Not every nonsense can be blamed to actual infinity. But most.)
§ 147 Every measurable in our reality has a Planck value that is a combination of three
fundamental constants: c (the velocity of light), g (the gravitational constant), and h (Planck's
constant). Planck values for time, length, area, volume, mass and any other measurable aspect
of reality limit the maximal precision by which meaningful measurements may be taken. One
Planck length is about 1.6ÿ10-35 m; the shortest interval of time is about 10-43 sec; volume cannot
be broken into pieces smaller than about 10-99 cm3 [...] Planck‘s limits have some uncertainty,
and the values may be refined in time, but what is clear is that real numbers applied to our reality
are granular, i.e., they are discrete. All reals are counting numbers at the quantum level.
[B. L. Crissey: "Unreal Irrationals: Turing Halts Cantor"]
http://www.briancrissey.info/Research/Resume_files/Unreal%20Irrationals.pdf
§ 148 A one-to-one mapping (i.e. a bijection) between natural numbers, and real numbers
between 0 & 1, is constructed; the mapping formula is simple, direct, and easy to calculate and
work with. The traditional Cantor diagonal argument is then traced through: but at each step we
use the mapping formula to show that the number generated thus far, is present in our one-toone mapping; thus contradicting the traditional conclusion of said diagonal argument. We then
extend the range of the mapping to the full set of real numbers, using the traditional tangentfunction approach. The existence of the bijection naturally appears to contradict results in
Cantor’s analysis. The present author speculates - but doesn’t pursue or prove here - that the
apparent contradiction may be due to how the mappings here and in Cantor’s analysis, are
constructed; and may be analogous to phenomena seen in rates of convergence of, &/or in
naive rearrangements of conditionally-convergent, infinite series.
[Edward Grattan: "A One-to-One Mapping from the Natural Numbers to the Real Numbers"
(2012)]
http://www.cruziero.com/conglom/11nr-vcurr.pdf
§ 149 It might be objected that no contradiction results from taking the real numbers to form a
definite totality. There is, however, no ground to suppose that treating an indefinitely extensible
concept as a definite one will always lead to inconsistency; it may merely lead to our supposing
ourselves to have a definite idea when we do not ...
[M. Dummett: "The seas of language", Oxford University Press (1993) p. 442]
http://books.google.de/books/about/The_Seas_of_Language.html?id=dzf_nYwI5IC&redir_esc=y
§ 150 There are many mathematicians who will accept the Garden of Eden, i.e. the theory of
functions as developed in the 19th century, but will, if not reject, at least put aside the theory of
transfinite numbers, on the grounds that it is not needed for analysis. {{In reality nearly all
mathematicians do so.}} Of course, on such grounds, one might also ask what analysis is
needed for; and if the answer is basic physics, one might then ask what that is needed for. When
it comes down to putting food in one’s mouth, the 'need' for any real mathematics becomes
somewhat tenuous. Cantor started us on an intellectual journey. One can peel off at any point;
but no one should make a virtue of doing so. {{Neither of doing not so.}}
[W.W. Tait: "Cantor’s Grundlagen and the Paradoxes of Set Theory" (2000) p. 21f]
http://home.uchicago.edu/~wwtx/cantor.pdf
§ 151 The (truly) infinite, I claim, can never be subjugated. Indeed I would go further: the (truly)
infinite, as a unitary object of thought, does not and cannot exist.
This is not to say that the concept of the infinite has no legitimate use. One such use, if I am
right, is precisely to claim that the infinite does not exist. Another such use, I would further argue,
is to claim that there are infinitely many possibilities (including endlessly recurring possibilities of
set membership) afforded by all the finite things that do exist. And to claim these things is, in a
suitably neoteric way, to repudiate the actual infinite and to acknowledge the potential infinite -
the very thing that Aristotle was teaching us to do some two and a half millennia ago. {{This view
is not neoteric but simply sensible, rational and mathematical.}}
[A.W. Moore: “The Infinite”, 2nd ed., Routledge, New York (2005) p. XV]
§ 152 Consider the following sequence of decimal numbers, consisting of digits 0 and 1
01.
0.1
010.1
01.01
0101.01
010.101
01010.101
0101.0101
...
which, when indexed by natural numbers, looks like this:
0 21 1.
02.11
041302.11
0413.0211
06150413.0211
061504.130211
0817061504.130211
08170615.04130211
...
What is the limit of the sequence of the sets of indexes on the left hand side? What is the limit of
the decimal numbers?
§ 153 A charismatic speaker well-known for his clarity and wit, he once delivered a lecture
giving an account of Gödel's second incompleteness theorem, employing only words of one
syllable. [George Boolos: "Gödel's Second Incompleteness Theorem - Explained in Words of
One Syllable", Mind, 103, Jan. 1994, p. 1ff]
http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf
At the end of his viva, Hilary Putnam asked him, "And tell us, Mr. Boolos, what does the
analytical hierarchy have to do with the real world?" Without hesitating Boolos replied, "It's part
of it". {{If present at that illustrious moment I would have added another question: And tell us, Mr.
Boolos, does every part of the real world have to observe its constraints? Unfortunately we don't
know the answer. But the constraints of the speech would have been kept by a simple "yes".}}
http://en.wikipedia.org/wiki/George_Boolos
The talk ended: So, if math is not a lot of bunk, then, though it can't be proved that two plus two
is five, it can't be proved that it can't be proved that two plus two is five.
By the way, in case you'd like to know: yes, it can be proved that if it can be proved that it
can't be proved that two plus two is five, then it can be proved that two plus two is five. {{Spelled
out clearly: If math is not a lot of bunk, then math is a lot of bunk. And this obvious nonsense not
only has been accepted in matheology, but is sacred as a touchstone of the intellectual capacity
of their disciples and as a fixing of their belief in finished infinity. - Because, as Gödel himself
already noted, without actual infinity his theorems are invalid.}}
§ 154 Consistency Proof!
The long missed solution of an outstanding problem came from a completely unexpected side:
Social science proves the consistency of matheology by carrying out a poll.
As reported in § 152 mathematics and matheology lead to different values of the continued
fraction
100
+ 101
10
+ 102
10
+ 103
10
1/
+ ... = 0 (Cauchy)
10
100
+ 101
10
+ 102
10
+ 103
10
+ ... > 1 (Cantor)
1/
10
But 100 % of all matheologians who responded to our poll said that this difference is not
surprising since different methods have been applied, namely the mathematical calculation
invented by Cauchy and the matheological method invented by Cantor. Although both names
begin with a C (like certainty (and even with a Ca (like can and cannot))) the following letters are
completely different.
The general opinion is that it is not surprising to find different results when applying different
methods. Even the application of the same method by different people may yield different results
as we see daily in our elementary schools.
This attitude implies some consequences with respect to the human rights. We should no longer
talk of mistakes and errors in calculations and punish pupils who deviate from the majority or
main stream, but we should only note beside the result who applied what method and possibly
also location and time because experience shows that the result of a calculation may depend on
such details.
For, he reasons pointedly: That which must not, can not be. (C. Morgenstern)
§ 155 At first it seems obvious, but the more you think about it, the stranger the deductions
from this axiom seem to become; in the end you cease to understand what is meant by it.
(Bertrand Russell about the Axiom of Choice)
[Naum Yakovlevich Vilenkin: "In search of infinity", Birkhäuser, Boston (1995) p. 123]
http://books.google.de/books?id=cU3HQFek7L0C&printsec=frontcover&source=gbs_v2_summa
ry_r&cad=0#v=onepage&q=&f=false
The axiom of choice is obvious. But there are no uncountable sets. Therefore the impossible
task vanishes that elements must be well-orderable without the possibility to distinguish and
identify them.
§ 156 The object of this paper is to give a satisfactory account of the Foundations of
Mathematics in accordance with the general method of Frege, Whitehead and Russell. Following
these authorities, I hold that mathematics is part of logic, and so belong to what may be called
the logical school as opposed to the formalist and intuitionist schools. I have therefore taken
Principia Mathematica as a basis for discussion and amendment; and believe myself to have
discovered how, by using the work of Mr. Ludwig Wittgenstein, it can be rendered free from the
serious objections which have caused its rejection by the majority of German authorities, who
have deserted altogether its line of approach.
In this chapter we shall be concerned with the general nature of pure mathematics, and how it is
distinguished from other sciences. (Footnote: In the future by 'mathematics' will always be meant
'pure mathematics'.) Here there are really two distinct categories of things of which an account
must be given -- the ideas or concepts of mathematics, and the propositions of mathematics.
This distinction is neither artificial nor unnecessary, for the great majority of writers on the
subject have concentrated their attention on the explanation of one or other of these categories,
and erroneously supposed that a satisfactory explanation of the other would immediately follow.
Thus the formalist school, of whom the most eminent representative is now Hilbert, have
concentrated on the propositions of mathematics, such as '2 + 2 = 4'. They have pronounced
these to be meaningless formulae to be manipulated according to certain arbitrary rules, and
they hold that mathematical knowledge consists in knowing what formulae can be derived from
what others consistently with the rules. Such being the propositions of mathematics, their
account of its concepts, for example the number 2, immediately follows. '2' is a meaningless
mark occurring in these meaningless formulae. But, whatever may be thought of this as an
account of mathematical concepts, it is obviously hopeless as a theory of mathematical
concepts; for these occur not only in mathematical propositions, but also in those of everyday
life. Thus '2' occurs not merely in '2 + 2 = 4', but also in 'It is 2 miles to the station', which is not a
meaningless formula, but a significant proposition, in which '2' cannot conceivably be a
meaningless mark. Nor can there be any doubt that '2' is used in the same sense in the two
cases, for we can use '2 + 2 = 4' to infer from 'It is two miles to the station and two miles on to
the Gogs' that 'It is four miles to the Gogs via the station', so that these ordinary meanings of two
and four are clearly involved in '2 + 2 + 4'. So the hopelessly inadequate formalist theory is, to
some extent, the result of considering only the propositions of mathematics and neglecting the
analysis of its concepts, on which additional light can be thrown by their occurrence outside
mathematics in the propositions of everyday life.
[F.P. Ramsey: "The Foundations of Mathematics", Proc. Lond. Math. Soc., Vol.s2-25, Issue 1
(1926) 338-384]
http://www.hist-analytic.org/Ramsey.htm
http://www-history.mcs.st-and.ac.uk/Mathematicians/Ramsey.html
§ 157 Finitism is usually regarded as the most conservative standpoint for the foundations of
mathematics. Induction is justified by appeal to the finitary credo: for every number x there exists
a numeral d such that x is d. It is necessary to make this precise. We cannot express it as a
formula of arithmetic because "there exists" in "there exists a numeral d" is a metamathematical
existence assertion, not an arithmetical formula beginning with $.
The finitary credo can be formulated precisely using the concept of the standard model of
arithmetic: for every element ξ of Ù there exists a numeral d such that it can be proved that d is
equal to the name of ξ, but this brings us into set theory. The finitary credo has an infinitary
foundation. {{There is a sober fraction of set theory, namely finite set theory.}}
The use of induction goes far beyond the application to numerals. It is used to create new
kinds of numbers (exponential, superexponential, and so forth) in the belief that they already
exist in a completed infinity. If there were a completed infinity Ù consisting of all numbers, then
the axioms of {{PA}} would be valid assertions about numbers and {{PA}} would be consistent.
[E. Nelson: "Outline, Against finitism"]
http://www.math.princeton.edu/~nelson/papers/outline.pdf
§ 158 One user of transfinite set theory, i.e., a man who arrives where he cannot arrive, is
Mohamed El Naschie.
http://www.el-naschie.net/bilder/file/Photo-Gallery.pdf
Then came the next quantum jump, around 1990, when M.S. El Naschie who was originally
working on elastic and fluid turbulence began to work on his Cantorian version of fractal spacetime. He showed that the n-dimensional triadic Cantor set has the same Hausdorff dimension as
the dimension of a random inverse golden mean Sierpinski space to the power n-1. [...]
Sometime later El Naschie using the work of Prigogine on irreversibility showed that the arrow of
time may be explained in a fractal space-time. A few years later two of El Naschie’s papers on
the subject were noted by Thompson essential science indicators as the most cited New frontier
paper in physics and as Hot paper in engineering. {{That seems to come a bit too early. At the
University of Applied Sciences Augsburg the theories of El Naschie have not yet been taught.}}
[...]
In E-infinity theory El Naschie admit formally infinite dimensional "real” space-time. This infinity
is hierarchical in a strict mathematical way and he was able to show that E-infinity has finite
number of dimensions when observed from a distance. At low resolution or equivalently at low
energy the E-infinity Cantorian space-time appear as a four dimensional space-time manifold.
[...] The eigenvalues like equation have a very simple interpretation: Dim E8 E8 = 496 represent
all fundamental interactions. Thus it is equal to particle physics 339 symmetries plus the R(4) =
20 of gravity plus ¡0. From that we deduce ¡0 = 496 - 339 - 20 = 137. {{The correct value
137.036 has exorcized some number mysticists and numerologists.}} [...] The author is indebted
to the many members of the fractal-Cantorian space-time community {{Cantor's idea of
countably many body-atoms and uncountably many ether-atoms gains new impetus. The fatal
Space-Time-Community grows, it seems, above all limits.}}
[L. Marek-Crnjac: "A short history of fractal-Cantorian space-time", Chaos, Solitons and Fractals
41 (2009) 2697–2705]
http://www.msel-naschie.com/pdf/cantorian-history.pdf
§ 159 Hellman, Maddy, and Steel are all impressed by the possibility that set-theoretically
substantial mathematics might one day be needed in scientific applications. But of course the
mere possibility of future applications provides no support whatever for the indispensability
argument. Indeed, one could say of virtually any formal system that future applications are
possible. We would only find this possibility noteworthy if we had separate reasons for being
interested in the particular system in question. This is really the opposite of an indispensability
argument, because ZFC is not gaining credibility from its scientific applications — at present it
has none — but rather is seen as a good candidate for future applications because evidently it is
already felt to be credible on some other grounds. I must add, however, that given our current
understanding of basic physics, the prospect of set-theoretically substantial mathematics ever
becoming essential to meaningful scientific applications appears extremely unlikely. This should
be obvious to anyone with a basic knowledge of mathematical physics and an understanding of
the scope of predicative mathematics. An essential incorporation of impredicative mathematics
in basic physics would involve a revolutionary shift in our understanding of physical reality of a
magnitude which would dwarf the passage from classical to quantum mechanics (after all, both
of these theories are completely predicative). I would rate the likelihood of ZFC turning out to be
inconsistent as much higher than the likelihood of it turning out to be essential to basic physics.
[Nik Weaver: "Is set theory indispensable?" (2009)]
http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.1680v1.pdf
In principle, yes, cp. § 154, for instance. But these faint sounds are always shouted down by
There's no contra-dic-tion!
There's no contra-dic-tion!
There's no contra-dic-tion!
§ 160 {{Another application of set theory?}} One of the remarkable observations made by the
Voyager 2 probe was of the extremely fine structure of the Saturn ring system. [...] The Voyager
1 and 2 provided startling images that the rings themselves are composed of thousands of
thinner ringlets each of which has a clear boundary separating it from its neighbours.
This structure of rings built of finer rings has some of the properties of a Cantor set. The
classical Cantor set is constructed by taking a line one unit long, and erasing its central third.
This process is repeated on the remaining line segments, until only a banded line of points
remains. {{Materialized points are certainly not available in the Saturn ring system.}}
[H. Takayasu: "Fractals in the physical sciences", Manchester University Press (1990) p. 36]
http://books.google.de/books?id=NRYNAQAAIAAJ&pg=PA180&lpg=PA180&dq=Takayasu:+%2
2Fractals+in+the+physical+sciences%22&source=bl&ots=_jQrSNVTs&sig=ttuEDGX_6381_T2AdLEBT8HIT20&hl=de&sa=X&ei=Yb6GTiFDcvUsgao6ITPBg&sqi=2&ved=0CDMQ6AEwAQ#v=onepage&q=Takayasu%3A%20%22Fract
als%20in%20the%20physical%20sciences%22&f=false
Mandelbrot conjectures that radial cross-sections of Saturn's rings are fat Cantor sets. For
supporting evidence, click each picture for an enlargement in a new window.
http://classes.yale.edu/fractals/labs/paperfoldinglab/fatcantorset.html
http://www.youtube.com/watch?v=Ztgqa_5vumI
§ 161 {{Yet another application of set theory?}} I propose here, then, first to illustrate, and then
to discuss theoretically, the nature and ideal outcome of any recurrent operation of thought, and
to develope, in this connection, what one may call the positive nature of the concept of Infinite
Multitude.
Prominent among the later authors who have dealt with our problem from the mathematical side,
is George Cantor. [...] With this theory of the Mächtigkeiten I shall have no space to deal in this
paper, but it is of great importance for forming the conception of the determinate Infinite.
A map of England, contained within England, is to represent, down to the minutest detail, every
contour and marking, natural or artificial, that occurs upon the surface of England.
Our map of England, contained in a portion of the surface of England, involves, however, a
peculiar and infinite development of a special type of diversity within our map. For the map, in
order to be complete, according to the rule given, will have to contain, as a part of itself, a
representation of its own contour and contents. In order that this representation should be
constructed, the representation itself will have to contain once more, as a part of itself, a
representation of its own contour and contents; and this representation, in order to be exact, will
have once more to contain an image of itself; and so on without limit. We should now, indeed,
have to suppose the space occupied by our perfect map to be infinitely divisible, even if not a
continuum.
That such an endless variety of maps within maps could not physically be constructed by men,
and that ideally such a map, if viewed as a finished construction, would involve us in all the
problems about the infinite divisibility of matter and of space, I freely recognize.
Suppose that, for an instance, we had accepted this assertion as true. Suppose that we then
attempted to discover the meaning implied in this one assertion. We should at once observe that
in this one assertion, "A part of England perfectly maps all England, on a smaller scale," there
would be implied the assertion, not now of a process of trying to draw maps, but of the
contemporaneous presence, in England, of an infinite number of maps, of the type just
described. The whole infinite series, possessing no last member, would be asserted as a fact of
existence.
We should, moreover, see how and why the one and the infinitely many are here, at least within
thought's realm, conceptually linked. Our map and England, taken as mere physical existences,
would indeed belong to that realm of "bare external conjunctions." Yet the one thing not
externally given, but internally self-evident, would be that the one plan or purpose in question,
namely, the plan fulfilled by the perfect map of England, drawn within the limits of England, and
upon a part of its surface, would, if really expressed, involve, in its necessary structure, the
series of maps within maps such that no one of the maps was the last in the series.
This way of viewing the case suggests that, as a mere matter of definition, we are not obliged to
deal solely with processes of construction as successive, in order to define endless series. A
recurrent operation of thought can be characterized as one that, if once finally expressed, would
involve, in the region where it had received expression, an infinite variety of serially arranged
facts, corresponding to the purpose in question.
[Josiah Royce: "The world and the individual", MacMillan, London (1900) p. 500ff]
http://www.archive.org/stream/worldindividual00royciala#page/n0/mode/2up
http://www.archive.org/stream/worldindividual00royciala/worldindividual00royciala_djvu.txt
The repeated application of the fotocopier has been proposed as a cheap replacement for
expensive electron microscopes. Unfortunately I have forgotten the name of the inventor of this
idea.
§ 162 About limits of real sequences.
The limit of an infinite sequence (ak) of real numbers ak is determined solely by the finite terms
of the sequence. Otherwise, the limit would not have to be computed but would have to be
created. Analysis is concerned with analyzing, i.e., with finding.
To give an example, we can state with absolute certainty that in the real numbers the
sequence 0.1, 0.11, 0.111, ... has the limit 0.111... = 1/9. That is independent of the method
which is used to analyze the sequence.
But there are different aspects of the limit, namely the numerical value of the limit, the set of
coefficients of the power series, its cardinal number, the set of indexes which belong to a digit 1,
its cardinal number, the set of indexes which belong to a digit 2, its cardinal number, the set of
different digits appearing in the limit, and many further aspects.
If any of these aspects is computed by another than the analytical method and turns out as
deviating from the analytical result, then the other method is not suitable for analytical purposes.
§ 163 First hidden necessary condition of Cantor's proof. - In the middle of the XX c., metamathematics announced Cantor's set theory "naive" and soon the very mention of the term
"actual infinity" was banished from all meta-mathematical and set theoretical tractates. The
ancient logical, philosophical, and mathematical problem, which during millenniums troubled
outstanding minds of humankind, was "solved" according to the principle: "there is no term there is no problem". So, today we have a situation when Cantor's theorem and its famous
diagonal proof are described in every manual of axiomatic set theory, but with no word as to the
"actual infinity". However, it is obvious that if the infinite sequence of Cantor's proof is potential
then no diagonal method will allow to construct an individual mathematical object, i.e., to
complete the infinite binary sequence. Thus, just the actuality of the infinite sequence is a
necessary condition (a Trojan Horse) of Cantor's proof, and therefore the traditional, settheoretical formulation of Cantor's theorem is, from the standpoint of classical mathematics,
simply wrong and must be re-written as follows without any contradiction with any logic.
[A.A. Zenkin: "Scientific Intuition of Genii Against Mytho-'Logic' of Cantor‘s Transfinite Paradise"
Procs. of the International Symposium on “Philosophical Insights into Logic and Mathematics,”
Nancy, France, 2002, p. 2]
http://www.ccas.ru/alexzen/papers/CANTOR-2003/Zenkin%20PILM2002.doc
§ 164 Because of this obsession with "rigorous" (or "formal") proofs, Mathematics has gotten
so specialized, where no one can see the forest, and even most people can't see the whole tree
they sit on. All they can see is their tiny branch. Even in specialized conferences, many people
skip the invited talks and only go to their own doubly-specialized session. […]
Computer algebra, and experimental mathematics, has the potential to become the new
unifying "religion". There is still room for some proofs, especially nice ones […] but "formal"
proofs should lose their centrality. They are an obsolete relic from a bygone age, just like printjournals, and using a typist to convert your hand-written manuscript to a .tex file. There is so
much mathematical knowledge out there that can be discovered empirically (like in the natural
sciences, of course it should still be theory-laden, or else it won't go very far). Once we convert
to this new religion, we would understand the big picture so much better, and have much more
global insight (those that tell me that the purpose of proofs is insight make me laugh, true, the
top one percent of proofs give you (local) insight, but the bottom 99 percent are just formal
verifications, many of which can already be done by computer, and the rest soon will be [if you
are stupid enough to want them]).
Proofs are Dead, Long Live Algorithms (and Meta-Algorithms!).
[Doron Zeilberger: "Opinion 113: Mathematics is Indeed a Religion, But It has too Many Sects!
Let's Unite Under the New God of Experimental Mathematics" (2011)]
http://www.math.rutgers.edu/~zeilberg/Opinion113.html
§ 165 Thus the conquest of actual infinity may be considered an expansion of our scientific
horizon no less revolutionary than the Copernican system or than the theory of relativity, or even
of quantum and nuclear physics.
[.A. Fraenkel, A. Levy: "Abstract Set Theory" North Holland, Amsterdam (1976), p. 240]
Inaccessible cardinals on the same scientific level with transistor, laser, computer, NMR, GPS,
or space probes? No, rather with Star Trek or Dallas or, as it is a genuinely German invention,
with Schwarzwaldklinik.
§ 166 The fact that some discrete items might lack a determinate number, this being connected
with the possibility of them being given as a complete whole, was, of course, the traditional,
Aristotelian point of view, which Intuitionists, more recently, have still held to. But many others
now doubt this fact. Is there any way to show that Aristotle was right? I believe there is.
For when discrete items do clearly collect into a further individual, and we have a finite set, then
we determine the number in that set by counting. But what process will determine what the
number is, in any other case? The newly revealed independence of the Continuum Hypothesis
shows there is no way to determine the number in certain well known infinite sets. [...] The key
question therefore is: if there is a determinate number of natural numbers, then by what process
is it determined? Replacing 'the number of natural numbers' with 'Aleph zero' does not make its
reference any more determinate. The natural numbers can be put into one-one correspondence
with the even numbers, it is well known, but does that settle that they have the same number?
We have equal reason to say that they have a different number, since there are more of them.
So can we settle the determinate number in a set of discrete items just by stipulation?
Indeed, if all infinite sets could be put into one-one correspondence with each other, one would
be justified in treating the classification 'infinite' as an undifferentiated refusal of numerability. But
given Cantor's discovery that there are infinite sets which cannot be put into one-one
correspondence with each other, this conclusion is less compelling.
For Dedekind defined infinite sets as those that could be put into one-one correlation with proper
subsets of themselves, so the criteria for 'same number' bifurcate: if any two such infinite sets
were numerable, then while, because of the correlation, their numbers would be the same, still,
because there are items in the one not in the other, their numbers would be different. Hence
such 'sets' are not numerable, and one-one correlation does not equate with equal numerosity
[...]
[H. Slater: "The Uniform Solution of the Paradoxes" (2004)]
http://www.philosophy.uwa.edu.au/about/staff/hartley_slater/publications/the_uniform_solution_o
f_the_paradoxes
§ 167 If, in order to decorate one of my books,
https://portal.dnb.de/opac.htm?method=showFirstResultSite&currentResultId=auRef%3D10963
5876%26any&selectedCategory=any
I had a straight choice between aleph and the Ishango-bone
http://en.wikipedia.org/wiki/Ishango_bone
I would choose the latter. Contrary to the former the bone contains mathematics.
§ 168 "However, this negative attitude towards Cantor's set theory, and toward classical
mathematics, of which it is a natural generalization, is by no means a necessary outcome of a
closer examination of their foundations, but only the result of a certain philosophical conception
of the nature of mathematics, which admits mathematical objects only to the extent in which they
are interpretable as our own constructions of our own mind, or at least, can be completely given
in mathematical intuition. For someone who considers mathematical objects to exist
independently of our constructions and of our having an intuition of them individually, and who
requires only that the general mathematical concepts must be sufficiently clear for us to be able
to recognize their soundness and the truth of the axioms concerning them, there exists, I
believe, a satisfactory foundation of Cantor´s set theory in its whole original extent and
meaning."
[K. Gödel: "Collected Works II" (1964) OUP/1990, p. 258]
{{Nevertheless (cp. § 026) Gödel held Robinson in high esteem and favoured Robinson as his
successor.}}: Meanwhile, he was meeting with Gödel at Princeton, where they discussed their
mutual interests in mathematics and logic. Gödel was especially impressed by nonstandard
analysis and its potential applications in other parts of mathematics. He had suggested in fact
that Robinson come to the Institute for an extended period of time, and even hoped that
Robinson might one day be his successor.
[Robinson to Gödel, April 14, 1971; Gödel papers #011957, Princeton University archives; cited
in Dauben, 1995, p. 458]
§ 169 If God has mathematics of his own that needs to be done, let him do it himself.
[Errett Bishop: "Foundations of constructive analysis", McGraw-Hill, New York, (1967)
Introduction]
Can He have a list of all real numbers? If not, can He store the real numbers in His Infinite
Eternity or Eternal Infinity? - each number pinned down by a rational space-time-quadruple? - or
quintuple or even centuple? If so, then He could also compute 2 + 2 = 8 and he could conclude
that Cantor was right. It does not seem that He has mathematics of His own.
§ 170 The infinite triangle formed by the sequence
0.1
0.11
0.111
...
has height ¡0 but width less than ¡0 (because the limit 1/9, the first line with ¡0 digits, does not
belong to the triangle). This lack of symmetry is disturbing for a physicist.
But it would be completely unclear, what side of the infinite triangle is the first one to complete
infinity ¡0, when constructed in the following manner:
a
a
bb
c
ac
bbc
d
dc
dac
dbbc
d
dc
dac
...
dbbc
eeeee
§ 171 Every set can be well-ordered.
[E. Zermelo: "Beweis, daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904) 514
Zermelo, E.: "Neuer Beweis für die Möglichkeit einer Wohlordnung", Math. Ann. 65 (1908) 107]
Well-ordering elements requires to identify them. For that sake we need physical elements or
at least labels, less than 10100 of which are available in the whole universe. But well-orderability
is claimed for uncountable sets too.
Does enlightenment never touch matheology?
Is that degree of energy saving necessary???
§ 172 Wallis in 1684 […] accepts, without any great enthusiasm, the use of Stevin's decimals.
He still only considers finite decimal expansions and realises that with these one can
approximate numbers (which for him are constructed from positive integers by addition,
subtraction, multiplication, division and taking nth roots) as closely as one wishes. However,
Wallis understood that there were proportions which did not fall within this definition of number,
such as those associated with the area and circumference of a circle:
Real numbers became very much associated with magnitudes. No definition was really thought
necessary, and in fact the mathematics was considered the science of magnitudes. Euler, in
Complete introduction to algebra (1771) wrote in the introduction: "Mathematics, in general, is
the science of quantity; or, the science which investigates the means of measuring quantity." He
also defined the notion of quantity as that which can be continuously increased or diminished
and thought of length, area, volume, mass, velocity, time, etc. to be different examples of
quantity. All could be measured by real numbers.
Cauchy, in Cours d'analyse (1821), did not worry too much about the definition of the real
numbers. He does say that a real number is the limit of a sequence of rational numbers but he is
assuming here that the real numbers are known. Certainly this is not considered by Cauchy to
be a definition of a real number, rather it is simply a statement of what he considers an "obvious"
property. He says nothing about the need for the sequence to be what we call today a Cauchy
sequence and this is necessary if one is to define convergence of a sequence without assuming
the existence of its limit.
[J.J. O'Connor and E.F. Robertson: "The real numbers: Stevin to Hilbert"]
http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_2.html
§ 173 The question of language is also particularly relevant, for Brouwer would continue to
emphasize that mathematics does not depend on language or logic, being prior to language and
logic. Language is merely an instrument of social domination, and it makes impossible a real
communication: “nobody has ever communicated his soul to someone else by means of
language.” In 1908, this line of thought would derive into a denounce of classical logic and of
axiomatic systems, which obviously from his standpoint cannot be the real foundation of
mathematics. [...] Brouwer stressed that he had been elaborating these ideas since 1907, before
his involvement with topology, and mentioned how (in his opinion) they also forced him to
disagree with Hilbert’s conviction that all mathematical problems are solvable. He emphasized
that the foundations for set theory provided both by the logicists and Zermelo were to be
rejected.
[José Ferreirós: "Paradise Recovered? Some Thoughts on Mengenlehre and Modernism",
(2008)]
§ 174 Literature, art, jurisprudence, medicine, and religion are not restricted by reality? But if
someone is going to write a novel which on 10100 pages describes 10100 characters, if someone
announces a painting in the style of van Gogh with 10100 strokes, if someone expects that the
German tax law will contain 10100 paragraphs by the end of the millennium, if someone trusts in
the dilution D100 as most helpful in homoeopathy, if someone believes in a final battle between
10100 apes on earth at the end of time --- then he will be considered a fool.
But if someone "proves" with mathematical certainty that 10100 different elements can be
distinguished and well-ordered, and if this same person is demanding to be taken serious as a
scientist --- what can we say?
§ 175 Until then, no one envisioned the possibility that infinities come in different sizes, and
moreover, mathematicians had no use for “actual infinity.” The arguments using infinity, including
the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets.
T. Jech: "Set Theory", Stanford Encyclopedia of Philosophy (2002)
http://plato.stanford.edu/entries/set-theory/
There are only countably many names.
An uncountable set of names cannot be well-ordered - because it does not exist.
A set of numbers cannot be well-ordered unless all the numbers have names.
This seems to contradict Cantor's diagonal argument - but only if infinite set are complete.
Conclusion: Infinities do not come come in different sizes. In fact mathematicians have never
had use for actual infinity because they could not. All they could is to believe that they had use
for actual infinity, i. e., for numbers that have no names and cannot be used.
--- That's called matheology.
§ 176 Here's a paradox of infinity noticed by Galileo in 1638. It seems that the even numbers
are as numerous as the evens and the odds put together. Why? Because they can be put into
one-to-one correspondence. The evens and odds put together are called the natural numbers.
The first even number and the first natural number can be paired; the second even and the
second natural can be paired, and so on. When two finite sets can be put into one-to-one
correspondence in this way, they always have the same number of members.
Supporting this conclusion from another direction is our intuition that "infinity is infinity", or that all
infinite sets are the same size. If we can speak of infinite sets as having some number of
members, then this intuition tells us that all infinite sets have the same number of members.
Galileo's paradox is paradoxical because this intuitive view that the two sets are the same size
violates another intuition which is just as strong {{and as justified! If it is possible to put two sets
A and B in bijection but also to put A in bijection with a proper subset of B and to put B in
bijection with a proper subset of A, then it is insane to judge the first bijection as more valid than
the others and to talk about equinumerousity of A and B.}}
[Peter Suber: "Infinite Reflections", St. John's Review, XLIV, 2 (1998) 1-59]
http://www.earlham.edu/~peters/writing/infinity.htm#galileo
§ 177 {{I have often denied any benefit of the distinction between countable and uncountable.
But the advantage for this particular application cannot be overlooked:}} English nouns are often
described as "countable" or "uncountable".
http://www.englishclub.com/grammar/nouns-un-countable.htm
§ 178 The whole history of the Mathematische Annalen conflict was quietly incorporated into
the oral tradition of European mathematics. Little is known of the aftermath; the Göttinger had
won the battle, and they may have been tempted to pick a bone or two with some of the minor
actors. […]
For Brouwer the matter had, in my opinion, far more serious consequences. His mental state
could, under severe stress, easily come dangerously close to instability. Hilbert's attack, the lack
of support from old friends, the (real or imagined) shame of his dismissal, the cynical ignoring of
his undeniable efforts for the Annalen; each and all of these factors drove Brouwer to a selfchosen isolation. […] After the Annalen affair, little zest for the propagation of intuitionism was
left in Brouwer; he continued to work in the field, but on a very limited scale with only a couple of
followers. Actually, his whole mathematical activity became rather marginal for a prolonged
period. During the thirties Brouwer hardly published at all (only two small papers on topology);
he undertook all kinds of projects that had nothing to do with mathematics or its foundations.
[Dirk van Dalen: The Mathematical Intelligencer 12,4 (1990) 17-31]
§ 179 The great fascination that contemporary mathematical logic has for its devotees is due,
in large measure, to the ever increasing sophistication of its techniques rather than to any
definitive contribution to our understanding of the foundations of mathematics. Nevertheless, the
achievements of logic in recent years are relevant to foundational questions and it behooves the
logician, at least once in a while, to reflect on the basic nature of his subject and perhaps even to
report his conclusions. In an address given some years ago the present writer stated a point of
view on the foundations of mathematics which may be summed up as follows. (1) Infinite
totalities do not exist and any purported reference to them is, literally, meaningless; (2) this
should not prevent us from developing mathematics in the classical vein involving the free use of
infinitary concepts; and (3) although an infinitary framework such as set theory, or even only
Peano number theory cannot be regarded as the ultimate foundation for mathematics, it appears
that we have to accept at least a rudimentary form of logic and arithmetic as common to all
mathematical reasoning.
[A. Robinson: "From a formalist's point of view", Dialectica 23 (1969) 45-49]
http://onlinelibrary.wiley.com/doi/10.1111/j.1746-8361.1969.tb01177.x/abstract?
§ 180 A. S. Yessenin-Volpin: "About infinity, finiteness and finitization (in connection with the
foundations of mathematics)", Lecture Notes in Mathematics, 873 (1981) pp. 274-313
http://www.springerlink.com/content/76q2110gx555h660/
Probably interesting literature but certainly painstaking to understand. I have not read it and
cannot judge whether reading it is rewarding. A much simpler, but as valid conclusion is this:
Every form of information transfer (and what else are sequences of digits?) requires an end of
file signal. Infinite sequences of digits are therefore unsuitable for mathematical purposes. (That
they are unsuitable for all other other purposes is well-known anyway.)
Example: The decimal representation of 1/3 = 0.333... is never given by an infinite string of
digits but it is always given by a finite word with an end signal, a full stop or period in form of a
point - it is absolutely indispensable in correct formal expressions, although sometimes, like in
the present case, it is following only after a while - namely here .
§ 181 This paper gives a counterexample to the impossibility, by Gödel's second
incompleteness theorem, of proving a formula expressing the consistency of arithmetic in a
fragment of arithmetic on the assumption that the latter is consistent. This counterexample gives
rise to a new type of metamathematical paradox, to be called the Gödel-Wette paradox, which E.
Wette claims to have established since some time ago (see
[Wette, 1971]: Wette, Eduard W., On new paradoxes in formalized mathematics, Journal of
Symbolic Logic, vol. 36, pp. 376-377.
[Wette, 1974]: Wette, Eduard W., Contradiction within Pure Number Theory because of a
System Internal ’Consistency’-Deduction, International Logic Review, N. 9, 1974, pp. 51-62.
). Nevertheless, our work is independent of Wette's since we have failed to understand the
details of his work {{same happened to Paul Bernays some time before, cp. Kalenderblatt
090804
http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf
Paul Bernays: "Zum Symposium über die Grundlagen der Mathematik", Dialectica, 25:171-195,
1971. (Translation by: Steve Awodey)]
http://www.phil.cmu.edu/projects/bernays/Pdf/bernays28_2003-05-19.pdf
}} although we recognize the possibility of the correctness of the latter. Furthermore, the GödelWette paradox is not the only foundational anomaly which the framework of our approach has
uncovered but new questions concerning the decision problem, completeness problem, truth
definitions and the status of Richard's paradox in arithmetic and set theory (including type
theory) have arisen as well. This work will, eventually be unified into a single monograph.
[A.S. Yessenin-Volpin, C. Hennix: "Beware of the Gödel-Wette paradox", arXiv (2001)]
http://arxiv.org/abs/math/0110094
§ 182 Ultrafinitists don’t believe that really large natural numbers exist. The hard part is getting
them to name the first one that doesn’t.
[John Baez: "The Inconsistency of Arithmetic", September 30, 2011]
The problem is not the size of the number but its information contents. On a pocket calculator,
you can multiply 1030 by 1050, but you cannot add or multiply two numbers with more than 10
digits.
In real life, you can do superexponentiation, but you cannot use a sequence of more than
10100 digits that lack a finite expansion rule like 0.101010… or Σ1/n2.
[WM: Re: The Inconsistency of Arithmetic, September 30, 2011]
http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039531
§ 183 Let's suppose that ZFC is inconsistent. Should anyone here feel shame? I don't see
why. [Jesse F. Hughes, 28 Sep 2011]
Because there cannot be more infinite paths in the Binary Tree than points where paths get
distinct, i.e, nodes where they split. It is a very simple calculation:
|
o
/ \
Every point increases the number of distinct paths by 1. A countable number of points limits the
set of all distinct paths to a countable number. Therefore the elements of a set of uncountable
paths cannot be distinct.
Further the subset of real numbers without a finite definition does not allow to choose a certain
element from it. It could not be defined - being tantamount to the matheological statement: it
could be defined only in a language that nobody can speak, learn, and understand. That makes
Zermelo's axiom of choice obsolete - and his "proof" of well-ordering every set too.
Quite a lot of simple mistakes to feel ashamed.
http://groups.google.com/group/sci.math/msg/5ccdfbcc58a07f66?dmode=source
§ 184 Cantor coined the word "cardinal number" on January 22, 1886 in a letter to Cardinal
Johann Baptist Franzelin - presumably in order to change the Cardinal's mind with respect to
transfinite numbers. Cantor hoped, still in vain, to get an affirmative response of the Cardinal
about transfinite numbers which, even 12 years after their invention (my goodness, how
stubborn Cantor must have been), were completely refused by his contemporary
mathematicians.
§ 185 Sir Arthur Eddington (1882 - 1944): There was just one place where (Einstein's) theory
did not seem to work properly, and that was - infinity. I think Einstein showed his greatness in
the simple and drastic way in which he disposed of difficulties at infinity. He abolished infinity. ...
Since there was no longer any infinity, there could be no difficulties at infinity.
[Eli Maor: "To Infinity and Beyond. A Cultural History of the Infinite", Birkhäuser, Basel (1987) p.
221]
§ 186
Axiom 1: It is possible to choose every subset of a given set and to choose a first element of this
subset, unless the chosen subset is empty.
Axiom 2: It is possible to select a subset of natural numbers with cardinality larger than 10 and
sum of elements less than 10.
What is the epistemological difference of these axioms which are equally true?
§ 187 Whatever the choice of language, there will only be a countable infinity of possible texts,
since these can be listed in size order, and among texts of the same size, in alphabetical order.
{{Here is a simple example:
0
1
00
01
10
11
000
...
}}
This has the devastating consequence that there are only a denumerable infinitely of such
"accessible" reals, and therefore the set of accessible reals has measure zero.
So, in Borel's view, most reals, with probability one, are mathematical fantasies, because there
is no way to specify them uniquely. Most reals are inaccessible to us, and will never, ever, be
picked out as individuals using any conceivable mathematical tool, because whatever these
tools may be they could always be explained in French, and therefore can only "individualize" a
countable infinity of reals, a set of reals of measure zero, an infinitesimal subset of the set of all
possible {{interesting question: what are possible properties of possible/}} reals.
Pick a real at random, and the probability is zero that it's accessible - the probability is zero
that it will ever be accessible to us as an individual mathematical object. {{How can we pick? By
picking it, a real number would be finitely defined already. That means an undefined real number
can never be picked mathematically. And with finger or beak nobody could succeed.}}
[Gregory Chaitin: "How real are real numbers?" (2004)]
http://arxiv.org/abs/math.HO/0411418
The enumeration of all rational numbers is tantamount to an infinite sum of units. One gets the
divergent sequence of all finite cardinal numbers and maintains that a limit exists. That is a
mistake. The fact that we can count up to every number does not imply that we can count all
numbers. After every finite cardinal number there are infinitely many - but not after all. In a
similar way it is impossible to sum all terms of the series Σ1/2n. But contrary to a diverging
sequence, the sequence of partial sums of this series deviates from 1 less and less. Therefore 1
can be called the limit.
§ 188 In 1960 the physicist Eugene Wigner published an influential article on "The
unreasonable effectiveness of mathematics in the natural sciences". [E. P. Wigner: "The
unreasonable effectiveness of mathematics in the natural sciences", Communications on pure
and applied mathematics, 13 (1960)] I counter the claim stated in its title with an interpretation of
science in which many of the uses of mathematics are shown to be quite reasonable, even
rational, although maybe somewhat limited in content and indeed not free from ineffectiveness.
The alternative view emphasizes two factors that Wigner largely ignores: the effectiveness of the
natural sciences in mathematics, in that much mathematics has been motivated by
interpretations in the sciences, and still is; and the central place of theories in both mathematics
and the sciences, especially theory-building, in which analogies drawn from other theories play
an important role.
[Ivor Grattan-Guinness: "Solving Wigner's Mystery: The Reasonable (Though Perhaps Limited)
Effectiveness of Mathematics in the Natural Sciences" Springer Science+Business Media, Inc.,
Volume 30, Number 3 (2008)]
All correct mathematics has to orient itself by means of reality, i.e., natural sciences.
Mathematics is applied physics. Cantor intended to follow that scheme with his transfinite set
theory, which he, by his own protestation, had devised in order to apply it in natural sciences.
Alas his idea of reality was so bad (in contrast to most of his contemporaries he rejected
atomism and Darwinism), that it could yield only wrong mathematics.
§ 189 The Liar paradox [...] does not mention totalities at all. Russell held there to be a
common cause of, and solution to, all the paradoxes of self-reference. He therefore had to
manipulate the Liar paradox into a form where the theory of orders could be applied to it. He did
this by parsing the Liar sentence as: there is a proposition that I am affirming and that is false,
i.e., $ p(I assert p and p is false). If the quantifier in this proposition has order i, it, itself, is of
order i+1, and so does not fall within the scope of the quantifier. This breaks the argument to
contradiction. Russell's parsing, by insisting that the self-reference involved be obtained by
quantification, strikes one as totally artificial. For a start, the Liar does not have to be asserted to
generate a contradiction. But, more fundamentally, the self-reference required may be obtained
by ways other than quantification, for example, by a demonstrative: this proposition (or
sentence) is false
Notoriously, some 40 years after Tarski's proposal, there is no evidence to show that English is
a hierarchy of metalanguages - indeed, there is evidence to show that it is not. Nor is there any
reason to suppose that the extensions of words like "true" are context-dependent, in the way
that, for example "past" is. [...] It is certainly true that the domains of some quantifiers are
contextually determined ("everyone has had lunch"); but, equally, those of others are not ("every
natural number is odd or even"), and Parsons gives no reason independent of the paradoxes to
suppose that the quantifiers in question are context-dependent. Finally, the claim that different
sentences of the same non-indexical type can have different truth values is patently ad hoc.
Tarski obtains the fact that the Liar sentence at level n is true at level n+1, and not at level n,
purely by definition: the way the hierarchy is defined, the sentence just is a sentence of level
n+1, and not n. But, unless this is pure legerdemain, the question remains as to why things
should be defined in this way.
Thirdly, and crucially, the parameterisation does not avoid the paradox, merely relocates it. [...]
Suppose that this is true in some context/ tokening, then it follows that it is not true in that
context/tokening. Hence it is true in no contex/tokening. I.e., it is true in this context/tokening,
and so in some context/tokening.
[G. Priest: "Beyond the Limits of Thought", Clarendon Press, Oxford (2006) pp. 143, 153ff]
§ 190 The Binary Tree can be constructed by ¡0 finite paths.
0
/
\
1
2
/ \ / \
3 4 5 6
/ \ ...
7 ...
But wait! At each level the number of nodes doubles. We start with the (empty) finite path at
level 0 and get 2n+1 - 1 finite paths within the first n levels. The number of all levels of the Binary
Tree is called ¡0 although there is no level number ¡0. But mathematics uses only the number of
terms of the geometric sequence. That results in 2¡0+1 - 1 = 2¡0 finite paths.
The bijection of paths that end at the same node proves 2¡0 = ¡0.
This is the same procedure with the terminating binary representations of the rational numbers
of the unit interval. Each terminating binary representation q = 0,abc...z is an element out of
2¡0+1 - 1 = 2¡0.
Or remember the proof of divergence of the harmonic series by Nicole d'Oresme. He
constructed ¡0 sums (1/2) + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... requiring 2¡0+1 - 1 = 2¡0 natural
numbers. If there were less than 2¡0 natural numbers (or if 2¡0 was larger than ¡0) the harmonic
series could not diverge and mathematics would deliver wrong results.
Beware of the set-theoretic interpretation which tries to contradict these simple facts by
erroneously asserting ¡0 ∫ 2¡0.
§ 191 The complete infinite Binary Tree can be constructed by first constructing all ¡0 finite
paths and then appending to each path all ¡0 finitely definable tails from 000... to 111... This
Binary Tree contains ¡0ÿ¡0 = ¡0 infinite paths.
If there were further discernible paths, someone should be able to discern one of them. But
since all possible combinations of nodes (including all possible diagonals and anti-diagonals of
possible Cantor-lists) that can occur in the mathematical discourse already are present, a human
being cannot discern anything additional.
Matheologians may claim that God can discern more. But God is not present in mathematics.
Mathematicians have no pipeline to God, as Brouwer put it. At least God does never reveal
mathematical secrets. Or has any reader ever heard God tell a mathematical secret?
§ 192 We first consider the total amount of energy that one can harvest centrally. [...] one finds
Emax = 3.5ÿ1067 J, comparable to the total rest-mass energy of baryonic matter within today’s
horizon. This total accessible energy puts a limit on the maximum amount of information that can
be registered and processed at the origin in the entire future history of the Universe. [...] Dividing
the total energy by this value yields a limit on the number of bits that can be processed at the
origin for the future of the Universe: Information Processed [...] = 1.35ÿ10120. [..] It is remarkable
that the effective future computational capacity for any computer in our Universe is finite,
although, given the existence of a global event horizon, it is not surprising. Note that if the
equation of state parameter w for dark energy is less than -1, implying that the rate of
acceleration of the Universe increases with time, then similar although much more stringent
bounds on the future computational capacity of the universe can be derived. In this latter case,
distributed computing is more efficient than local computing (by a factor as large as 1010 for w =
-1.2, for example), because the Hawking-Bekenstein temperature increases with time, and thus
one gains by performing computations earlier in time. [...] On a more concrete level, perhaps,
our limit gives a physical constraint on the length of time over which Moore’s Law can continue
to operate. In 1965 Gordon Moore speculated that the number of transistors on a chip, and with
that the computing power of computers, would double every year. Subsequently this estimate
was revised to between 18 months and 2 years, and for the past 40 years this prediction has
held true, with computer processing speeds actually exceeding the 18 month prediction. Our
estimate for the total information processing capability of any system in our Universe implies an
ultimate limit on the processing capability of any system in the future, independent of its physical
manifestation and implies that Moore’s Law cannot continue unabated for more than 600 years
for any technological civilization. {{Not a breathtakingly large number.}}
[Lawrence M. Krauss, Glenn D. Starkman: "Universal Limits on Computation" (2004)]
http://arxiv.org/PS_cache/astro-ph/pdf/0404/0404510v2.pdf
Therefore it is not only theoretically wrong that a process can always be completed when every
single step can, but it is already practically impossible to perform a step the identification of
which requires more than 10130 bits. At least genuine mathematicans would hesitate to accept
steps that in principle are impossible - that is reserved for matheologians and lunatics.
§ 193 {{In 1927 David Hilbert gave a talk at Hamburg university, where he explained his opinions
about the foundations of mathematics.}} It is a great honour and at the same time a necessity for
me to round out and develop my thoughts on the foundations of mathematics, which was
expounded here one day five years ago {{compare Kalenderblatt 101212 to 101214
http://www.hs-augsburg.de/~mueckenh/KB/
}} and which since then have constantly kept me most actively occupied. With this new way of
providing a foundation for mathematics, which we may appropriately call a proof theory, I pursue
a significant goal, for I should like to eliminate once and for all the questions regarding the
foundations of mathematics [...]
I have already set forth the basic features of this proof theory of mine on different occasions, in
Copenhagen [1922], here in Hamburg [1922], in Leipzig [1922], and in Münster [1925]; in the
meantime much fault has been found with it, and objections of all kinds have been raised against
it, all of which I consider just as unfair as it can be. [...]
Poincaré already made various statements that conflict with my views; above all, he denied
from the outset the possibility of a consistency proof for the arithmetic axioms, maintaining that
the consistency of the method of mathematical induction could never be proved except through
the inductive method itself. [...] Regrettably Poincaré, the mathematician who in his generation
was the richest in ideas and the most fertile, had a decided prejudice against Cantor's theory,
which prevented him from forming a just opinion of Cantor's magnificent conceptions. Under
these circumstances Poincaré had to reject my theory, which, incidentally, existed at that time
only in its completely inadequate early stages. Because of his authority, Poincaré often exerted
a one-sided influence on the younger generation. {{Not to a sufficient degree, unfortunately. --Then Hilbert discusses the objections by Russell and Whitehead and finally Brouwer. Hilbert
concludes:}} I cannot for the most part agree with their tendency; I feel, rather, that they are to a
large extent behind the times, as if they came from a period when Cantor's majestic world of
ideas had not yet been discovered. {{A world discovered by a man who was behind his times,
who did not recognize atoms in the late 19th century, but rejected evolution, who believed in an
infinite set of angels and took the basis of his mathematics from the holy bible: "in infinity and
beyond".}}
[E. Artin et al. (eds.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar
Univ. Hamburg, vol. 6, Teubner, Leipzig (1928) 65-85. English translation: J. van Heijenoort:
"From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479]
§ 194 For many years I have in the hours of leisure granted me, given much study of the Life
and Works of Francis Bacon, who in my eyes is one of the greatest geniuses of Christianity. By
this I have become persuaded, that the opinion so ridiculed by most scholars, of Francis Bacon
being the writer of the Shakespearian Dramas, is founded on truth [...] The proofs, I believe I
have found, are purely historical, and I propose gradually to publish all the material in question I
have at command. [...] Therein Francis Bacon is designated not only as the Creator of the
Elisabethean Period, but indeed is addressed as Shakespeare, for <Quirinus> (found in the
seventeenth distich) denotes clearly in English <Spear-Swinger> or <-Shaker>. [Cantor's
Preface of the Resurrecti divi Quirini Francisci Baconi edited by Cantor, 1896, acccording to
Purkert, Ilgauds: "Georg Cantor 1845 - 1918", Birkhäuser, Basel (1987) p. 85]
§ 195 The weakest of the "platonistic" assumptions introduced by arithmetic is that of the
totality of integers. The tertium non datum for integers follows from it; viz.: if P is a predicate of
integers, either P is true of each number, or there is at least one exception.
By the assumption mentioned, this disjunction is an immediate consequence of the logical
principle of the excluded middle; in analysis it is almost continually applied.
For example, it is by means of it that one concludes that for two real numbers a and b, given
by convergent series, either a = b or a < b or b < a; and likewise: a sequence of positive rational
numbers either comes as close as you please to zero or there is a positive rational number less
than all the members of the sequence.
At first sight, such disjunctions seem trivial, and we must be attentive in order to notice that an
assumption slips in. But analysis is not content with this modest variety of platonism; it reflects it
to a stronger degree with respect to the following notions: set of numbers, sequence of numbers,
and function. It abstracts from the possibility of giving definitions of sets, sequences, and
functions. These notions are used in a "quasi combinatorial" sense, by which I mean: in the
sense of an analogy of the infinite to the finite.
Consider, for example, the different functions which assign to each member of the finite series
1, 2, ..., n a number of the same series. There are nn functions of this sort, and each of them is
obtained by n independent determinations. Passing to the infinite case, we imagine functions
engendered by an infinity of independent determinations which assign to each integer an
integer, and we reason about the totality of these functions.
In the same way, one views a set of integers as the result of infinitely many independent acts
deciding for each number whether it should be included or excluded. We add to this the idea of
the totality of these sets. Sequences of real numbers and sets of real numbers are envisaged in
an analogous manner. From this point of view, constructive definitions of specic functions,
sequences, and sets are only ways to pick out an object which exists independently of, and prior
to, the construction.
The axiom of choice is an immediate application of the quasi-combinatorial concepts in
question. {{And all that gets lost if there is no God or if he was too dull to create all real numbers
(because he knew he could not remember all - neither in his brain nor in a list). The axiom of
choice is natural and obviously correct. But since we can choose only what we can name, the
axiom of choice supplies one of the strongest contradictions of uncountable sets.}}
[Paul Bernays: "On Platonism in Mathematics", (1934)]
http://www.phil.cmu.edu/projects/bernays/Pdf/platonism.pdf
§ 196 Platonism about mathematics (or mathematical platonism) is the metaphysical view that
there are abstract mathematical objects whose existence is independent of us and our language,
thought, and practices. Just as electrons and planets exist independently of us, so do numbers
and sets. {{No there is a difference. All electrons and planets exist, but ideas do not exist unless
someone has them. If all sets would exist as complete sets, then obviously they would exist in
the platonic shelter. But then this shelter would contain all sets - and its cardinality would be
greater than its cardinality. If, however, the shelter would not exist as a complete set, but only as
a class or so, why then should any set be complete?}} And just as statements about electrons
and planets are made true or false by the objects with which they are concerned and these
objects' perfectly objective properties, so are statements about numbers and sets. Mathematical
truths are therefore discovered, not invented. {{This is a proof by naive belief.}} The most
important argument for the existence of abstract mathematical objects derives from Gottlob
Frege and goes as follows {{Gottlob Frege: "Foundations of Arithmetic", Blackwell, Oxford,
Translation by J.L. Austin (1953)}}. The language of mathematics purports to refer to and
quantify over abstract mathematical objects. And a great number of mathematical theorems are
true. But a sentence cannot be true unless its sub-expressions succeed in doing what they
purport to do. So there exist abstract mathematical objects that these expressions refer to and
quantify over. {{This argument is similar to Kant's ontological proof of God (1763). Contrary to
Frege Kant noticed his slip during his lifetime (in 1781).}}
[Øystein Linnebo: "Platonism in the Philosophy of Mathematics", Stanford Encyclopedia of
Philosophy (2009)]
http://plato.stanford.edu/entries/platonism-mathematics/
§ 197 "The global unity of mathematics with religion is central in Plato's work, and in his
followers' such as Plotinus and Proclus, but also much later in modern times." [Mathematics and
the Divine. A Historical Study edited by Teun Koetsier and Luc Bergmans, Amsterdam, Elsevier,
2005, Hardbound, 716 pp., US $250, ISBN-$3: 978-0-444-50328-2, ISBN-IO: 0-444-50328-5
Rewieved by Jean-Michel Kantor in The Mathematical Intelligencer 30, 4 (2008) 70-71]
Compare Goedel's proof of God and Cantor's arguing in favour of uncountable numbers and
Hilbert's laudatio of Cantor's work. My often cursed noun matheology does not seem to be really
far fetched.
§ 198 How can we distinguish between that infinite Binary Tree that contains only all finite initial
segments of the infinite paths and that complete infinite Binary Tree that in addition also contains
all infinite paths?
Let Lk denote the kth level of the Binary Tree. The set of all nodes of the Binary Tree is given
by the union
∞
∪ k =0 (L1, L2,..., Lk ) of all finite initial segments (L1, L2, ..., Lk) of the sequence of
levels. It contains (as subsets) all finite initial segments of all infinite paths. Does it contain (as
subsets) the infinite paths too?
How could both Binary Trees be distinguished by levels or by nodes?
Most mathematicians have no answer and know this. They agree that an impossible task is
asked for. But some of them offer really exciting ideas.
One of them proposed to distinguish between the trees 2<ω and 2§ω. "Not all nodes of the tree
2§ω are finite. Nodes at level ω are not elements of the binary tree 2<ω, but they are elements of
the binary tree 2§ω. And yes", he added, "I can state with confidence that nearly all of the
experts here support my ideas on this matter."
Another one assisted him, addressing me: "You’ve demonstrated copiously over the years in
numerous venues that the indistinguishability of 2<ω and 2§ω is an article of faith for you, and
that you are either unwilling or unable to learn better. One tree is 2<ω; the other is 2§ω, which
has 2ω as its top level, sitting above the levels of 2<ω."
In case you have not yet figured out what is under discussion, here is a simple explanation: Try
to distinguish the set of all terminating decimal fractions and the set of all real numbers of the
unit interval by digits.
When you have understood, here is another task: Try to explain why Cantor's diagonal
argument is said to apply to actually infinite decimal representations only. Try to understand,
why I claim that everything in Cantor's list happens exclusively within finite initial segments, such
that, in effect, Cantor proves the uncountability of the countable set of terminating decimals.
The Binary Tree can be constructed by a sequence such that in every step one node and with it
one finite path is added. If nevertheless all infinite paths exist in the Binary Tree after all nodes
have been constructed, then it is obvious that infinite paths can creep in without being noticed. If
that is proven possible in the tree, then we can also assume that after every line of a Cantor-list
has been constructed and checked to not contain the anti-diagonal, nevertheless all real
numbers and all possible anti-diagonals can creep into the list in the same way as the infinite
paths have crept into the Binary Tree.
[Mathematics StackExchange and MathOverflow, Jan. 23, 2013 (meanwhile deleted)]
§ 199 Gödel makes a rather strong comparison between "the question of the objective existence
of the objects of mathematical intuition" and the "question of the objective existence of the outer
world" which he considers to be "an exact replica."
Gödel's rejection of Russell's "logical fictions" may be seen as a refusal to regard mathematical
objects as "insignificant chimeras of the brain."
Gödel's realism, although similar to that of Locke and zz, places emphasis on the fact that the
"axioms force themselves upon us as being true." This answers a question, untouched by Locke
and Leibniz, why we choose one system, or set of axioms, and not another; that the choice of a
mathematical system is not arbitrary.
Gödel, in the "Supplement to the Second Edition" of "What is Cantor's Continuum Problem?"
remarked that a physical interpretation could not decide open questions of set theory, i.e. there
was (at the time of his writing {{and that did never change}}) no "physical set theory" although
there is a physical geometry.
[Harold Ravitch: "On Gödel's Philosophy of Mathematics"]
http://www.friesian.com/goedel/
http://www.friesian.com/goedel/chap-2.htm
§ 200 We know that the real numbers of set theory are very different from the real numbers of
analysis, at least most of them, because we cannot use them. But it seems, that also the natural
numbers of analysis 1, 2, 3, ... are different from the cardinal numbers 1, 2, 3, ...
This is a result of the story of Tristram Shandy, mentioned briefly in § 077 already, who,
according to Fraenkel and Levy ["Abstract Set Theory" (1976), p. 30] "writes his autobiography
so pedantically that the description of each day takes him a year. If he is mortal he can never
terminate; but if he lived forever then no part of his biography would remain unwritten, for to
each day of his life a year devoted to that day's description would correspond."
This result is counter-intuitive, but set theory needs the feature of completeness for the
enumeration of all rational numbers. If not all could be enumerated, the equality of cardinality of
– and Ù could not be proved.
However recently a formal contradiction with the corresponding limit of real analysis could be
shown here:
http://planetmath.org/?op=getobj&from=objects&id=12607
and here:
http://www.hsaugsburg.de/medium/download/oeffentlichkeitsarbeit/publikationen/forschungsbericht_2012.pdf
on p. 242 - 244
The limit of remaining undescribed days is infinite according to analysis whereas Fraenkel's
story is approved by set theory.
Nevertheless, matheologians violently deny every contradiction. One of them, Michael
Greinecker (as a self-proclaimed watchdog and bouncer in MathOverflow
http://meta.mathoverflow.net/discussion/1296/crank-post-to-flag-as-spam/#Item_0
an interbreeding of Tomás de Torquemada and Lawrenti Beria) stated: "there is no
contradiction. Just a somewhat surprising result. And there is no a apriory reason why one
should be able to plug in cardinal numbers in arithmetic formulas for real numbers and get a
sensible result."
This means the finite positive integers differ significantly from the finite positive cardinals or the
finite positive integers, as Cantor called them. Well, maybe, sometimes evolution yields strange
results. But if they differ, how can set theory any longer be considered to be the basis of
analysis?
§ 201 Two Commandments of Matheology with Explanations
First Commandment: Ù contains not more than all (finite initial segments {0, 1, 2, ..., n} of the
sequence of) natural numbers.
n
Explanation: TRUNC(real) can always be written as
∑a
k =0
k
⋅10k .
Second Commandment: Ù contains more than all (finite initial segments {0, 1, 2, ..., n} of the
sequence of) natural numbers.
n
Explanation: FRAC(real) can not always be written as
∑a
k =1
−k
⋅10− k .
§ 202 The bulk of Frege's critique of Hilbert consists of criticizing Hilbert's lack of terminological
clarity, {{That kind of accusation occurs often. Something is accused to be not formal enough or
unclear or completely meaningless. This objection is always advisable if the text is not
comprehensible by the reader or inacceptable to him, however without a counter argument being
available presently or in general. Hint: I do not want to judge in the Frege-Hilbert controversy. I
only aim at the dwarfs who believe to stand on the shoulders of these giants.}} particularly as
this applies to the differences between sentences and various collections of thoughts. He takes
Hilbert to task for misleadingly using the same sentences to express different thoughts, and
points out repeatedly that Hilbert's use of axioms as definitions needs considerably more-careful
treatment than Hilbert affords it. The more-substantial criticism flows naturally from this
terminological
critique: Frege takes it that once one disentangles Hilbert's terminology, it becomes clear that he
is simply not talking about the axioms of geometry at all, since the sets of thoughts he actually
deals with are the misleadingly-expressed thoughts about e.g. real
numbers. And, adds Frege, one cannot infer the consistency of the geometric axioms proper
from that of the thoughts Hilbert treats. [Patricia Blanchette (2009)]
http://plato.stanford.edu/entries/frege-hilbert/
§ 203 Differences of "all" and "every" in impredicative statements about infinite sets.
Consider the following statements:
A) For every natural number n, P(n) is true.
B) There does not exist a natural number n such that P(n) is false.
C) For all natural numbers P is true.
A implies B but A does not imply C.
Examples for A:
1) For every n œ Ù, there is m œ Ù with n < m.
2) For every n œ Ù, the set (1, 2, ..., n) is finite.
3) For every n œ Ù, the construction of the first n nodes of a tree adds n paths to the tree.
4) For every n œ Ù, the anti-diagonal of a Cantor-list is not in the lines L1 to Ln.
§ 204 Of today's literature on the foundations of mathematics, the doctrine that Brouwer
advanced and called intuitionism forms the greater part. Not because of any inclination for
polemics, but in order to express my views clearly and to prevent misleading, conceptions of my
own theory, I must look more closely into certain of Brouwer's assertions.
Brouwer declares (just as Kronecker did in his day) that existence statements, one and all, are
meaningless in themselves unless they also contain the construction of the object asserted to
exist; for him they are worthless scrip, and their use causes mathematics to degenerate into a
game. [...]
What, now, is the real state of affairs with respect to the reproach that mathematics would
degenerate into a game? [...] The formula game that Brouwer so depreciates has, besides its
mathematical value {{matheology à la Banach-Tarski-paradox contains no value at all - the
mathematical value of the proofs that Hilbert believs in can at most be measured in small
fractions of lira which not even a single cent will be paid for}}, an important general philosophical
significance. For this formula game is carried out according to certain definite rules, in which the
technique of our thinking is expressed. These rules form a closed system that can be discovered
and definitively stated {{like other religious systems too: Buddhism, Christianity, Hinduism, Islam,
Judaism, ... In Islam they enjoy apostasy-punishment by death. Heretics are usually killed. In
matheology heretics are called cranks and attempts are made to remove them from their
academic poitions. So matheology is not quite as intolerant as Islam but less tolerant than
Buddhism and Hinduism.
There is another parallel: Moslems are not allowed to contact God in any other language than
the Arabic. Allah seems to be less educated than Jahwe or God who accept prayers in every
language. In matheology every important prayer must be uttered in a certain formal language.
The God of matheology and his disciples seem to be very limited too.}}
[E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar
Univ. Hamburg, Bd. 6, Teubner, Leipzig (1928) 65-85. English translation in J. van Heijenoort:
"From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479.]
http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm
§ 205 The fundamental idea of my proof theory is none other than to describe the activity of our
understanding, to make a protocol of the rules according to which our thinking actually proceeds.
Thinking, it so happens, parallels speaking and writing: we form statements and place them one
behind another. If any totality of observations and phenomena deserves to be made the object
of a serious and thorough investigation, it is this one - since, after all, it is part of the task of
science to liberate us from arbitrariness, sentiment, and habit and to protect us from the
subjectivism that already made itself felt in Kronecker's views and, it seems to me, finds its
culmination in intuitionism. {{The erroneous opinion that a sphere never doubles itself really
should be opposed with severeness! Folks, distribute marbles or blow up balloons and condoms
in order to prove that!}}
Intuitionism's sharpest and most passionate challenge is the one it flings at the validity of the
principle of excluded middle [...] Existence proofs carried out with the help of the principle of
excluded middle usually are especially attractive because of their surprising brevity and
elegance. Taking the principle of excluded middle from the mathematician would be the same,
proscribing the telescope to the astronomer or to the boxer the use of his fists. [...]
Not even the sketch of my proof of Cantor's continuum hypothesis has remained uncriticised. I
would therefore like to make some comments on this proof.
[E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar
Univ. Hamburg, Bd. 6, Teubner, Leipzig (1928) 65-85. English translation in J. van Heijenoort:
"From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479.]
{{The interests of marxists are rather philosophical and social and less mathematical. But with
respects to Hilbert's "proof of the continuum hypothesis" they have hit the nail right on the
head:}} "The whole of Hilbert selection for series reproduced here, minus some inessential
mathematical formalism."
http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm
(For an evaluation of Hilbert's logic by Zermelo compare § 117.)
§ 206 From my presentation you will recognise that it is the consistency proof that determines
the effective scope of my proof theory and in general constitutes its core. The method of W.
Ackermann permits a further extension still. For the foundations of ordinary analysis his
approach has been developed so far that only the task of carrying out a purely mathematical
proof of finiteness remains. Already at this time I should like to assert what the final outcome will
be: mathematics is a presuppositionless science. To found it I do not need God {{I do not need
Hilbert, said Gott - and created Gödel.}}, as does Kronecker {{There is some correction required:
Kronecker's sentence "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist
Menschenwerk" probably has been meant ironically like the sentence "Gott Würfelt nicht" of the
pronounced atheist Albert Einstein. Hilbert, on the other hand needs God, among others for his
paradise.}}, or the assumption of a special faculty of our understanding attuned to the principle of
mathematical induction, as does Poincaré {{Here we need nothing but a little bit of common
sense: If a theorem is valid for the number k, and if from its validity for the number n + k the
validity for n + k + 1 can be concluded with no doubt, then n can be replaced by n + 1, and the
validity for n + k + 2 is proven too. This is the foundation of mathematics. To prove anything
about this principle is as useless as the proof that 1 + 1 = 2.}}, or the primal intuition of Brouwer,
or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which
in fact are actual, contentual assumptions that cannot be compensated for by consistency
proofs. {{That is correct. The axiom of infinity is simply an inconsistent assumption like the
assumption of a set of natural numbers with cardinality 10 and sum 10. Therefore it cannot be
proved from mathematics. But everything is proved after assuming it as an axiom.}}
I would like to note further that P. Bernays has again been my faithful collaborator. He has not
only constantly aided me by giving advice but also contributed ideas of his own and new points
of view, so that I would like to call this our common work. {{His newest point of view is this: "If we
pursue the thought that each real number is defined by an arithmetical law, the idea of the
totality of real numbers is no longer indispensable." (Paul Bernays, 1934, cp. § 144)}}
[E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar [E.
Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar Univ.
Hamburg, Bd. 6, Teubner, Leipzig (1928) 65-85. English translation in J. van Heijenoort: "From
Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479.]
http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm
§ 207 Towards the end of his Address on the Unity of Knowledge, delivered at the 1954
Columbia University bicentennial celebrations, Weyl enumerates what he considers to be the
essential constituents of knowledge. At the top of his list comes "…intuition, mind's ordinary act
of seeing what is given to it." (Weyl 1954, 629)
In particular Weyl held to the view that intuition, or insight - rather than proof - furnishes the
ultimate foundation of mathematical knowledge. {{What else should furnish it? A formal proof
can be given for every stupidity, and be it infinite.}} Thus in his Das Kontinuum of 1918 he says:
"In the Preface to Dedekind (1888) we read that 'In science, whatever is provable must not be
believed without proof.' This remark is certainly characteristic of the way most mathematicians
think. Nevertheless, it is a preposterous principle. As if such an indirect concatenation of
grounds, call it a proof though we may, can awaken any 'belief' apart from assuring ourselves
through immediate insight that each individual step is correct. In all cases, this process of
confirmation - and not the proof - remains the ultimate source from which knowledge derives its
authority; it is the 'experience of truth'” (Weyl 1987, 119) {{like Zermelos "proof" of the wellordering assertion is the experience of untruth}}.
[John L. Bell: "Hermann Weyl", Stanford Encyclopedia of Philosophy (2009)]
http://plato.stanford.edu/entries/weyl/index.html
§ 208 In Consistency in Mathematics (1929), Weyl characterized the mathematical method as
"the a priori construction of the possible in opposition to the a posteriori description of what is
actually given". {{Above all, mathematics has to be consistent. And there is only one criterion for
consistency: The "model" of reality.}}
The problem of identifying the limits on constructing “the possible” in this sense occupied Weyl
a great deal. He was particularly concerned with the concept of the mathematical infinite, which
he believed to elude “construction” in the naive set-theoretical sense.
Again to quote a passage from Das Kontinuum: "No one can describe an infinite set other than
by indicating properties characteristic of the elements of the set…. The notion that a set is a
'gathering' brought together by infinitely many individual arbitrary acts of selection, assembled
and then surveyed as a whole by consciousness, is nonsensical; 'inexhaustibility' is essential to
the infinite."
[...] Small wonder, then, that Hilbert was upset when Weyl joined the Brouwerian camp.
[John L. Bell: "Hermann Weyl", Stanford Encyclopedia of Philosophy (2009)]
http://plato.stanford.edu/entries/weyl/index.html
§ 209 In Das Kontinuum Weyl says: "The states of affairs with which mathematics deals are,
apart from the very simplest ones, so complicated that it is practically impossible to bring them
into full givenness in consciousness and in this way to grasp them completely."
Nevertheless, Weyl felt that this fact, inescapable as it might be, could not justify extending the
bounds of mathematics to embrace notions, such as the actual infinite, which cannot be given
fully in intuition even in principle. He held, rather, that such extensions of mathematics into the
transcendent are warranted only by the fact that mathematics plays an indispensable role in the
physical sciences, in which intuitive evidence is necessarily transcended. As he says in The
Open World: "… if mathematics is taken by itself, one should restrict oneself with Brouwer to the
intuitively cognizable truths … nothing compels us to go farther. But in the natural sciences we
are in contact with a sphere which is impervious to intuitive evidence; here cognition necessarily
becomes symbolical construction. Hence we need no longer demand that when mathematics is
taken into the process of theoretical construction in physics it should be possible to set apart the
mathematical element as a special domain in which all judgments are intuitively certain; from this
higher standpoint which makes the whole of science appear as one unit, I consider Hilbert to be
right" {{me too.}}
Weyl soon grasped the significance of Hilbert's program, and came to acknowledge its
"immense significance and scope". Whether that program could be successfully carried out was,
of course, still an open question. But independently of this issue Weyl was concerned about
what he saw as the loss of content resulting from Hilbert's thoroughgoing formalization of
mathematics. "Without doubt", Weyl warns, "if mathematics is to remain a serious cultural
concern {{a mathematician should never give up this premise}}, then some sense must be
attached to Hilbert's game of formulae."
[John L. Bell: "Hermann Weyl", Stanford Encyclopedia of Philosophy (2009)]
http://plato.stanford.edu/entries/weyl/index.html
§ 210 An accessible number, to Borel, is a number which can be described as a mathematical
object. The problem is that we can only use some finite process to describe a real number so
only such numbers are accessible. We can describe rationals easily enough, for example either
as, say, one-seventh or by specifying the repeating decimal expansion 142857. Hence rationals
are accessible. We can specify Liouville's transcendental number easily enough as having a 1 in
place n! and 0 elsewhere. Provided we have some finite way of specifying the n-th term in a
Cauchy sequence of rationals we have a finite description of the resulting real number. However,
as Borel pointed out, there are a countable number of such descriptions. Hence, as Chaitin
writes: "Pick a real at random, and the probability is zero that it's accessible - the probability is
zero that it will ever be accessible to us as an individual mathematical object."
[J.J. O'Connor and E.F. Robertson: "The real numbers: Attempts to understand"]
http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_3.html
But how to pick this dark matter of numbers? Only accessible numbers can get picked.
Unpickable numbers cannot appear anywhere, neither in mathematics nor in Cantor's lists.
Therefore Cantor "proves" that the pickable numbers, for instance numbers that can appear as
an antidiagonal of a defined list, i.e., the countable numbers, are uncountable.
§ 211 The belief in the universal validity of the principle of the excluded third in mathematics is
considered by the intuitionists as a phenomenon of the history of civilization of the same kind as
the former belief in the rationality of π, or in the rotation of the firmament about the earth {{or the
assumption that every particle has definite position and velocity at every time.}} The intuitionist
tries to explain the long duration of the reign of this dogma by two facts: firstly that within an
arbitrarily given domain of mathematical entities the non-contradictority of the principle for a
single assertion is easily recognized; secondly that in studying an extensive group of simple
every-day phenomena of the exterior world, careful application of the whole of classical logic
was never found to lead to error. [This means de facto that common objects and mechanisms
subjected to familiar manipulations behave as if the system of states they can assume formed
part of a finite discrete set, whose elements are connected by a finite number of relations.]
{{Unfortunately this principle, without any justification, has been applied to infinite sets.}}
[L.E.J. Brouwer: "Lectures on Intuitionism - Historical Introduction and Fundamental Notions"
(1951), Cambridge University Press (1981)]
http://www.marxists.org/reference/subject/philosophy/works/ne/brouwer.htm
§ 212 A synopsis of Brouwer's position yields two statements:
- Classical mathematics is contradictory.
- Infinite remains potential. It just means that you can go on and on.
Classical mathematics, i.e., Cantor's set theory cannot be reconciled with constructivism. There
are two big lies of matheology. The first is to call transfinite set theory "classical mathematics".
The second is to imply that this classical mathematics is as valid as constructivism. This is not a
matter of taste but provably wrong as becomes clear from the following statements:
According to Hilbert, not to believe in tertium non datur is the most glaring disbelief that we find
in the human history. [D. Hilbert: "Die Grundlegung der elementaren Zahlenlehre", Vortrag in
Hamburg (1930), Mathematische Annalen 104 (1933) 485-494]
Brouwer calls the belief in tertium non datur a disappearing superstition. [Brouwer-lecture in
Berlin, reported by A. Weil in Dirk van Dalen: "Mystic, Geometer, and Intuitionist: The Life of
L.E.J. Brouwer, Vol. 2", Clarendon Press, Oxford (2005) 643]
§ 213 Zermelo’s proof had not indicated how to determine the covering γ uniquely, and yet one
needed to be certain that γ remained the same throughout the proof. How could one be sure?
Moreover, even if such a covering γ existed and could be defined, it was doubtful that one could
use γ in the way that Zermelo had; for the subsets M’ of M were not defined in a unique way.
Indeed, Lebesgue doubted that one would ever be able to state a general method for wellordering a given set. {{That was very wise. But it shows one fact above all: The possibility of a
well-ordering of the reals had been expected within reach at that time. Today its impossibility for
the reals is well known. It is not admitted that this is contradicting Zermelo's proof because there
cannot be a contradiction in ZFC which stands for Zero Falsifying Contradictions.}}
[Gregory H. Moore: "The Origins of Zermelos Axiomatization of Set Theory", Journal of
Philosophical Logig 7 (1978) 307-329]
http://www.jstor.org/discover/10.2307/30226178?uid=3737864&uid=2&uid=4&sid=21101627338
593
§ 214 What’s wrong with the axiom of choice?
Part of our aversion to using the axiom of choice stems from our view that it is probably not
‘true’. {{In fact it is true for existing sets - but there it is not required as an axiom but is a selfevident truth.}} A theorem of Cohen shows that the axiom of choice is independent of the other
axioms of ZF, which means that neither it nor its negation can be proved from the other axioms,
providing that these axioms are consistent. Thus as far as the rest of the standard axioms are
concerned, there is no way to decide whether the axiom of choice is true or false. This leads us
to think that we had better reject the axiom of choice on account of Murphy’s Law that "if
anything can go wrong, it will". This is really no more than a personal hunch about the world of
sets. We simply don’t believe that there is a function that assigns to each non-empty set of real
numbers one of its elements. While you can describe a selection function that will work for finite
sets, closed sets, open sets, analytic sets, and so on, Cohen’s result implies that there is no
hope of describing a definite choice function that will work for "all" non-empty sets of real
numbers, at least as long as you remain within the world of standard Zermelo-Fraenkel set
theory. And if you can’t describe such a function, or even prove that it exists without using some
relative of the axiom of choice, what makes you so sure there is such a thing?
Not that we believe there really are any such things as infinite sets, or that the ZermeloFraenkel axioms for set theory are necessarily even consistent. Indeed, we’re somewhat
doubtful whether large natural numbers (like 805000, or even 2200) exist in any very real sense,
and we’re secretly hoping that Nelson will succeed in his program for proving that the usual
axioms of arithmetic -and hence also of set theory - are inconsistent. (See E. Nelson. Predicative
Arithmetic. Princeton University Press, Princeton, 1986.) All the more reason, then, for us to
stick with methods which, because of their concrete, combinatorial nature, are likely to survive
the possible collapse of set theory as we know it today.
[Peter G. Doyle, John Horton Conway: "Division by Three" 1994, ARXIV math/0605779v1]
http://arxiv.org/abs/math/0605779v1
§ 215 The set of all sets and the devil have one property in common: Both terrify only their
believers.
§ 216 One remark that Penelope Maddy makes several times in Naturalism in Mathematics, is
that if the indispensability argument was really important in justifying mathematics, then set
theorists should be looking to debates over quantum gravity to settle questions of new axioms.
Since this doesn’t seem to be happening, she infers that the indispensability argument can’t play
the role Quine and Putnam (and perhaps her earlier book?) argued that it does. [...]
I don’t know much about the details, but from what I understand, physicists have conjectured
some deep and interesting connections between seemingly disparate areas of mathematics, in
order to explain (or predict?) particular physical phenomena. These connections have rarely
been rigorously proved, but they have stimulated mathematical research both in pursuing the
analogies and attempting to prove them. Although the mathematicians often find the physicists’
work frustratingly imprecise and non-rigorous, once the analogies and connections have been
suggested by physicists, mathematicians get very interested as well.
If hypothetically, one of these connections was to turn out to be independent of ZFC, I could
imagine that there would at least be a certain camp among mathematicians that would take this
as evidence for whatever large cardinal (or other) principle was needed to prove the connection.
[Kenny Easwaran: "Set Theory and String Theory" (2006)]
http://antimeta.wordpress.com/2006/10/29/set-theory-and-string-theory/
Of course, every physical result is independent of ZFC.
§ 217 Whatever may play a role in mathematics, symbols, numbers, operators, definitions,
theorems: All together belongs to a countable set. The remaining is matheology. Alas, contrary
to theology matheology is not suitable to awaken the hope of happiness and luck after death in
souls - not even in the souls of matheologians.
§ 218 What is Mathematics? Most mathematicians don't know and don't care. Mathematics is
what mathematicians do. [...] In fifty years (at most) human mathematicians will be like lamplighters and ice-delivery men. All serious math will be done by computers. Let's hope that human
philosophy will still survive, but we need to adjust naturalism to the practice of math in the future
and to the way it will be done by machines. Of course, we don't know exactly how, so let's put
this project of Naturalist mathematical philosophy on hold and wait to see how things turn out in
fifty years.
Tim Gowers said that we are all formalists, but most of us don't know it (and if we knew, we
wouldn't care). I kind of agree, but this is only a corollary of a more profound truth: Everything is
Combinatorics. Classify Lie Algebras? It is just root systems and Dynkin Diagrams. Finite
Groups? The Monster is a Combinatorial Design. Even when it is not obviously combinatorics, it
could be made so. If it is too hard for us, then we need a computer! But computer science is all
Discrete Math, alias combinatorics. In a way Logic is too. But Logic is so low-level, like machine
language. It is much more fun and gratifying to work in Maple, and do higher-level
combinatorics.
I am also a trivialist. [...] We humans, and even our computers, can only prove trivial results.
Since all knowable math is ipso facto trivial, why bother? So only do fun problems, that you
really enjoy doing. It would be a shame to waste our short lives doing "important" math, since
whatever you can do, would be done, very soon (if not already) faster and better (and more
elegantly!) by computers. So we may just as well enjoy our humble trivial work.
[Doron Zeilberger: "Opinion 69: Roll Over Platonism, Logicisim, Formalism, Intuitionism,
Constructivism, Naturalism and Humanism! Here Comes Combinatorialism and Trivialism."
(2005)]
§ 219 Applied Math is an indispensable part of various engineering disciplines because its
application and usefulness in predictive models has been validated against real-world conditions
again and again and again.
If, for argument’s sake, PA was proven inconsistent, then math merely becomes a defacto
natural science like biology or chemistry, in the sense that the “validity” of math no longer stems
from axioms, but rather validation against real world conditions and observations.
[Paul AC Chang, Re: The Inconsistency of Arithmetic, The n-Category Café, October 2, 2011]
http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039531
§ 220 PA {{Peano-Arithmetik}} already tells us that the universe is infinite, but PA “stops” after
we have all the natural numbers. {{No, PA never stops because it never reaches an end. Here
potential and actual infinity are confused.}} ZFC goes beyond the natural numbers; in ZFC we
can distinguish different infinite cardinalities such as “countable” and “uncountable”, and we can
show that there are infinitely many cardinalities, uncountably many, etc. {{and we can show that
there is nothing of that kind other than in dreams, but not in logic.}}
[Saharon Shelah: "Logical Dreams" (2002)]
http://arxiv.org/PS_cache/math/pdf/0211/0211398v1.pdf
§ 221 As I get older I seem to be getting more and more relaxed about foundational issues. I’m
happy to see people formalize and explore all imaginable attitudes toward the foundations of
mathematics. I feel confident that the more interesting axiom systems will eventually attract more
researchers, while the less interesting ones will remain marginal. I am not eager for one system
to prevail over all others … nor do I feel any desire for systems I dislike to go completely extinct.
It’s a lot like my fondness for biodiversity. I enjoy the diversity of life, and am very happy there
are tigers, and would be sad for them to go extinct, even though I wouldn’t want a bunch running
around in my back yard.
In particular, I’m glad there are ultrafinitists, because I suspect that only someone with views
like that could be motivated to prove the inconsistency of (say) Peano arithmetic, and seek
plausible strategies for doing it.
If everyone believes Peano arithmetic is consistent, and it’s not, we’re in big trouble because
it’ll take us a long time to discover it. So we need a few lonely people working on the other side
of this issue. I don’t think they’ll succeed, but I’m glad they’re trying. Even if they don’t succeed,
there could be some interesting concepts and theorems that only they are likely to find.
Finally, I don’t expect these people to take the same ‘relaxed, balanced’ attitude that I have. I
suspect that only someone with strong opinions could possibly be motivated to spend a lot of
time developing ultrafinitism, or trying to prove the inconsistency of Peano arithmetic. Expecting
them to share my relaxed attitude is a bit like expecting a tiger to be an environmentalist.
[John Baez: "The Inconsistency of Arithmetic", n-Category-Cafe, September 30, 2011]
§ 222 Consider a Cantor-list with entries an and anti-diagonal d:
" n (for every n) œ Ù: (an1, an2, ..., ann) ∫ (d1, d2, ..., dn).
" n (for every n) œ Ù: (an1, an2, ..., ann) is terminating.
" n (for every n) œ Ù: (d1, d2, ..., dn) is terminating.
" n (for all n) œ Ù: (an1, an2, ..., ann) ∫ (d1, d2, ..., dn).
" n (for all n) œ Ù: (an1, an2, ..., ann) is terminating.
" n (for all n) œ Ù: (d1, d2, ..., dn) is not terminating.
That's the origin of matheology.
§ 223 How obvious a contradiction has to result from an additional axiom in order to reject it?
The Axiom of Choice (AC) states that every set can be well-ordered. In order to well-order an
uncountable set, an uncountable alphabet is required, since a countable alphabet is not
sufficient to label uncountably many elements (compare the Binary Tree, § 190). But an alphabet
is a linearly ordered set (otherwise you would never find most letters of the alphabet - compare
the telephone book). And linear ordering implies well-ordering.
So the Axiom of Choice contradicts the other ZF-axioms. (This has already been shown by
Hausdorff-Banach-Tarski with the result that by means of AC we can prove that, after some
turning and twisting, but without any addition or subtraction of even one single point, the
measurable set V is identical with the measurable set 2V.)
With equal right we can introduce the Axiom of Meagre Sum (AMS) stating: There is a set of n
positive natural numbers with sum nÿn/2. This axiom is not constructive, since nobody can
construct such a set. But the disproof by the well known fact that the sum of n different positive
natural numbers is never less than n(n+1)/2 is not less obvious than the disproof of AC.
§ 224 Consider a tabletop supported by three legs. The tabletop is stable if its center of mass
lies in the triangle formed by the supporting points. No special one of the three points is
necessary. But we know, that three legs exist, if the table is stable. If we have support by four (or
more) suitable legs, we can show that one (ore more) are superfluous, i.e., not necessary. Could
we prove that of three legs one is not required, then reality would be contradictory. But since
reality proves its consistency by simple existence (and not by inconsistent sets of axioms), such
a proof is impossible.
Now consider the list of finite initial segments of natural numbers
1
1, 2
1, 2, 3
...
According to set theory it contains all ¡0 natural numbers in its lines. But is does not contain a
line containing all natural numbers. Therefore it must be claimed that more than one line is
necessary to contain all natural numbers. This means at least two lines are necessary. There
are no special lines necessary, but there must be at least two. In this case, however, we can
prove, by the construction of the list, that every union of a pair of lines is contained in one of the
lines. This contradicts the assertion that all natural numbers exist and are in lines of the list.
The solution of this paradox is a potentially infinite list that has always a last line (which is the
only line necessary to contain all natural numbers that are in the list) but this last line cannot be
fixed.
§ 225 Axiom der Auswahl. - Man kann das Axiom auch so ausdrücken, daß man sagt, es sei
immer möglich, aus jedem Elemente M, N, R , ..., von T ein einzelnes Element m, n, r, ...
auszuwählen und alle diese Elemente zu einer Menge S1 zu vereinigen. [E. Zermelo:
"Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908)
261-281]
So the axiom of choice says that it is always possible to choose an element from every nonempty set and to union the chosen elements into a set S1.
"Choosing" something means pointing to or showing this something, or, if this something has
no material existence, defining or labelling it by a finite word.
For uncountable sets this is known to be as impossible as to find a second prime number triple
besides 3, 5, 7.
Would matheologians accept the axiom "there is a second prime number triple" if necessarily
required to save matheology? Yes, I am sure.
§ 226 There is an isomorphism between the field of all non-negative binary representations r of
real numbers (—, +, ÿ) and the filed of all paths p of the extended Binary Tree (BT, +, ÿ) such that
for every r in — there is a p = f(r) in BT (and vice versa) and we have for all a, b, r, s in — (a and
b can also be taken from BT): f(ar + bs) = af(r) + bf(s).
The extended Binary Tree is obtained by extending the ordinary Binary Tree to all integers:
...
0101
\/ \/
0 1
\/
.
/ \
0 1
/\ /\
0101
...
§ 227 {{Fields medalist Voevodsky}} stated the theorem as follows [...]: It is impossible to prove
the consistency of any formal reasoning system which is at least as strong as the standard
axiomatization of elementary number theory ("first order arithmetic").
So he failed to inform his audience that the impossibility that Goedel actually established was
the impossibility of proof-in-S of a sentence expressing the consistency of S, for any consistent
and sufficiently strong system S.
As we know, Gentzen's proofs of the consistency of PA are among the most important results
in proof-theory, second only to Goedel's results themselves and perhaps Prawitz' normalization
results. (For a great overview of Gentzen's proofs, see von Plato's SEP entry.) What I find most
astonishing about Gentzen's proofs, based on transfinite induction, is that the theory obtained by
adding quantifier free transfinite induction to primitive recursive arithmetic is not stronger than
PA, and yet it can prove the consistency of PA (it is not weaker either, obviously; they just prove
different things altogether). One may raise eyebrows concerning transfinite induction (and
apparently this is what lies behind Voevodsky's dismissal of Gentzen's results), but apparently
most mathematicians and logicians seem quite convinced of the cogency of the proof. {{Most
astrologers are convinced of the consistency of astrology.}} [...]
So Voevodsky seems to seriously entertain the possibility of PA's inconsistency. Is it because
he doesn't understand Goedel's results, or Gentzen's results, or both? Or is there something
else going on? [...]
Now, within the bigger picture of things, the consistency of PA is actually a tangential,
secondary issue. Voevodsky’s seemingly polemic statement concerning the potential
inconsistency of PA in fact seems to amount to the following: all the currently available proofs of
the consistency of PA in fact rely on the very claim they prove, namely the consistency of PA, on
the meta-level. [...] So if PA was inconsistent, these proofs would still go through; in other words,
there is a sense in which such proofs are circular in that they presuppose the very fact that they
seek to prove. {{And nobody has noticed hitherto?}}
[Catarina Dutilh Novaes (2011)]
http://m-phi.blogspot.com/2011/05/voevodsky-consistency-of-pa-is-open.html
http://m-phi.blogspot.com/2011/06/latest-news-on-inconsistency-of-pa.html
§ 228 From 1969 until 1973 I worked to delineate mathematical methods devoid of any
unprovable aspects.
I began by observing that some logicians disagree with most mathematicians on one important
point: The logicians insist that false assumptions must lead to both the proof and disproof of
every meaningful statement within the logical system in question. The math people believe that
they can make up sets of axioms that make sense to them (true in the world or not) and that the
resultant math will be free of logical flaws.
Here are some of the conclusions I came to during this time:
1. Calculus does not need any concept of infinity in order to provide limits, derivatives,
integrals, and differentials.
2. No infinity is possible unless axioms assert it. That is, no infinity can be derived without
being presupposed.
3. No even roots of -1 are needed except one (i = the square root).
4. All of the above facts are known to many mathematicians.
5. Real numbers that are not rational numbers can be expressed as limits of functions of
rational numbers. This means that quantities like π and e need not be regarded as numbers and
they may be formally handled just as computer programs handle them. It also means that the
concept of a process replaces the real numbers that are not rational. The complex numbers (a +
bi) become processes where a and b numbers or processes.
6. Mathematics needs to be synthetic (founded upon definitions which, in turn, are founded
upon undefined but perceived meanings in the common language). Axioms are unnecessary
and harmful. No axiomatic system works anyway if we don't agree on the meaning of such
fundamental terms as single, pair, the, associated with, and the like. Good systems would rely
on a minimum of such terms and would explicitly recognize them. Formal math comes from our
perceptions of reality, not the other way around.
7. The phenomenon of the conditional branch (if incorporated into math proper), represents a
giant advance in the power of math to solve problems.
That is the short story. We get every kind of functionality in the whole world without any logical
flaws and without esoteric and spooky contradictions (many mathematicians are in awe of them,
but they can all be easily fixed).
[Jim Trek (1999)]
http://members.chello.nl/~n.benschop/finite.htm
§ 229 The difference between potential and actual infinity can even be photographed: Infinity,
to find use in set theory, must split off. The following pictures of a movie of an everexpanding
square show this for the first time:
o
oo
oo
ooo
ooo
ooo
oooo
oooo
oooo
oooo
...
For each finite square we find height = width. For the infinite square however, height > width,
namely an infinite sequence of finite lines (scale changed):
.
.
.
.
.
.
.
There seem to be some gravity effects involved in transfinite set theory. Otherwise the (actually)
infinite set of natural numbers cannot be gathered.
[WM: "Gravity effects detected in transfinite set theory", sci.logic, sci.math, 8 Oct 2011]
http://groups.google.com/group/sci.math/msg/d48b3d3e7581802d?dmode=source
§ 230 Cantor's theory of infinite sets, developed in the late 1800's, was a decisive advance for
mathematics, but it provoked raging controversies and abounded in paradox. One of the first
books by the distinguished French mathematician Emile Borel (1871-1956) was his Lecons sur
la Théorie des Fonctions [Borel, 1950], originally published in 1898, and subtitled Principes de la
théorie des ensembles en vue des applications à la théorie des fonctions.
This was one of the first books promoting Cantor's theory of sets (ensembles), but Borel had
serious reservations about certain aspects of Cantor's theory, which Borel kept adding to later
editions of his book as new appendices. The final version of Borel's book, which was published
by Gauthier-Villars in 1950, has been kept in print by Gabay. That's the one that I have, and this
book is a treasure trove of interesting mathematical, philosophical and historical material.
One of Cantor's crucial ideas is the distinction between the denumerable or countable infinite
sets, such as the positive integers or the rational numbers, and the much larger
nondenumerable or uncountable infinite sets, such as the real numbers or the points in the plane
or in space. Borel had constructivist leanings, and as we shall see he felt comfortable with
denumerable sets, but very uncomfortable with nondenumerable ones. [...]
The idea of being able to list or enumerate all possible texts in a language is an extremely
powerful one, and it was exploited by Borel in 1927 [Tasic, 2001, Borel, 1950] in order to define
a real number that can answer every possible yes/no question!
You simply write this real in binary, and use the nth bit of its binary expansion to answer the
nth question in French.
Borel speaks about this real number ironically. He insinuates that it's illegitimate, unnatural,
artificial, and that it's an "unreal" real number, one that there is no reason to believe in.
Richard's paradox and Borel's number are discussed in [Borel, 1950] on the pages given in the
list of references, but the next paradox was considered so important by Borel that he devoted an
entire book to it. In fact, this was Borel's last book [Borel, 1952] and it was published, as I said,
when Borel was 81 years old. I think that when Borel wrote this work he must have been thinking
about his legacy, since this was to be his final book-length mathematical statement. The
Chinese, I believe, place special value on an artist's final work, considering that in some sense it
contains or captures that artist's soul. If so, [Borel, 1952] is Borel's "soul work." [...]
Here it is: Borel's "inaccessible numbers:" Most reals are unnameable, with probability one.
Borel's often-expressed credo is that a real number is really real only if it can be expressed, only
if it can be uniquely defined, using a finite number of words. It's only real if it can be named or
specifed as an individual mathematical object. [...] So, in Borel's view, most reals, with
probability one, are mathematical fantasies, because there is no way to specify them uniquely.
{{In Borel's view only reals that can be named belong to mathematics. Uncountability is not part
of mathematics.}}
Borel, E. [1950] Lecons sur la Théorie des Fonctions (Gabay, Paris) pp. 161, 275.
Borel, E. [1952] Les Nombres Inaccessibles (Gauthier-Villars, Paris) p. 21.
Tasic, V. [2001] Mathematics and the Roots of Postmodern Thought (Oxford University Press,
New York) pp. 52, 81-82.
[Gregory Chaitin: "How real are real numbers?" (2004)]
http://arxiv.org/abs/math.HO/0411418
§ 231 One philosophically important way in which numbers and sets, as they are naively
understood, differ is that numbers are physically instantiated in a way that sets are not. Five
apples are an instance of the number 5 and a pair of shoes is an instance of the number 2, but
there is nothing obvious that we can analogously point to as an instance of, say, the set {{«}}.
[Nik Weaver: "Is set theory indispensable?" (2009)]
http://arxiv.org/abs/0905.1680
§ 232 In June 1905 Nelson sent a letter to Hessenberg {{who invented the set that contains a
certain element only if it does not contain that element}} commenting on Hilbert’s lecture "Über
die Grundlagen der Arithmetik", and he expressed his disappointment about Hilbert’s ideas.
Rather perplexed he wrote: “To remove the contradictions in set theory, he [i.e., Hilbert] intends
to reform (not set theory but) logic. Well, we shall see, how he will do it.” Hessenberg answered
quite to the point: "I do not at all consider it as paradoxical that one has to reform logic in order
to make set theory free of contradictions. First of all it is not yet possible to separate logic
sharply from arithmetical considerations. Secondly, however: If there are paradoxes in set
theory, then either the inferences are not correct or the concepts generated are contradictory."
{{Trust in infallibility of set theory has been always well formed.}}
In both cases, Hessenberg continued, it is a logical task to uncover the mistakes. According to
the laws of logic a thing a falls under the concept b or not. No other principle is needed for the
concept of a set. {{That is not always the case. An infinite path cannot be localized in the Binary
Tree although each of its nodes can be localized there. Whether or not the path belongs to the
Binary Tree depents on the intentions of the path finder (scout).}} Hessenberg stressed that
Hilbert very much strengthened the requirements for building concepts in order to avoid the
resulting paradoxes.
[Volker Peckhaus: "Paradoxes in Göttingen", p. 10f]
http://kw.uni-paderborn.de/fileadmin/kw/instituteeinrichtungen/humanwissenschaften/philosophie/personal/peckhaus/Texte_zum_Download/pg.p
df
§ 233 The set of all termination decimals is a subset of –. If the set of all terminating decimals
of the unit interval is arranged as set of all terminating paths of the decimal tree, unavoidably all
irrationals are written as infinite paths too. But we know that it is impossible to write the path of
even one single irrational number, let alone of several or infinitely many or uncountably many.
So belief in the above requires strong faith.
A view without faith is this: There is no irrational path at all. But that would destroy the pet
dogma of matheology, namely uncountability.
The question is, how come uncountably many irrational paths into being during the countable
process of constructing the complete decimal tree by constructing all its countably many nodes.
Provably none of the irrationals is constructed in any step.
And an additional question for skilled matheologians: If we delete all paths containing digits 2,
3, 4, 5, 6, 7, 8, and 9 from the decimal tree of finite paths: Do all irrationals remain? Is infinity the
only necessary condition of matheological belief?
§ 234 Enumerate all rational numbers to construct a Cantor-list. Replace the diagonal digits ann
by dn in the usual way to obtain the anti-diagonal d. Beyond the n-th line there are f(n) rational
numbers the first n digits of which are the same as the first n digits d1, d2, d3, ..., dn of the antidiagonal. " n in Ù: f(n) > k for every k in Ù. Define for every n in Ù the function g(n) = 1/f(n) = 0.
In analysis the limit of this function is limnض g(n) = 0.
So matheology with its limit limnض f(n) = 0 is incompatible with analysis. Since analysis is a
branch of mathematics, matheology is incompatible with mathematics.
§ 235 Rough set theory has an overlap with many other theories dealing with imperfect
knowledge {{a similarity that it shares with transfinite set theory}}.
[Zdzisław Pawlak: "Rough set theory and its applications", J. Telecomm. Information Theory
(3/2002) 7-9]
http://www.nit.eu/czasopisma/JTIT/2002/3/7.pdf
§ 236 Tarski’s theorem: (For all infinite sets X there exists a bijection of X to XäX) fl (Axiom of
Choice). [...] Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication
between two well known propositions is not a new result. Lebesgue wrote that an implication
between two false propositions is of no interest.
[Jan Mycielski: "A System of Axioms of Set Theory for the Rationalists", Notices of the AMS 53,2
(2006) 206-213]
http://www.ams.org/notices/200602/fea-mycielski.pdf
§ 237 The Cantor's set theory is a Trojan Horse of the mathematics-XX: on the one hand, it is a
natural, visual, universal language to describe mathematical objects, their properties and
relations, originating from the famous Euler's "logical circles", and just therefore this language
was accepted by all mathematicians with a natural enthusiasm. However, on the other hand,
together with the language, Cantor's transfinite conceptions and constructions (like the
actualization of all infinite sets, a distinguishing of infinite sets by the number of their elements
(i.e., their cardinalities), the hierarchy of ordinal and cardinal transfinite numbers, continuum
hypothesis, etc.) went into the mathematics-XX. Just the Cantor's actualization of infinite sets
generated a lot of set-theoretical paradoxes and, ultimately, the Third Great Crisis in foundations
of mathematics in the beginning of the XX c. The theme itself of the present conference shows
that the problem of the actual infinity is not closed and the Third Great Crisis in foundations of
mathematics goes on hitherto.
[A.A. Zenkin: "Scientific Intuition of Genii Against Mytho-'Logic' of Cantor‘s Transfinite Paradise"
Procs. of the International Symposium on “Philosophical Insights into Logic and Mathematics,”
Nancy, France, 2002, p. 1]
http://www.ccas.ru/alexzen/papers/CANTOR-2003/Zenkin%20PILM2002.doc
§ 238 Sir, I just came across your paper on "Cantor's Theorem" that there is no bijection from a
set to its power set. I think you are right about the set M of "non-generators" being paradoxical.
[...] I have been troubled about set theory since they told me in school that there is a rational
between every two irrationals, yet more irrationals than rationals. It is obvious that "between
every two irrationals there is a rational" implies that there are as many rationals as irrationals.
However, I am frustrated that this could be so hard to prove while being immediately obvious to
the intuition. I am also convinced that there cannot be more distinct Dedekind cuts than distinct
rational numbers. Just drawing a sketch of some Dedekind cuts convinces me. The Dedekind
cuts are 1) nonempty and 2) totally ordered by the relation "is a proper subset of". For finite sets
it is easy to see, and prove by induction, that for such a collection of sets there are no more sets
than elements. But I do not know how to make this a transfinite induction. Thank you for reading
my long email. I hope that people are listening to your arguments!
Dear NN, You were right. The reason is: There are at most countably many finite definitions like
e = Σ1/n!. That is undisputed. So if there should be uncountably many reals, most of them
cannot be defined - or can only be defined by infinite sequences. But that means the same as
being undefined, because none of those sequences defines a number unless you know the last
digit - which is impossible. So those "reals" cannot be used in mathematics (which means
communication) because they cannot be communicated. They are not really real. And here
comes a simple proof that the notion of uncountablility is in fact nonsense: Construct all real
numbers of the unit interval as infinite paths of the complete infinite Binary Tree. It contains all
real numbers between 0 and 1 as infinite paths i. e. infinite sequences of bits. [...] The complete
tree contains all infinite paths. The structure of the Binary Tree excludes that are any two initial
segments, Bk and Bk+1, which differ by more than one infinite path. (In fact no Bk does contain
any infinite path - but that is not important for the argument.) Hence either there are only
countably many infinite paths. Or uncountably many infinite paths come into the tree after all
finite steps of the sequence have been done. But if so, then it is by far more probable to assume
that the single Cantor-diagonal comes into the Cantor-list after all lines at finite places have been
searched. And then we have no reason to assume the existence of uncountable sets.
§ 239 We have, it is true, a clear idea of division, as often as we think of it; but thereby we have
no more a clear idea of infinite parts in matter, than we have a clear idea of an infinite number,
by being able still to add new numbers to any assigned numbers we have: endless divisibility
giving us no more a clear and distinct idea of actually infinite parts, than endless addibility (if I
may so speak) gives us a clear and distinct idea of an actually infinite number; they both being
only in a power still of increasing the number, be it already as great as it will. So that of what
remains to be added (wherein consists the infinity) we have but an obscure, imperfect, and
confused idea, from or about which we can argue or reason with no certainty or clearness, no
more than we can in arithmetic, about a number of which we have no such distinct idea as we
have of 4 or 100; but only this relative obscure one, that compared to any other, it is still bigger:
and we have no more a clear positive idea of it when we say or conceive it is bigger, or more
than 400,000,000, than if we should say it is bigger than 40, or 4; 400,000,000 having no nearer
a proportion to the end of addition, or number, than 4. For he that adds only 4 to 4, and so
proceeds, shall as soon come to the end of all addition, as he that adds 400,000,000 to
400,000,000. And so likewise in eternity, he that has an idea of but four years, has as much a
positive complete idea of eternity, as he that has one of 400,000,000 of years: for what remains
of eternity beyond either of these two numbers of years is as clear to the one as the other; i. e.
neither of them has any clear positive idea of it at all. For he that adds only four years to 4, and
so on, shall as soon reach eternity, as he that adds 400,000,000 of years, and so on; or, if he
please, doubles the increase as often as he will: the remaining abyss being still as far beyond
the end of all these progressions, as it is from the length of a day or an hour. For nothing finite
bears any proportion to infinite; and therefore our ideas, which are all finite, cannot bear any.
Thus it is also in our idea of extension, when we increase it by addition, as well as when we
diminish it by division, and would enlarge our thoughts to infinite space. After a few doublings of
those ideas of extension, which are the largest we are accustomed to have, we lose the clear
distinct idea of that space: it becomes a confusedly great one, with a surplus of still greater;
about which, when we would argue or reason, we shall always find ourselves at a loss; confused
ideas in our arguings and deductions from that part of them which is confused always leading us
into confusion.
[J. Locke: "The Works of John Locke in Nine Volumes", 12th ed., Vol. 1. Chapter XXIX, §16: "Of
Clear and Obscure, Distinct and Confused Ideas", Rivington, London (1824)]
http://oll.libertyfund.org/title/761/80774/1923786
§ 240 Consider a Cantor-list that contains a complete sequence (qk) of all rational numbers qk.
The first n digits of the anti-diagonal d are d1, d2, d3, ..., dn. It can be shown for every n that the
Cantor-list beyond line n contains infinitely many rational numbers qk that have the same
sequence of first n digits as the anti-diagonal d.
Proof: There are infinitely many rationals qk with this property. All are in the list by definition. At
most n of them are in the first n lines of the list. Infinitely many must exist in the remaining part of
the list. So we have obtained:
" n $ k: d1, d2, d3, ..., dn = qk1, qk2, qk3, ..., qkn
This theorem it is not less important than Cantor's theorem: For all " k: d ∫ qk
Both theorems contradict each other with the result that finished infinity as presumed for
transfinite set theory is not a valid mathematical notion.
§ 241 The aim of the production was to find visual and theatrical ways of expressing the idea of
infinity. The audience did learn some math, but the main impact was at the experiential level.
The scenes each concerned aspects of infinity. The first showed Hilbert's hotel. This is a hotel
with an infinite amount of rooms. Even if each room is occupied, it can accommodate a new
guest: each of the present guests move one room along the line. This does not make life easier
for the hotel owner, but is clearly possible given the concept of infinity – which may nonetheless
be too complicated for efficient hotel management! The hotel was seen as endless doors lining
the wall stretching into the rafters of the warehouse.
Another scene deals with eternal life. The audience sees very, very old people in wheelchairs or
under hairdryers, reading and trying to pass the time. The surroundings are black and enclosed,
so that a stifled, monotonous atmosphere is created. Barrow describes it thus: "It makes us think
about living forever, exploring the social, religious and human implications of infinite life for
everything from life insurance, how to set punishment for crime and recompense for negligence
when an infinite future is taken away, and what to make of religion that promises everlasting life.
[…] The action takes place mostly above the audience with old chrones conveyed in chairs on
monorails."
The third scene takes place in a large space full of corridors with mirrors at the end, dramatizing
Jorge Luis Borges' parable of the library of Babel. The audience wanders through the corridors
which are filled with empty bookcases, and the actors around them are identically masked and
clothed, repeating the same words, seemingly an endless amount of identical people. The
audience feels they are wandering through an infinite universe where anything that can happen
will happen. {{Can and will it happen also that everywhere nothing happens?}}
The fourth scene (finally, the mathematicians will sigh) brings us Georg Cantor himself, the
father of our modern concept of infinity. Set in a hospital, it dramatizes Cantor's conflict with
Kronecker about the nature of infinity. Cantor, covered with bandages, sits in a wheelchair as
Kronecker rants at him. {{This stresses that the author of the play did not know the historical
facts.}}
[Judy Kupferman: Infinity in Theatre]
http://www.jewish-theatre.com/visitor/article_display.aspx?articleID=579
§ 242 We should not say that the least number not definable in less than 19 words is 'definable'
in less than 19 words [...] Of course one could replace 'definable' in the phrase with a bare
'referrable to' and then it might seem that the paradox would reappear in another guise: the least
number not referrable to in 19 words is clearly referrable to in less than 19 words. But now
Donnellan's Distinction comes into its own: for there is no paradox in the man with martini in his
glass having no martini in his glass - once one appreciates the difference between reference and
attribution.
[H. Slater: "The Uniform Solution of the Paradoxes" (2004)]
http://msc.uwa.edu.au/philosophy/about/staff/hartley_slater/publications/the_uniform_solution_of
_the_paradoxes
§ 243 So any reasonably complete account of what mathematics is, or what mathematical
activity is, must ultimately confront the issue of what mathematicians are trying to accomplish at least if it is to be relevant to actual mathematical activity.
This is very difficult to get a hold of, especially in light of the fact that mathematicians are not in
anything like full agreement as to what they are trying to accomplish. What makes matters more
difficult still, is that writing about "what is mathematics, and what are we trying to accomplish" is
not considered normal professional activity among mathematicians. This statement is not so
negative. After all, "what is philosophy, and what are we trying to accomplish" is rarely a topic of
the leading philosopher's papers either. I gather that philosophers did not like it when Rorty
wrote about this. I asked Kripke if he would write about this and related matters, and he made it
clear that he wouldn't touch it with a xxxxxxxx foot pole! {{The length is given in the unary
system. To be absolutely safe?}}
[...] I do get the feeling that this situation is better in physics. This may be partly due to the
apparent fact(?) that it is relatively clear from the outset "what physics is and what physicists
are trying to accomplish" than "what mathematics is and what mathematicians are trying to
accomplish".
[Harvey M. Friedman: "Philosophical Problems in Logic" (2002)]
http://www.math.ohio-state.edu/~friedman/manuscripts.html
http://www.math.ohio-state.edu/~friedman/pdf/Princeton532.pdf
A flimsy attempt to mystify mathematics. Algebradabra! And yet it is so simple. Mathematicians
calculate in order to obtain correct results, i.e., results which agree with reality. If the
experiments are complicated and some physics is required in addition, we call it applied or impure mathematics. If the experiments are simple enough to be carried out with marbles, building
blocks or strings, we call it pure or dis-plied mathematics. Both kinds of mathematics are housed
under the roof of the faculty of science. And Cantor himself has done quite a lot of im-pure
mathematics, namely his lessons on mechanics:
Auch die drei Wintervorlesungen: analytische Mechanik [...] werden mich von den anderen
Gebieten fern halten. [Cantor to Mittag-Leffler, 6 Sept. 1885]
[...] und so fange ich denn morgen 4. Mai meine angekündigte 5 stündige Vorlesung über
analyt. Mechanik an. [Cantor to Jourdain, 3 May 1905]
His last lesson in winter semester 1910/11 was on analytical mechanics {{my translation}}. [W.
Purkert, H.J. Ilgauds: "Georg Cantor 1845-1918", Birkhäuser, Basel (1987) p. 165]
By the way: Lectures in General Sciences are not an invention of Bavarian Universities of
Applied Sciences (laudably they are obligatory here, in contrast to other countries and nations).
Even with over 70 Cantor planned lessons, which unfortunately could not be realized. In his
estate an announcement of 3 May 1917 has been found: In this summer semester I intend to
read privatim Aristotelian logic for all four faculties, Wednesday and Saturday 9 to 10 o'clock
{{my translation}}. [loc. cit.]
§ 244 Consider the diagonal d = d1, d2, d3, ... of a Cantor list, constructed by some appropriate
digit-substitution dn ∫ ann. For instance, if ann > 5 let dn = 2, and if ann < 6 let dn = 8.
If we have: " n: dn is an even digit. Can it be then, that d contains any odd digit?
If we have: " n: d1, d2, d3, ... dn the first n digits are even. Can it be then, that d contains any
odd digit?
If we can prove: " n: dn has property P. Can it be then that there is a digit with the property ŸP?
If we can prove: " n: d1, d2, d3, ... dn have the property P. Can it be then that there is a digit with
the property ŸP?
Cantor speaks of the Inbegriff (set) of all positive numbers n, which should be denoted by the
symbol (n). Does that differ from what today is denoted by Ù or "all n œ Ù"?
Zermelo requires a set that with a also contains {a}. What is the difference to an inductive set
that can be put in bijection with Ù?
Zermelo defines identity: If M ΠN and N ΠM, then M = N.
Is diagonal d an ordered set of all digits dn such that the Inbegriff (dn) differs from {dn | n œ Ù}?
§ 245 In the present paper I would like to develop a different point of views on the continuum.
[...] As a background this point-set theoretic concept is influenced by individualism in modern
civilization. 19th and 20th centuries are the centuries of individualism and the individualism
played an important role in the revelation of people and high advancement of science and
technology. Historically individualism came from liberalism, which in turn came from Reform of
Religion by earlier Protestants, and the fundamental roots can even go upstream to Apostle
Paul. Anyway by historical reason Protestantism performed an important role to the development
of civilization. It is marvelous, if it is taken into consideration that religion is conservative in
nature, that Protestantism contributed the advancement of science that sometimes contradicts
against Bible (This is caused because Protestantism abandoned to be a religion.) [...] New point
of view I am now going to propose is a "missing ring", whose trace can be seen in many part of
mathematics and philosophy, and these traces and "holes" will be fulfilled by the proposal
introduced below. First of all I propose that continuum is never a gathering of points and is a
thing that can never be counted out by points. Continuum and point, they can co-exist but are
very different concept and have no relations each other. [...] Of course we can embed numbers
(points) in the continuum. By doing so we sometimes measure the length of continuum or divide
continuum. But it is just embedding and not any more. Looking from the continuum embedded
points exist only ideally or as an intersecting limit of stringlets. So even though you may measure
continuum or do addition using continuum, it is just virtual, and what you are really doing is only
arithmetical operation on conventional point-set theory. [...] As a counterpart of point-set theory
string-set theory is proposed. It is asserted that the string-set is the essence of continuum in one
aspect [...] And importance of introducing string-set theoretical point of view not only to make
mathematics useful but also correct crippled modern civilization. {{That sounds promising.}}
[Akihiko Takizawa: "String Set Theory" (2002)]
http://www.geocities.co.jp/SweetHome-Ivory/6352/sub7/string.html
§ 246 Cantor's list contains real numbers r as binary or decimal fractions. Real numbers,
however, are limits of binary or decimal fractions.
For every terminating fraction of r, Cantor obtains a difference between r and the due
terminating fraction of the anti-diagonal d: rnn ∫ dn. He concludes that this remains true for the
limits of the list numbers r and d by using the argument: different sequences have different limits.
But it is well known that this argument is not admissible in proofs because it is false.
§ 247 The dependence of our systems of kinds on our theories, and the dependence of these,
in turn, on our interests, values, technology, and the like, make questionable, at best, the thesis
that our predicates pick out real properties or natural kinds whose existence, extension, and
metaphysical status are independent of any contribution of ours. And the claim that just these
kinds or properties are required to answer scientific questions or provide scientific explanations
supports that thesis only if backed by an account of why these questions or forms of explanation
have priority - an account that does not, in turn, appeal to the practices or institutions of which
they are a part, else all questions are begged.
[Catherine Z. Elgin: "With Reference to Reference", Hacket Publishing Company, Inc (US)
(1983) p. 35]
§ 248 Cantor's work was well received by some of the prominent mathematicians of his day,
such as Richard Dedekind. But his willingness to regard infinite sets as objects to be treated in
much the same way as finite sets was bitterly attacked by others, particularly Kronecker. There
was no objection to a "potential infinity" in the form of an unending process, but an "actual
infinity" in the form of a completed infinite set was harder to accept.
[H.B. Enderton, Elements of Set Theory". Academic Press, New York (1977) p. 14f]
Glossary.
Potential infinity: " n $ m : n < m
For every cardinal number, there exists a greater cardinal number.
Actual infinity: $ m " n : n § m
There exists a cardinal number such that no cardinal number is greater.
And this is only the beginning, invented for the "smallest" infinity. The whole sequence of
accessible cardinal numbers is again potentially infinite.
And this is only the beginning - cp. the essay about Archangels and inaccessible Cardinals:
Das Kalenderblatt 1090 of 28 May 2012 (in German)
http://www.hs-augsburg.de/~mueckenh/KB/KB%201001-1111.pdf
§ 249 When I was a first-year student at the Faculty of Mechanics and Mathematics of the
Moscow State University, the lectures on calculus were read by the set-theoretic topologist L.A.
Tumarkin, who conscientiously retold the old classical calculus course of French type in the
Goursat version. [...] These facts capture the imagination so much that (even given without any
proofs) they give a better and more correct idea of modern mathematics than whole volumes of
the Bourbaki treatise. [...] The emotional significance of such discoveries for teaching is difficult
to overestimate. It is they who teach us to search and find such wonderful phenomena of
harmony of the Universe.
The de-geometrisation of mathematical education and the divorce from physics sever these
ties. [...] teaching ideals to students who have never seen a hypocycloid is as ridiculous as
teaching addition of fractions to children who have never cut (at least mentally) a cake or an
apple into equal parts. No wonder that the children will prefer to add a numerator to a numerator
and a denominator to a denominator.
From my French friends I heard that the tendency towards super-abstract generalizations is
their traditional national trait. I do not entirely disagree that this might be a question of a
hereditary disease, but I would like to underline the fact that I borrowed the cake-and-apple
example from Poincaré {{who used to name a disease a disease too}}.
[V.I. Arnold: "On teaching mathematics" (1997), Translated by A.V. Goryunov]
http://pauli.uni-muenster.de/~munsteg/arnold.html
§ 250 Fibonacci-sequences with fatalities
The Fibonacci-sequence
f(n) = f(n-1) + f(n-2) for n > 2 with f(1) = f(2) = 1
the first recursively defined sequence in human history (Leonardo of Pisa, 1170 - 1240) should
be well known. A pair of rabbits that reproduces itself monthly as from the completed second
month on will yield 144 pairs after the first 12 months:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
If we assume that each pair reproduces itself after two months for the last time and dies
afterwards, we get a much more trivial sequencence:
1, 1, 1, ...
However the rabbits behind this numbers change. If we call them in the somewhat unimaginative
but effective manner of the old Romans, we get Prima, Secunda, Tertia, Quarta, Quinta, Sexta,
Septima, Octavia, Nona, Decima and so on.
A more interesting question is brought up, if the parent pair dies immediately after the birth of its
second child pair. Then the births in month n can be traced back to pairs who have been born in
months n-2 and n-3.
g(n) = g(n-2) + g(n-3).
The number f(n) of pairs in month n is given by those born in month n, i.e. g(n) and those
already present in month n-1, i.e., f(n-1), minus those who died in month n (i.e. those who were
born in month n-3:
f(n) = g(n) + f(n-1) - g(n-3) = g(n-2) + f(n-1)
g(n-2) = f(n) - f(n-1)
g(n-2) = g(n-4) + g(n-5)
= f(n-2) - f(n-3) + f(n-3) - f(n-4)
= f(n-2) - f(n-4)
For n > 4 we have with f(1) = 1, f(2) = 1, f(3) = 2, f(4) = 2.
f(n) = f(n-1) + f(n-2) - f(n-4)
The number of pairs during the first 12 months is
1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21.
The sequence grows less than the original one, but without enemys or other restrictions it will
grow beyond every threshold. If we wait ¡0 days (or use the trick that the duration of pregnancy
is halved in each step, facilitated by genetic evolution), we will get infinitely many pairs - a
nameless number, alas of nameless rabbits, because they cannot be distinguished. The set of
all Old-Roman names has been exhaused already, and even all of Peano's New-Roman names
S0, SS0, SSS0, ... have been passed over to pairs which already have passed away. That is
amazing, since none of the pairs of the original and much more abundant Fibonacci sequence
has to miss a name.
But this sequence with fatalities can also be obtained without fatalities (killings), namely if each
pair has to pause for two months after each bearth in order to breed again in the following
month. Mathematically, there is no difference. Set theory, however, yields a completely different
limit in this case. The limit set of living rabbits is no longer empty, but it is infinite - and every
rabbit has
a name.
So we obtain from set theory: The cultural assets of distinguishability of distinct objects by
symbols or thoughts do not belong to the properties of Cantor's paradise. Like in the book of
genesis, before Adam began to name the animals, we have a nameless paradise - but not
mathematics.
§ 251 Frege, too, as Lesniewski points out, attacks those mathematicians who introduce into
their theories such arbitrary 'inventions' as the empty set, merely because they prove expedient
for certain purposes. Lesniewski's own strictures in this respect are directed in particular against
axiomatic theories of sets such a were developed by Zermelo. These do not merely lack the sort
of naturalness that would dispose one to accept them; they lack also the intrinsic intelligibility
which would make their meaning clear, so that Lesniewski can in all honesty assert that he does
not understand what is meant by 'set' as this term is supposed to be 'implicitly defined' by
theories like Zermelo's.
Lesniewski himself, in contrast, starts not from 'inventions" or from axioms or hypotheses
selected for pragmatic reasons, but from what he calls intuitions, commonly accepted and
meaningful to all, relating to such concepts as whole, part, totality, object, identity, and so on.
The language of Lesniewski's theories is therefore an extrapolation of natural language, a
making precise of what, in natural language, is left inarticulate or indistinct. [...] Hence he is
mistrustful, too, of the model-theoretic semantics that has been built up on an abstract settheoretical basis, and he is opposed also to the work of those formalist logicians who embrace
an essentially abstract-algebraic approach to logic, or see logic as having to deal essentially with
uninterpreted formal systems.
[Barry Smith: "On the phases of reism" in: A. Chrudzimski, D. Lukasiewics (eds): "Actions,
Products, and Things", Ontos-Verlag Frankfurt (2006) p. 124]
§ 252 The table T
1
2, 1
3, 2, 1
...
n, ..., 3, 2, 1
...
is a sequence of lines Ln, finite initial segments (1, ..., n) of the sequence of natural numbers. It
contains every natural number that can be somewhere. Every number in the table T is in one
line Ln and in all further lines by construction of T (always the last line is added). Every number
in T is in the first column C (and in every other column too).
" n : (1, ..., n) Œ C fl (1, ..., n) œ T
" n : (1, ..., n) œ T fl (1, ..., n) Œ C
Therefore it is impossible that C contains more than T and more than any line Ln of T. But we
know that there is no line Ln with an actually infite set Ù of numbers (because T is a sequence of
finite lines Ln). Conclusion: An actually infinite set Ù cannot be in the first column either (and
nowhere else).
Same gets clear from the sets. All natural numbers can be found in each of the sets, unions of
sets and sequences of sets.
{1, 2, 3, ...} = Ù
{1} » {2} » {3} » ... = Ù
{1}, {1, 2}, {1, 2, 3}, ...
{1} U {1, 2} U {1, 2, 3} U ... = Ù
{1, 1, 2, 1, 2, 3, ...}
{1}, {1} U {1, 2}, {1} U {1, 2} U {1, 2, 3}, {1} U {1, 2} U {1, 2, 3} U ..., ...
Nothing of Ù is lacking in any. All of Ù is in all these sets or sequences, not only in those which
are officially denoted by Ù but also in the sequences of finite sets. Therefore Ù is not actually
infinite.
§ 253 Notice that if the result is a method that we do not quite recognize as mathematical,
{{then the reason is that mathematics like many social standards have been perverted.}}. [...]
What we have traced is a more or less simultaneous rise of pure mathematics and reevaluation
of applied mathematics. Before all these, back in Newton’s or Euler’s day, the methods of
mathematics and the methods of science were one and the same {{Mathematics was considered
as a science. Frequently theologians like Nicole Oresme, John Wallis, Bonaventura Cavalieri or
George Berkeley used to pursue it as an alternative to their professional occupation - today
mathematics does no longer offer an alternative. Mathematics and theology have merged.}}; if
the goal is to uncover the underlying structure of the world, if mathematics is simply the
language of that underlying structure, then the needs of celestial mechanics (for Newton) or
rational mechanics (for Euler) are the needs of mathematics. From this perspective, the
correctness of a new mathematical method – say the infinitary methods of the calculus or the
expanded notion of function – is established by its role in application. {{That's the philosophers
(touch-)stone.}}
[Penelope Maddy: "How applied mathematics became pure", Reviev Symbolic Logic 1 (2008)
16-41]
§ 254
1. Finite cannot comprehend, contain, the Infinite. - Yet an inch or minute, say, are finites, and
are divisible ad infinitum, that is, their terminated division incogitable.
2. Infinite cannot be terminated or begun. - Yet eternity ab ante ends now; and eternity a post
begins now. So apply to Space.
3. There cannot be two infinite maxima. - Yet eternity ab ante and a post are two infinite
maxima of time.
4. Infinite maximum if cut in two, the halves cannot be each infinite, for nothing can be greater
than infinite, and thus they could not be parts; nor finite, for thus two finite halves would make an
infinite whole.
5. What contains infinite quantities (extensions, protensions, intensions) cannot be passed
through, - come to an end. An inch, a minute, a degree contains these; ergo, &c. Take a minute.
This contains an infinitude of protended quantities, which must follow one after another; but an
infinite series of successive protensions can, ex termino, never be ended; ergo, &c.
6. An infinite maximum cannot but be all-inclusive. Time ab ante and a post infinite and
exclusive of each other; ergo, &c.
7. An infinite number of quantities must make up either an infinite or a finite whole. I. The
former. - But an inch, a minute, a degree, contain each an infinite number of quantities; therefore
an inch, a minute, a degree, are each infinite wholes; which is absurd. II. The latter. - An infinite
number of quantities would thus make up a finite quantity, which is equally absurd.
[John Stuart Mill: "An Examination of William Hamilton’s Philosophy", The Collected Works of
John Stuart Mill, Volume IX, CHAPTER XXIV: "Of Some Natural Prejudices Countenanced by
Sir William Hamilton, and Some Fallacies Which He Considers Insoluble" (1865), John M.
Robson (ed.), Routledge and Kegan Paul, London (1979)]
http://oll.libertyfund.org/?option=com_staticxt&staticfile=show.php%3Ftitle=240&chapter=40898
&layout=html#a_761210
§ 255 Let S = (1), (1, 2), (1, 2, 3), ... be a sequence of all finite initial sets si = (1, 2, 3, ..., i) of
natural numbers.
Every natural number n is in some term si of S: »si = Ù.
(0) " n $ i: n e si.
S contains si+1 after si. So we have
(1) " n " i : (n § i ñ n œ si) ⁄ (n > i ñ n – si).
There is no term si of S that contains all natural numbers. This condition requires that in every
term at least one natural number is missing.
(2) $ j, k, m, n : m œ sj ⁄ m – sk ⁄ n – sj ⁄ n œ sk.
(2) is in contradiction with (1).
§ 256 In his dissertation of 1907, Brouwer had actually explained how he could accept some of
Cantor’s ideas, including his transfinite numbers ω, ω + 1, … up to a certain point (as long as
they are denumerable and in a certain sense constructible {{i.e., given by a finite formula or
rule}}) but not the further concepts of a totality of all such denumerable numbers.[...]. And it was
not the set-theoretic paradoxes that caused his reaction. As he remarked in 1923, an incorrect
theory, even if it cannot be checked by any contradiction that would refute it, is none the less
incorrect, just as a criminal policy is none the less criminal even if it cannot be checked by any
court that would curb it. [...] The point for the intuitionists is that mathematics is a mental
construction erected freely by the mind. It is simply an illusion to conceive of mathematics as
dealing with independently existing objects, with an objective reality somehow external to the
mind. {{"God created man in His own image? Rather man created God in his. [Georg Christoph
Lichtenberg, Göttingen]. If there are Gods, then only such that are man-made. And if there are
numbers, then only such that are man-made. Mental constructions cannot exist without mind even if matheologians are prepared to prove the contrary (double meaning intended).}} But this
is what modern mathematics does: the objects of the theory are conceived as elements of a
totality or set that is regarded as given, totally independently of the thinking subject. This feature
is deeply embedded in the methods employed in mathematics, and (following Bernays, a key
collaborator of Hilbert) it is often called the “Platonism” of modern mathematics.
{{Nothing is so out of date as the utopies of yesterday: 1984, 2001, transfinite set theory.}}
Meanwhile, the constructivists’ treatment of mathematics – exemplified by intuitionism – is based
on careful consideration of the processes by which numbers, etc., are defined or constructed.
Each and every thing that a mathematician can legitimately talk about must have been explicitly
constructed in a mental activity.
As time went by, Brouwer realized that it was better to avoid talking of “sets” at all, and he
introduced new terminology (“species” and “spreads”). [...] As Brouwer’s reconstruction of
mathematics developed in the 1920s, it became more and more clear that intuitionistic analysis
was extremely subtle, complicated and foreign. Brouwer was not worried, for “the spheres of
truth are less transparent than those of illusion,” as he remarked in 1933.
[José Ferreirós: "Paradise Recovered? Some Thoughts on Mengenlehre and Modernism",
(2008)]
§ 257 The set of all possible computer programs is countable {{if going into concrete details of
computers, we can even say more restrictively: The number of all possible programs is finite and
certainly less than 210100. But of course Turing did not refer to real computers and did not know
anything about the memory space of an exploitable surronding. Therefore let us analyze his
approach here.}}, therefore the set of all computable reals is countable, and diagonalizing over
the computable reals immediately yields an uncomputable real. Q.E.D. {{Well, it is not always
that easy to diagonalize over finite sequences. The following list has no diagonal:
0
1
So we must take some more care, as will be done in the following.}}
Let's do it again more carefully.
Make a list of all possible computer programs. Order the programs by their size, and within
those of the same size, order them alphabetically. The easiest thing to do is to include all the
possible character strings that can be formed from the finite alphabet of the programming
language, even though most of these will be syntactically invalid programs.
Here's how we define the uncomputable diagonal number 0 < r < 1. Consider the kth program
in our list. If it is syntactically invalid, or if the kth program never outputs a kth digit, or if the kth
digit output by the kth program isn't a 3, pick 3 as the kth digit of r. Otherwise, if the kth digit
output by the kth program is a 3, pick 4 as the kth digit of r.
This r cannot be computable, because its kth digit is different from the kth digit of the real
number that is computed by the kth program, if there is one. Therefore there are uncomputable
reals, real numbers that cannot be calculated digit by digit by any computer
program. [...]
In other words, the probability of a real's being computable is zero, and the probability that it's
uncomputable is one. [Who should be credited for this measure-theoretic proof that there are
uncomputable reals? I have no idea. It seems to have always been part of my mental baggage.]
{{The error occurs always at the same place - that is common to all "uncountability-proofs". The
completed infinite number of all finite programs is presupposed. Otherwise the just defined
number would be generated by another finite process with a finite program. But the set of all
finite things is not a completed infinity, it is potentially infinite, it is not a set and does not contain
all progarms but only every finite program up to every length. After all you cannot use a program
that is longer than the longest program that is used. Nevertheless, up to every length, only a
non-measurably small share of all usable finite programs has bee used.}}
In spite of the fact that most individual real numbers will forever escape us, the notion of an
arbitrary real has beautiful mathematical properties and is a concept that helps us to organize
and understand the real world. Individual concepts in a theory do not need to have concrete
meaning on their own; it is enough if the theory as a whole can be compared with the results of
experiments. - So much for mathematics! {{Mathematics?}}
[Gregory Chaitin: "How real are real numbers?" (2004)]
http://arxiv.org/abs/math.HO/0411418
Some mathematicians write so beautifully that they should be poets. I mean, they should be
poets instead of being mathematicians. As poets, they wouldn't be doing any real harm.
Artistic license is out of place in mathematics.
[David Petry, sci.math, Matheology § 257, 22 April 2013]
http://groups.google.com/group/sci.math/browse_frm/thread/f7249ecd519bcfde?scoring=d&
§ 258 So what about Cantor’s much celebrated non-denumerable real? Where is it? Did Cantor
produce such a real number? No, he merely sketched out the logic for a nonterminal procedure
that would produce an infinitely long digit string representing a real number that would not be in
the input stream of enumerated reals. Cantor’s procedure, and with it his celebrated
nondenumerable, infinitely long real number, will appear with 100% certainty in the denumerable
list of procedures. {{That's the point: Every diagonal number can be distinguished at a finite
position from every other number. But if all strings are there to any finite dephts, as is easily
visualized in the Binary Tree, then there is no chance for distinction at a finite position - and
other positions are not available.}}
There is no non-denumerable real, and every source of real numbers is denumerable [...]
Implications throughout mathematics that build upon Cantor’s Diagonal Proof must now be
carefully reconsidered.
So Who Won? Professor Leopold Kronecker was right. Irrationals are not real {{ - at least they
have no real strings of digits, and only countably many of them can be defined in a language that
can be spoken, learned and understood}}. God made all the integers and Man made all the rest
{{and in addition something more - unfortunately.}}
[Brian L. Crissey: "Kronecker 1, Cantor 0: The End of a Hundred Years’ War"]
http://www.briancrissey.info/files/Kronecker1Cantor0.pdf
§ 259 A discussion in sci.logic*) yielded the following remarkable results with respect to a list
having infinitely many lines
{1}
{1} U {2}
{1} U {2} U {3}
...
Each line contains as many unioned sets as its line number indicates but does not contain a line
Ù, since each line has a finite last number n.
On the other hand, there are infinitely many lines and, as each line adds one natural number,
there are infinitely many natural numbers in the list. Since, by construction, every finite initial
segment sn = {1, 2, 3, ..., n} is in one single line, all finite initial segments are in one single line.
But Ù is not more than all its finite initial segments.
Otherwise there must exist at least two finite initial segments such that
$ j, k, m, n : m œ sj ⁄ m – sk ⁄ n – sj ⁄ n œ sk.
Further, if all lines of the list are written within one single line then Ù is in this single line.
Further if the list is prepared such that (for n > 1) after adding line sn the preceding line sn-1 is
removed, then the list, again consisting of a single line only, but by construction never being
empty, is empty.
[*) Matheology § 255, sci.logic, April 2013]
http://groups.google.com/group/sci.logic/browse_frm/thread/83ff0cf1d8f6e48a?scoring=d&
§ 260 Potential Infinity: " n $ m : m > n
For every string of the list
0.1
0.11
0.111
...
there is a longer one. This, however, does not prove the uncountability of strings, because every
longer string is in the list too.
A proof of uncountability needs a string that differs from every string of the list and, by
construction of the list, is longer than all. That is called
Actual Infinity: $ m " n : m > n
But it is impossible to construct such a string because, by definition, all digits 1 at natural indices
are already in the strings of the list. Therefore Cantor's argument fails to produce a string
different from all strings of the list at natural indices.
Example: There is no irrational number that can be distinguished by any sequence of its digits
from all rational numbers. For that sake always a finite definition is required. But finite definitions
cannot result from Cantor's argument.
§ 261 [...] when I found it, I thought in the beginning that it causes invincible problems for set
theory that would finally lead to the latter’s eventual failure; now I firmly believe, however, that
everything essential can be kept after a revision of the foundations, as always in science up to
now {{of course, just like the teachings of world spirit, geocentric system, phlogiston theory,
ether, or principle of causality}}. I have not published this contradiction {{what a bad word
escaped Hilbert's mouth}} [D. Hilbert: "Logische Principien des mathematischen Denkens,
lecture course in the summer term 1905, lecture notes by Ernst Hellinger", Library of the
Mathematics Seminar of the University of Göttingen, p. 204]
[...]
The paradox is based on a special notion of set which Hilbert introduces by means of two set
formation principles starting from the natural numbers. The first principle is the addition principle.
In analogy to the finite case, Hilbert argued that the principle can be used for uniting two sets
together “into a new conceptual unit [...], a new set that contains each element of either sets.”
This operation can be extended: “In the same way, we are able to unite several sets and even
infinitely many into a union.” The second principle is called the mapping principle. Given a set M,
he introduces the set MM of self-mappings of M to itself. [Hilbert used the German term
“Selbstbelegung” which is translated here by “selfmapping”.] A self-mapping is just a total
function which maps the elements of M to elements of M. [In classical logic, MM is isomorphic to
2M, and the set of all mappings from M to {0, 1} is isomorphic to P(M), the power set of M.]
Now, he considers all sets which result from the natural numbers “by applying the operations
of addition and self-mapping an arbitrary number of times.” By use of the addition principle which
allows to build the union of arbitrary sets one can “unite them all into a sum set U which is welldefined.” In the next step the mapping principle is applied to U, and we get F = UU as the set of
all self-mappings of U. Since F was built from the natural numbers by using the two principles
only, Hilbert concludes that it has to be contained in U. From this fact he derives a contradiction.
Since “there are ‘not more’ elements” in F than in U there is an assignment of the elements ui
of U to elements fi of F such that all elements of fi are used. Now one can define a self-mapping
g of U which differs from all fi. Thus, g is not contained in F. Since F was assumed to contain all
selfmappings we have a contradiction. In order to define g Hilbert used Cantor’s diagonalization
method. [...]
Hilbert finishes his argument with the following observation: "We could also formulate this
contradiction so that, according to the last consideration, the set UU is always bigger [of greater
cardinality] than U but, according to the former, is an element of U." [...]
Considering Cantor’s general definition of a set as the comprehension of certain welldistinguished objects of our intuition or our thinking as a whole, one can justly ask whether the
sets of all cardinals, of all ordinals or the universal set of all sets are sets according to this
definition, i. e., whether an unrestricted comprehension is possible. Cantor denies this.
Hilbert, on the other hand, introduces two alternative set formation principles, the addition
principle and the mapping principle, but they lead to paradoxes as well. In avoiding concepts
from transfinite arithmetic Hilbert believes that the purely mathematical nature of his paradox is
guaranteed. For him, this paradox appears to be much more serious for mathematics than
Cantor’s, because it concerns an operation that is part of everyday practice of working
mathematicians.
[Volker Peckhaus: "Paradoxes in Göttingen" (2003)] {{This headline is written in English. But it
acquires a good meaning when understood in German. - Unfortunately no longer online.}}
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.4718
§ 262 As an interesting detail let us add that in response to the paper (1927) by von Neumann
it was reacted critically by Stanisław Leśniewski who published the paper “Grundzüge eines
neuen Systems der Grundlagen der Mathematik” (1929) in which he critically analyzed various
attempts to formalize logic and mathematics. Leśniewski among others expresses there his
doubts concerning the meaning and significance of von Neumann’s proof of the consistency of
(a fragment of) arithmetic and constructs – to maintain his thesis – a “counterexample”, namely
he deduce (on the basis of von Neumann’s system) two formulas a and Ÿa, hence an
inconsistency. Von Neumann answered to Leśniewski’s objections in the paper “Bemerkungen
zu den Ausführungen von Herrn St. Leśniewski über meine Arbeit ‘Zur Hilbertschen
Beweistheorie’”(1931). Analyzing the objections of Leśniewski he came to the conclusion that
there is in fact a misunderstanding resulting from various ways in which they both understand
principles of formalization. {{No contradictions! Never! None!}} He used also the occasion to fulfill
the gap in his paper (1927). Add also that looking for a proof of the consistency of the classical
mathematics and being (still) convinced of the possibility of finding such a proof (in particular a
proof of the consistency of the theory of real numbers) von Neumann doubted whether there are
any chances to find such a proof for the set theory – cf. his paper (1929).
[Roman Murawski: "John von Neumann and Hilbert's school of foundations of mathematics",
Studies in logic, grammar and rhetoric 7,20 (2004) p. 49]
http://www.google.de/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=1&sqi=2&ved=0CDIQ
FjAA&url=http%3A%2F%2Flogika.uwb.edu.pl%2Fstudies%2Fdownload.php%3Fvolid%3D20%2
6artid%3Drm&ei=rBpsUf2eDsKqtAbk2oGQCA&usg=AFQjCNE1Em4kIz3pFqzC8wtfGwUoALATI
A&sig2=EwLxn6T5JatAx-ik-QyzAA&bvm=bv.45175338,d.Yms
§ 263 There are things. They exist by energy (Joule).
There are ideas. They exist by information (bit).
If a plumber asserts to have twenty thousand hammers in his toolbag, I would carefully
supervise his work, if done in in my house.
If a farmer asserts that his barn contains twenty billions potatoes, I will buy from him expecting
very big potatoes. *)
If a mathematician asserts that his model contains uncountably many indistinguishable but
distinct numbers, this will not cause a sensation.
Isn't that sensational?
*) The joke is based on the German proverb fortune favours foolish farmers by increasing the
diameter of their potatoes.
§ 264 Hilbert's Hotel, last chapter: checking out.
Please leave the room on the day of your departure at the latest till 11:00 a.m.
Hilbert's hotel is not luxurious but expensive and notorious for frequent change of rooms.
Therefore many guests prefer Math's Motel. Before checking out a guest must have occupied
room number 1 (because the narrow halls are often blocked). No problem, guests are
accustomed to that habit.
The guest of room number 1 checks out at half past 10 and all other guests change their
rooms such that all rooms remain occupied. The second guest checks out at quarter to 11. And
all guests switch rooms such that no room is empty. And so on. At 11 a.m. all guests have left
Hilbert's Hotel. Every room is occupied.
A fine result of set theory. It can be improved however to have a real mathematical application,
enumerating the sets of rational and of irrational algebraic numbers.
First enumerate the first two rationals q2 = 1/2 and q1 = 1/3. Then take off label 1 from 1/3 and
enumerate the first irrational x1. 1/3 will get remunerated and re-enumerated in the next round by
label 3, when 1/2 will lose its 2 but gain label 4 instead. So 1/2 and 1/3 will become q4 and
q3. Continue until you will have enumerated the first n rationals and the first n irrationals
q2n, q2n-1, ..., qn+1 and xn, xn-1, ..., x1
and if you got it by now, then go on until you will have enumerated all of them. Then you have
proved in ZFC that there are no rational numbers. (If you like you can also prove that there are
no algebraic irrational numbers. But that's not a contradiction, of course.)
§ 265 Abstract. This paper examines the possibilities of extending Cantor’s two arguments on
the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the
set of rational numbers. The paper proves that, unless certain restrictive conditions are satisfied,
both extensions are possible. It is therefore indispensable to prove that those conditions are in
fact satisfied in Cantor’s theory of transfinite sets. Otherwise that theory would be inconsistent.
[...]
We have just proved [...] the alternatives of Cantor's 1874-argument on the cardinality of the real
numbers can be applied to the set – of rational numbers, except the last one, that applies only if
the common limit of the sequences of left and right endpoints of the QP-intervals is rational.
Evidently, if Cantor’s 1874-argument could be extended to the rational numbers we would have
a contradiction: the set – would and would not be denumerable. Accordingly, in order to ensure
the impossibility of that contradiction, each of the following points have to be proved:
Whatsoever be the rational interval (a, b) and whatsoever be the reordering of <qi>, it must hold:
(1) The number of QP-intervals can never be finite.
(2) The sequences of endpoints <ai> and <bi> can never have different limits.
(3) The common limit of <ai> and <bi> can never be rational.
[...] Until those proofs be given, Cantor’s 1874-argument should be suspended, and the
possibility of a contradiction involving the foundation of (infinitist) set theory should be
considered.
[Antonio Leon Sanchez: "Cantor versus Cantor" (2010)]
http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.2874v3.pdf
On the other hand, the proof can feign the uncountability of a countable set. If, for instance, the
alternating harmonic sequence
ωn = (-1)n/n Ø 0
is taken [...], yielding the intervals (-1, 1/2), (-1/3, 1/4), ... we find that its limit 0 does not belong
to the sequence, although the set of numbers involved, Ù » {0}, is obviously denumerable [...]
The alternating harmonic sequence does not, of course, contain all real numbers, but this
simple example demonstrates that Cantor's first proof is not conclusive. Based upon this proof
alone, the uncountability of this and every other alternating convergent sequence must be
claimed. Only from some other information we know their countability (as well as that of –), but
how can we exclude that some other information, not yet available, in future will show the
countability of ” or —?
[W. Mueckenheim: "On Cantor's important proofs" (2003)]
http://arxiv.org/pdf/math.GM/0306200
§ 266 For every natural number n the sequence (10-1, 10-2, 10-3, ..., 10-n) can be reflected at
the 100-position to (10n, ..., 103, 102, 101).
Obviously this transformation does not depend on the number of terms and does not depend
on the number of exponents, but solely on the condition that all exponents are natural numbers.
Now try to reflect the sequence for 1/9. The claim is that all exponents are natural numbers
too. Has 1/9 a complete decimal representation with only natural exponents?
§ 267 ... our axioms [of set theory], if interpreted as meaningful statements, necessarily
presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even
produce the conviction that they are consistent.
[S. Feferman et al. (eds.): "Kurt Gödel, Collected Works, Vol. III, Unpublished Essays and
Lectures", Oxford University Press, Oxford (1995) p. 50]
§ 268 Can a matheologian disprove the existence of matheologians?
Here are the facts: Recently a matheologian wrote:
I actually question whether the existence of matheologians is consistent. As a matheologian is
either a man or a woman, it follows that a matheologian is a person. Now this person believes in
thoughts that nobody can think. I would surmise that thinking a thought is a prerequisite to
believing that same thought. So this person (this body) thinks a thought that nobody thinks.
[name withheld - from "How to distinguish between the complete and the incomplete infinite
binary tree?", a question in math.stackxchange, meanwhile deleted.]
In my opinion this only shows that not all matheologians know the fundamentals of their belief.
Of course the sober phrase "thinking a thought is prerequisite in believing that thought" does not
hold in matheology. On the contrary, it is just the touchstone for a matheologian to believe in
uncountably many thoughts, namely one for each real number as an individual, that obviously
nobody can think. - Is this acceptable?
§ 269 How to distinguish a decimal representation from a set of all terminating decimal
representations?
It seems impossible to accomplish this task by means of one ore more digits that have finite
indices, i.e., finite distances from the decimal point.
It is clear that an infinite decimal representation has more digits than every finite one. But it is
as clear that there are not more finite indices than all finite indices which are already used and
occupied by all possible sequences of digits to produce all finite decimal representations.
Is it possible to apply other tools?
§ 270 Gödel’s incompleteness results had great influence on von Neumann’s views towards
the perspectives of investigations on the foundations of mathematics. He claimed that “Gödel’s
result has shown the unrealizability of Hilbert’s program” and that “there is no more reason to
reject intuitionism” (cf. his letter to Carnap of 6th June 1931 – see Mancosu, 1999, 39–41). He
added in this letter:
Therefore I consider the state of the foundational discussion in Königsberg to be outdated, for
Gödel’s fundamental discoveries have brought the question to a completely different level. (I
know that Gödel is much more careful in the evaluation of his results, but in my opinion on this
point he does not see the connections correctly).
Incompleteness results of Gödel changed the opinions cherished by von Neumann and
convinced him that the programme of Hilbert cannot be realized. In the paper “The
Mathematician” (1947) he wrote:
My personal opinion, which is shared by many others, is, that Gödel has shown that Hilbert’s
program is essentially hopeless.
Another reason for the disappointment of von Neumann with the investigations in the
foundations of mathematics could be the fact that he became aware of the lack of categoricity of
set theory, i.e., that there exist various nonisomorphic models of the set theory. The latter fact
implies that it is impossible to describe the world of mathematics in a unique way. In fact there is
no absolute description, all descriptions are relative. Not only von Neumman was aware of this
feature of the set theory.
{{So it is good luck that set theory (including Gödel's results) is built upon finished infinity and
that this notion has been proven to be contradictory. Mathematicians should celebrate this as a
great triumph instead of racking their brains in order to desperately stick to matheology.}}
[Roman Murawski: "John von Neumann and Hilbert's school of foundations of mathematics",
Studies in logic, grammar and rhetoric 7,20 (2004) p. 50f]
http://www.google.de/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=1&sqi=2&ved=0CDIQ
FjAA&url=http%3A%2F%2Flogika.uwb.edu.pl%2Fstudies%2Fdownload.php%3Fvolid%3D20%2
6artid%3Drm&ei=rBpsUf2eDsKqtAbk2oGQCA&usg=AFQjCNE1Em4kIz3pFqzC8wtfGwUoALATI
A&sig2=EwLxn6T5JatAx-ik-QyzAA&bvm=bv.45175338,d.Yms
§ 271 Axiom of extensionality: A set X is equal to set Y if and only if they both have exactly the
same elements
The union of all finite initial sequences of natural numbers F(n) = (1, 2, 3, ..., n) is equal to the
union of all F without the first k F:
" k œ Ù : »nœ ÙF(n) = »nœ Ù(F(n) \ F(k))
" k œ Ù : »nœ ÙF(n) = »nœ Ù(F(n) \ »j§k F(j))
This is true because " k $ n : F(k) Õ F(n)
§ 272 In algorithmic information theory, the notion of Kolmogorov complexity is named after the
famous mathematician Andrey Kolmogorov even though it was independently discovered and
published by Ray Solomonoff a year before Kolmogorov. Li and Vitanyi, in "An Introduction to
Kolmogorov Complexity and Its Applications", write: Ray Solomonoff [...] introduced [what is now
known as] 'Kolmogorov complexity' in a long journal paper in 1964. [...] This makes Solomonoff
the first inventor and raises the question whether we should talk about Solomonoff complexity.
http://en.wikipedia.org/wiki/Matthew_effect_(sociology)
The idea is that a string is random if it cannot be compressed. That is, if it has no short
description. {{A string x of bits with |x| = n bit is incompressible, if no string p of bits with less
than n bits exists, which defines or generates the string x (for instance via a computer program.}}
Using {{Kolmogorov complexity}} C(x) we can formalize this idea via the following.
Theorem 1.2. For all n, there exists some x with |x| = n such that C(x) ¥ n. Such x are called
(Kolmogorov) random.
Proof. Suppose not. Then for all x, C(x) < n. Thus for all x there exists some px such that g(px) =
x and |px| < n. Obviously, if x ∫ y then px ∫ py.
But there are 2n - 1 programs of length less than n, and 2n strings of length n. {{Compare the
finite paths up to level n - 1 in the Binary Tree and the paths wit n nodes, i.e., those with one n-th
node beyond the level n - 1}}. By the pigeonhole principle, if all strings of length n have a
program shorter than n, then there must be some program that produces two different strings.
Clearly this is absurd, so it must be the case that at least one string of length n has a program of
length at least n.
[Lance Fortnow: "Kolmogorov Complexity" (2000)]
http://people.cs.uchicago.edu/~fortnow/papers/kaikoura.pdf
By the pigeonhole principle, if all of the first n natural numbers have a unary representation that
is shorter then n there must be some unary representation that defines two different natural
numbers. Clearly this is absurd, so it must be the case that at least one of ¡0 numbers has a
unary representation of length at least ¡0.
§ 273 In Sec. 3.1 we constructed an uncomputable real r. It must be uncomputable, by
construction. Nevertheless, as was the case in the Richard paradox, it would seem that we gave
a procedure for calculating Turing's diagonal real r digit by digit. {{That seems only so to
someone who thinks an infinite sequence could convey information.}} How can this procedure
fail? What could possibly go wrong?
The answer is this: The only noncomputable step has got to be determining if the kth computer
program will ever output a kth digit. If we could do that, then we could certainly compute the
uncomputable real r.
In other words, Sec. 3.1 actually proves that there can be no algorithm for deciding if the kth
computer program will ever output a kth digit.
And this is a special case of what's called Turing's halting problem. In this particular case, the
question is whether or not the wait for a kth digit will ever terminate. In the general case, the
question is whether or not a computer program will ever halt. {{It will not if there is an infinite
loop
00 Begin
10 Goto 20
20 Goto 10
30 Print "3"
40 End
An infinite loop will belong to a potential infinity of programs. Infinitely many programs will never
have been investigated with respect to this property. Therefore their number r will never be
defined. r is and remains undefined. Therefore r is not a number. This fact has no connection to
a proof of uncountability. The dyslogic of the condition "if the kth program never outputs a kth
digit" becomes easily visible by slightly paraphrasing it: In case that the case never occurs. Of
course that is undecidable until a numbers will be returned.}} The algorithmic unsolvability of
Turing's halting problem is an extremely fundamental meta-theorem. It's a much stronger result
than Gödel's famous 1931 incompleteness theorem. Why? Because in Turing's original 1936
paper he immediately points out how to derive incompleteness from the halting problem. {{Shit
happens. Incompleteness happens - for instance in every potential infinity. Uncountability does
not happen or exist.}}
A formal axiomatic math theory (FAMT) consists of a finite set of axioms and of a finite set of
rules of inference for deducing the consequences of those axioms. Viewed from a great
distance, all that counts is that there is an algorithm for enumerating (or generating) all the
possible theorems, all the possible consequences of the axioms, one by one, by systematically
applying the rules of inference in every possible way. This is in fact what's called a breadth-first
(rather than a depth-first) tree walk, the tree being the tree of all possible deductions.
So, argued Turing in 1936, if there were a FAMT that always enabled you to decide whether or
not a program eventually halts, there would in fact be an algorithm for doing so. You'd just run
through all possible proofs until you find a proof that the program halts or you find a proof that it
never halts.
So uncomputability is much more fundamental than incompleteness. Incompleteness is an
immediate corollary of uncomputability. But uncomputability is not a corollary of incompleteness.
The concept of incompleteness does not contain the concept of uncomputability.
[Gregory Chaitin: "How real are real numbers?" (2004)]
http://arxiv.org/abs/math.HO/0411418
Admit only such programs which within three minutes (on a certain computer) return a number.
In principle there are arbitrarily many such programs. But if you wish to construct a real number r
from the output, you have to fix a last program. Alas you get only a rational number which of
course can be calculated by a program. The inclusion of programs that do never halt sneaks a
logical trap into that matter. In case of a not occuring case no case can be recognized (and what
digit it would define).
§ 274 Mathematicians define real numbers to be an ordered field, yet when one applies
Turing‘s proof that the halting problem is unsolvable to the processes that generate infinitely
precise numbers (irrationals), it follows that some irrational pairs cannot be ordered without
solving the halting problem and thus are not real by definition.
The de facto acceptance of Cantor‘s diagonal proof means that mathematicians accept
1. infinitely precise numbers as real numbers and
2. output from nonterminal procedures as definitions of infinitely precise real numbers.
Inequality of unpredictable irrationals can be detected in time, but not equality. So
unpredictable irrational numbers cannot be real.
[B. L. Crissey: "Unreal Irrationals: Turing Halts Cantor"]
http://www.briancrissey.info/Research/Resume_files/Unreal%20Irrationals.pdf
§ 275 The term actual infinite is not a term of set theory, despite what crazy internet authors
like to say.
[Kwalish Kid (2006)]
http://forums.philosophyforums.com/threads/actual-infinity-and-potential-infinity-23597.html
Yes, set theorists try to avoid that term in order to remain attractive to sober minds - but set
theory needs actual or completed or, why not put it frankly, finished infinity and uses it heavily.
§ 276 Cantor's belief in the actual existence of the infinite of Set Theory still predominates in
the mathematical world today.
[A. Robinson:"The metaphysics of the calculus", in Imre Lakatos (ed.): "Problems in the
philosophy of mathematics", North Holland, Amsterdam (1967) p. 39]
http://www.amazon.de/Problems-Philosophy-
Mathematic/dp/0444534113/ref=sr_1_cc_1?s=aps&ie=UTF8&qid=1369999937&sr=1-1catcorr&keywords=0444534113#_
§ 277 It is clear that the theological considerations by which Cantor motivated his notion of the
actual infinite, were metaphysical in nature.
[A. Heyting: "Technique versus metaphysic in the calculus", in Imre Lakatos (ed.): "Problems in
the philosophy of mathematics", North Holland, Amsterdam (1967) p. 43]
http://www.amazon.de/Problems-PhilosophyMathematic/dp/0444534113/ref=sr_1_cc_1?s=aps&ie=UTF8&qid=1369999937&sr=1-1catcorr&keywords=0444534113#_
§ 278 If, for example, our set theory includes sufficient large cardinals, we might count
Banach–Tarski as a good reason to model physical space [...] From this I think it is clear that
considerations from applications are quite unlikely to prompt mathematicians to restrict the range
of abstract structures they admit. It is just possible that as-yet-unimagined pressures from
science will lead to profound expansions of the ontology of mathematics, as with Newton and
Euler, but this seems considerably less likely than in the past, given that contemporary set
theory is explicitly designed to be as inclusive as possible. More likely, pressures from
applications will continue to influence which parts of the set-theoretic universe we attend to, as
they did in the case of Dirac’s delta function; in contemporary science, for example, the needs of
quantum field theory and string theory have both led to the study of new provinces of the settheoretic universe {{with negative result. There is no meaningful application of a meaningless
theory possible}}.
[Penelope Maddy: "How applied mathematics became pure", Reviev Symbolic Logic 1 (2008) 16
- 41]
§ 279 Attempts to create "pure" deductive-axiomatic mathematics have led to the rejection of
the scheme used in physics (observation - model - investigation of the model - conclusions testing by observations) and its substitution by the scheme: definition - theorem - proof. It is
impossible to understand an unmotivated definition but this does not stop the criminal
algebraists-axiomatisators. For example, they would readily define the product of natural
numbers by means of the long multiplication rule. With this the commutativity of multiplication
becomes difficult to prove but it is still possible to deduce it as a theorem from the axioms. It is
then possible to force poor students to learn this theorem and its proof (with the aim of raising
the standing of both the science and the persons teaching it). It is obvious that such definitions
and such proofs can only harm the teaching and practical work.
It is only possible to understand the commutativity of multiplication by counting and re-counting
soldiers by ranks and files or by calculating the area of a rectangle in the two ways. Any attempt
to do without this interference by physics and reality into mathematics is sectarianism and
isolationism which destroy the image of mathematics as a useful human activity in the eyes of all
sensible people.
[V.I. Arnold: "On teaching mathematics" (1997) Translated by A.V. Goryunov]
http://pauli.uni-muenster.de/~munsteg/arnold.html
§ 280 There is a concept which is the corruptor and the dazzler of the others. I do not speak of
Evil, whose limited empire is ethics; I speak of the infinite.
[Jorge Luis Borges: "Los avatares de la tortuga", translated by by BombaMolotov: "The Avatars
of the Tortoise"]
http://bombamolotov.deviantart.com/art/The-Avatars-of-the-Tortoise-87015348
§ 281 So if actual infinities exist then there cannot be any discrete computational foundation to
reality but so far no actual infinites have ever been discovered.
[John Ringland]
http://www.anandavala.info/TASTMOTNOR/Infinity.html
§ 282 I think, second order categoricity results are deceiving: they serve only to puzzle ordinary
mathematicians who do not know enough logic to distinguish between first order and second
order methods. One can say humorously, while first order reasonings are convenient for proving
true mathematical theorems, second order reasonings are convenient for proving false
metamathematical theorems.
[L. Kalmár: "On the role of second order theories" in Imre Lakatos (ed.): "Problems in the
philosophy of mathematics", North Holland, Amsterdam (1967) p. 104]
http://www.amazon.de/Problems-PhilosophyMathematic/dp/0444534113/ref=sr_1_cc_1?s=aps&ie=UTF8&qid=1369999937&sr=1-1catcorr&keywords=0444534113#_
§ 283 One says that one quantity is the limit of another quantity, when the second can
approach the first more closely than any given quantity, however small, without the quantity
approaching, passing the quantity which it approaches; so that the difference between a quantity
and its limit is absolutely inassignable. [...]
The theory of limits is the basis of the true metaphysics of differential calculus. [...] Properly
speaking, the limit never coincides, or is never equal to the quantity of which it is the limit; but it
is approached more and more, and can differ by as little as one wants. The circle, for example,
is the limit of the inscribed and circumscribed polygons; because it never merges with them,
though they can approach it ad infinitum. This notion can serve to clarify many mathematical
propositions.
[Jean Le Rond d'Alembert, Jean de la Chapelle: "Limite", Encyclopédie ou Dictionnaire
raisonné des sciences, des arts et des métiers, vol. 9. Paris (1765) p. 542, translated by Jeff
Suzuki: "Limit", The Encyclopedia of Diderot & d'Alembert Collaborative Translation Project, Ann
Arbor: MPublishing, University of Michigan Library (2012)]
§ 284 Crudely put, a potential infinite is a collection which is increasing toward infinity as a limit,
but never gets there. Such a collection is really indefinite, not infinite. The sign of this sort of
infinity, which is used in calculus, is ¶. An actual infinite is a collection in which the number of
members really is infinite. The collection is not growing toward infinity; it is infinite, it is
"complete". The sign of this sort of infinity, which is used in set theory to designate sets which
have an infinite number of members, such as {1, 2, 3, ...}, is ¡0. Now [...] an actually infinite
number of things cannot exist. For if an actually infinite number of things could exist, this would
spawn all sorts of absurdities.
[W. L. Craig: "The Existence of God and the Beginning of the Universe", Truth: A Journal of
Modern Thought 3 (1991) 85-96]
http://www.leaderu.com/truth/3truth11.html
§ 285 In this article, I argue that it is impossible to complete infinitely many tasks in a finite time.
A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task
remaining, when tasks are done 1-by-1. I suggest that the only way to deny this premise is by
begging the question, that is, by assuming that supertasks are possible.
1 By definition, completing infinitely many tasks requires getting the number of tasks remaining
down to 0.
2 If tasks are done 1-by-1, then the only way to get to 0 tasks is from 1 task, because if more
than 1 task remains, then performing a task does not leave 0 tasks. (This reasoning holds in
both the finite and infinite cases.)
3 When infinitely many tasks are attempted 1-by-1, there is no point at which 1 task remains.
4 Then from 2 and 3, there is no point at which 0 tasks remain.
5 Then from 1 and 4, it is not possible to complete infinitely many tasks.
[Jeremy Gwiazda: "A proof of the impossibility of completing infinitely many tasks", Pacific
Philosophical Quarterly, 93,1 (March 2012)
http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0114.2011.01412.x/full
Therefore it is not possible to enumerate all rational numbers (always infinitely many remain) by
all natural numbers (always infinitely many remain) or to traverse the lines of a Cantor list
(always infinitely many remain).
§ 286 Cantor completely contradicted the Aristotelian doctrine proscribing actual, “completed”
infinities, and for his boldness he was rewarded with a lifetime of controversy, including
condemnation by many of the most influential mathematicians of his time. This reaction stifled
his career and may ultimately have destroyed his mental health. It also, however, gained him a
prominent and respected place in the history of mathematics, for his ideas were ultimately
vindicated, and they now form the very foundation of contemporary mathematics.
[Math Academy Online: "You can't get there from here"]
http://www.mathacademy.com/pr/minitext/infinity/
{{No, Canotor's delusions have been contradicted. Alas matheologians refuse to understand
these contradictions. Here is an example:}}
Construct a Cantor-list containing all rational numbers of the unit interval. Replace the diagonal
digits ann by dn in the usual way to obtain the anti-diagonal d. Beyond the n-th line there are f(n)
rational numbers the first n digits of which are the same as the first n digits d1, d2, d3, ..., dn of
the anti-diagonal.
f(n) is infinite for every n. So we can safely say that it is possible to find at least n duplicates of
d1, d2, d3, ..., dn in entries below line n. Define for every n the sequence g(n) = 1/n. g(n) has limit
0 in analysis. So in the limit there are infinitely many duplicates of the anti-diagonal d1, d2, d3, ...
in the list. [user81183, June 2013]
{{This shows that infinitely many numbers are in the list that up to every digit are identical to
the anti-diagonal. Two numbers that are identical up to every digit are identical - in anlysis. But a
student of set theory cannot risk to understand that:}}
"I still don't understand the relevancy of these functions to the 'anti-diagonal'". [Asaf Karagila,
June 2013]
http://math.stackexchange.com/questions/412757/what-is-the-difference-between-these-limitsin-set-theory-and-analysis
§ 287 Cantor's transfinite universe became the infinite paradigm during the 20th Century. This
affected educational studies, which tended to view children's responses against Cantorian ideas.
Robinson's non-standard universe (Robinson, 1966) is equally authoritative (though not as well
known) and it is a different paradigm. It offers researchers a release from a single paradigm and
allows them to interpret children's ideas with reference to children's ideas instead of with
reference to Cantorian ideas.
[J. Monaghan: "Young peoples' ideas of infinity", Educational Studies in Mathematics 48,2
(2001) 239-257]
§ 288 Here are the differences in the premises which lead to differences in the results of
mathematics and matheology.
Matheology requires:
1) The Binary Tree
0.
/ \
0
1
/ \ / \
0 10 1
...
containing all rational numbers of the unit interval also contains all irrational numbers. If the
rationals are written in the usual manner this is not the case.
2) The triangle constructed in 3-symmetry is equilateral.
d
dc
dac
dbbc
...
If however, the triangle is constructed such that always one and the same side is expanded,
then it loses 3-symmetry "in the limit".
a
bb
ccc
...
3) For the union of the sequence of unions of preceding sets
»({1}, {1, 2}, {1, 2, 3} , ..., {1, 2, 3, ..., n}) = {1, 2, 3, ..., n}
equality holds - but not "in the limit".
In mathematics all these premises lead to different results:
1) The Binary Tree containing all rational numbers of the unit interval does not contain any
irrational number (it does not even contain periodic rationals).
2) The triangle constructed in 3-symmetry is and always remains equilateral.
3) For the union of the sequence of unions of preceding sets equality holds always.
§ 289 According to Fontenelle, none of the geometers who had invented or employed the
calculus of infinity had given a general theory of it; that is what he proposed to do {{in his
Élémens de la géométrie de l’infini (1727)}} ... There was a great deal of discussion in the
scientific community about this work, in which mathematicians found numerous paradoxes.
Johann I Bernoulli, for example, in his correspondence with Fontenelle allowed his criticisms to
show through his praise: he did not understand what was meant by finis indéterminables.
Fontenelle attempted to defend his theory and above all his distinction between metaphysical
infinity and geometric infinity: one must ignore the metaphysical difficulties in order to further
geometry, and the finis indéterminables ought to be considered “as a type of hypothesis
necessary until now in order to explain several phenomena of the calculus” (letter to Johann I
Bernoulli, 29 June 1729). “The orders of infinite and indeterminable quantities, like the
magnitudes that they represent, are only purely relative entities, hypothetical and auxiliary.
["Fontenelle, Bernard Le Bouyer (or Bovier) De", Complete Dictionary of Scientific Biography
(2008) Encyclopedia.com. 15 Jun. 2013]
http://www.encyclopedia.com/doc/1G2-2830901469.html
According to Cantor none of the geometers who had invented or employed the infinite had given
a correct theory of it ...
§ 290 [...] the future is only a potential infinite; one can keep adding future time to it infinitely,
but it is never a complete infinite set. Whereas an infinite past consists of an actual infinite
number of events that have occurred.
[Shrunk: "Can an actual infinite exist?", 17 Aug. 2011]
http://www.rationalskepticism.org/mathematics/can-an-actual-infinite-exist-t24806.html
Therefore there is no infinite past. Somewhen it must have done the step from finity to infinity.
But that step never happens.
§ 291 Only someone who (like the intuitionist) denies that the concepts and axioms of classical
set theory have any meaning (or any well-defined meaning) could be satisfied with such a
solution {{undecidability of the continuum hypothesis}}, not someone who believes them to
describe some well-determined reality. For this reality Cantor's conjecture must be either true or
false, and its undecidability from the axioms known today can only mean that these axioms do
not contain a complete description of this reality; [...] not one plausible proposition is known
which would imply the continuum hypothesis. Therefore one may on good reason suspect that
the role of the continuum problem in set theory will be this, that it will finally lead to the discovery
of new axioms which will make it possible to disprove Cantor's conjecture.
[Kurt Godel: "What is Cantor's Continuum Problem?", The American Mathematical Monthly, 54,9
(1947) p. 520, 524]
http://www.personal.psu.edu/ecb5/Courses/M475W/Readings/Week06IntoTheTwentiethCentury-108/Supplementary/What%20is%20Cantor's%20Continuum%20Problem,%20by%20Kurt%20God
el.pdf
§ 292 Ultrafinitism does not mean confining mathematics to a segment of the natural numbers,
or to a particular hereditarily finite set [...] Instead, ultrafinitism looks for nonclassical objects, and
it looks to nature -- the world of natural appearances or phenomena -- for them.
[Robert Tragesser: "FOM: Part I: Ultrafinitism, Naturalism, Vagueness", 10 April 1998]
http://www.cs.nyu.edu/pipermail/fom/1998-April/001825.html
§ 293 [...] parallel considerations would force us to conclude, not merely that a series of
discrete, successive events must have a first member, but also that such a series must have a
final member. Anyone who thinks that an end-less series of events is possible must therefore
reject this popular line of argument against the possibility of an actual infinite.
[Wes Morriston: "Beginningless Past, Endless Future, and the Actual Infinite"]
http://www.academia.edu/647482/Beginningless_Past_Endless_Future_and_the_Actual_Infinite
Endless for ever or endless at the moment are two different things. The time passed since the
big bang is not endless. The time to pass may be endless. But the passed time will never report
"infinity reached".
729
§ 294 Does the Bernays' number 67257 actually belong to every set which contains 0 and is
closed under the successor function? The conventional answer is yes but we have seen that
there is a very large element of fantasy in conventional mathematics which one may accept if
one finds it pleasant, but which one could equally sensibly (perhaps more sensibly) reject.
[R. Parikh: "Existence and feasibility in arithmetic", Journal of Symbolic Logic 36 (1971) 494-508]
§ 295 When God, at the end of all time, will check what of his creation has been worthwile, he
will also consider the set of natural numbers that ever have been used by his creatures. And he
will find that only a very small subset has been applied. (This Idea goes back to Borel, cp. § 80.)
For every usable number we have a finite set of predecessors and an infinite set of ¡0
successors. So there is no usable natural number behind some borderline, although that border
line cannot be determined yet.
Is it, in principle, possible to find circumstantial evidence for the existence of the ¡0
inaccessible numbers - in order to satisfy platonists like Gödel? Or is postulating them by the
axiom of infinity the only way to lay hold of them?
§ 296 The following three characteristics of the horizon are now important for our theme.
Firstly, we do not understand the horizon as the boundary of the world, but as a boundary of our
view. So the world continues even beyond the horizon. Secondly, the horizon is not some line
drawn and fixed in the world but it moves depending on the view in question, specifically on the
degree of its sharpness. The further we manage to push the horizon, the sharper the view.
Thirdly, for a phenomenon situated in front of the horizon, the closer it is to the horizon, the less
definite it is.
[P. Vopenka: "The philosophical foundations of alternative set theory", International Journal Of
General System 20,1 (1991) 115-126.]
§ 297 What would correspond more to the spirit of physics would be a mathematical theory of
the integers in which numbers, when they became very large, would acquire, in some sense, a
"blurred" form and would not be strictly defined members of the sequence of natural numbers as
we consider it. The existing theory is, so to speak, over-accurate: adding unity changes the
number, but what does the addition of one molecule to the gas in a container change for the
physicist? If we agree to accept these considerations even as a remote hint of the possibility of a
new type of mathematical theory, then first and foremost, in this theory one would have to give
up the idea that any term of the sequence of natural numbers is obtained by the successive
addition of unity - an idea which is not, of course, formulated literally in the existing theory, but
which is provoked indirectly by the principle of mathematical induction. It is probable that for
"very large" numbers, the addition of unity should not, in general, change them (the objection
that by successively adding unity it is possible to add on any number is not quoted, by force of
what has been said above).
[P.K. Rashevskii: "On the dogma of the natural numbers", Russian Mathematical Surveys 28,4
(1973) 143-148]
Compare the pocket calculator with 1010 + 1 = 1010.
And forget Cantor's impracticable ω + 1 > 1 + ω.
§ 298 The scheme of construction of a mathematical theory is exactly the same as that in any
other natural science. First we consider some objects and make some observations in special
cases. Then we try and find the limits of application of our observations, look for counterexamples which would prevent unjustified extension of our observations onto a too wide range of
events [...].
I even got the impression that scholastic mathematicians (who have little knowledge of
physics) believe in the principal difference of the axiomatic mathematics from modelling which is
common in natural science and which always requires the subsequent control of deductions by
an experiment.
Not even mentioning the relative character of initial axioms, one cannot forget about the
inevitability of logical mistakes in long arguments (say, in the form of a computer breakdown
caused by cosmic rays or quantum oscillations). Every working mathematician knows that if one
does not control oneself (best of all by examples), then after some ten pages half of all the signs
in formulae will be wrong and twos will find their way from denominators into numerators.
The technology of combatting such errors is the same external control by experiments or
observations as in any experimental science and it should be taught from the very beginning to
all juniors in schools.
{{Great scepticisms appears appropriate when something simultaneously is asserted to be
attainable and simultaneously is asserted to be unattainable. Experimental verification of
matheological results is not possible. You are depending on the statements of logicians.
By the way, according to Thomas Mann, literati are people who find writing difficult.
Transferred to logic and set theory, logicians are people who find thinking difficult.}}
[V.I. Arnold: "On teaching mathematics" (1997) Translated by A.V. Goryunov]
http://pauli.uni-muenster.de/~munsteg/arnold.html
§ 299 There are many examples of "soritic properties" for which mathematical induction does
not hold ("number of grains in a heap", "number that can be written down with pencil and paper
in decimal notation", "macroscopic number", ... ), but mathematicians traditionally take no
account of them in their theories, with the excuse that such properties are vague. We present
here a mathematically rigorous theory in which a soritic property is put to constructive use.
[Karel Hrbacek, Olivier Lessmann, Richard O'Donovan: "Analysis with ultrasmall numbers",
Amer. Math. Monthly 117,9 (2010) 801-816]
§ 300 Let F(n) = {1, 2, 3, ..., n} be the n-th finite initial segment of the set of natural numbers.
Then the sequence (an) defined by an = min{ 10100, |Ù \ F(n)| } = 10100 has limit 0 in matheology
but limit 10100 in mathematics. So matheology broke the bands of mathematics.
§ 301 MadOverflow
Richard Dawkins' phrase "I regard Islam as one of the great evils in the world, and I fear that we
have a very difficult struggle there" would be very hard to defend in Pakistan. Probably Dawkins
would be deleted. That would prove him right. But he hardly would enjoy it.
I regard matheology [i.e., the belief in finished infinity and in numbers which cannot be (finitely how else?) defined, which cannot be used as individuals in mathematics and elsewhere and
which cannot be known at all] as one of the great evils in academia, and I fear that we have a
very hard time to clear the brains from that confusion. Fortunately this leads only to deletion of
my explanatory contributions to MathOverflow, a forum of "research-level mathematics" where
quite a lot of strange things happened. Why? Well, one of the moderators has already provided
the answer: "This is a field particularly prone to incompetence so severe as to make recognizing
one's own incompetence impossible." [Scott Morrison, Jul 11 at 16:56]
http://meta.mathoverflow.net/questions/435/the-association-bonus
In order to dismiss accusations of incompetence let me mention that I wrote three text books of
mathematics, one of them available in seventh edition meanwhile and another one having
acquired bestseller status with a renowned publisher. Let me mention further that I have
collected more than 1000 reputation points in MatheOverflow within less than three months of
activity - just for fun and of course under pseudonym because my signed contributions usually
have not a long lifetime in that forum.
I will report here in sci.logic 50 paragraphs of those events which really are so strange that one
could believe the reports stem from a forum called MadOverflow.
§ 302 Can family planning change the equipartition of boys and girls?
In a country in which people only want boys every family continues to have children until they
have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the
proportion of boys to girls in the country?
http://www.businessinsider.com/answers-to-15-google-interview-questions-that-will-make-youfeel-stupid-2009-11#in-a-country-in-which-people-only-want-boys-3
Google claims that a 50/50 population will be maintained. Family planning cannot influence birthprobability.
The easiest proof is this: Imagine that boys and girls to be born come on a great conveyor belt,
distributed at a 50/50-ratio. In case a woman wants to have a child she takes the next one from
that conveyor belt. So it is clear that the ratio will not depend on the question whether or not the
woman had already born one or more girls.
This simple fact is understood by all biologists and all other scientists which I know of - alas not
by all users of MathOverflow, since it is not "research-level" mathematics. In MathOverflow
things look different. Douglas Zare in a "great answer" has calculated the percentage of girls,
G/(G+B) per family, and has found that the average of this ratio deviates from the correct answer
and depends on the number of couples involved.
http://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-onlywant-boys-closed
Instead, however, doubting his result, he writes: “So, for a large population such as a country,
the official answer of 1/2 is approximately correct, although the explanation is misleading. In
particular, for 10 couples, the expected percentage of girls is 47.51% contrary to what the official
answer suggests.” This "great answer” is not even approximately correct (simple example: if you
want to know whether the sum of numerators is equal to the sum of denominators, you cannot
calculate the average of fractions) but it has been approved by far more than 100 "professional”
mathematicians and logicians in the self-proclaimed elite forum MathOverflow.
Zare is strongly supported by Steven Landsburg, who even accepts bets over 5000 $ and has
accused the Harvard string theorist Lubos Motl (as well as others) of “not knowing enough
mathematics”. Landsburg asks in his blog: Are You Smarter Than Google? There’s a certain
country where everybody wants to have a son. Therefore each couple keeps having children
until they have a boy; then they stop. What fraction of the population is female?
http://www.thebigquestions.com/2010/12/21/are-you-smarter-than-google/
Landsburg explains the terms of the bet: The best grad studens of the top 10 elite universities
should write simulations to get results with reliable statistics. Obviously he has not recognized
that every random binary sequence will do which can be easily obtained just by throwing a coin.
A mathematician calling himself Monty has figured that out: Consider a very long random
sequence of bits. If only partial sequences ending with a 1 (put 1 for boy, 0 for girl) are taken,
like in this sequence: 01|01|001|1|001|0001|1|01|1|01|01|1|1| then this sequence would simulate
many countries with only one couple each {{as well as one country with many couples and every
other mixture}}. Every sequence stops when a 1 is returned. The average of the ratios G/(G+B)
is in fact as caculated by Douglas Zare about 31 %.
Alas, giving this answer does not mean to be smarter than Google! It answers a completely
different question. The question asks for the expected ratio of all girls (or all boys) to all girls and
boys, and not for the average of these ratios per family. Zare could have learned the difference
from Monty, who was about to ask why Zare had used G/(G+B) per family and not B/G per
family. In the latter case the average would have been infinity, because those families who
stopped after having a boy as the first child contribute B/G = ¶. Douglas Zare and Steven
Landsburg and their over one hundred supporters then would have learned that they are far
from being smarter than Google. But Monty's answer had already been deleted before he could
provide his explanation.
§ 303 The digits of π
I know it's likely that, given a finite sequence of digits, one can find that sequence in the digits of
π, but is there a proof that this is possible for all finite sequences? Or is it just very probable?
[sep332, March 2010]
http://mathoverflow.net/questions/18375/is-there-any-finitely-long-sequence-of-digits-which-isnot-found-in-the-digits-of/132035#132035
The answer is no: It is neither provable nor even probable because it is provably impossible to
find out.
The set of n-digit-strings contains at least one string that has Solomonoff- or Kolmogorovcomplexity n. See http://people.cs.uchicago.edu/~fortnow/papers/kaikoura.pdf
The complexity of a program is limited by the ressources available in the memory of the
computer. The accessible universe contains an upper limit of less than 10100 bit. Therefore it is
even impossible, and will remain so forever, to define "every finite string of bits" and it would be
unscientific speculation to argue whether all finite strings are in the decimal representation of π.
It can even be excluded that a natural number with a complexity of 10100 bit can be defined and
applied, let alone the corresponding digit of the decimal expansion of π. [AGreen, May 2013]
This answer of course is not what "research-level-professionals" like to enjoy. It would hurt their
belief in countable and uncountable sets if they realized that they have not even the chance to
use one infinite set in mathematics, i.e., such that every element has its own name and identity.
They confuse the possibility to calculate the limit of an infinite sequence with the impossibility to
calculate infinitely many terms of that sequence. Therefore the answer got a lot of downvotes
and finally was deleted.
And Douglas Zare, the genius who had proven that family planning can change the ratio of girls
in a country (see § 302), again shows his superiority by stating:
en.wikipedia.org/wiki/Champernowne_constant can be proven to be normal. I find it very odd
that 3 people have voted up a supposed answer with a huge error like this. [Douglas Zare, May
2013]
Obviously Zare does not or cannot distinguish between the two propositions: "There exist normal
numbers like the Champernowne constant" and "all irrational numbers including π are normal".
§ 304 Everything is consistent in any case
"It is impossible in principle to well-order the reals in a definable manner."
To be more precise, the belief I am talking about is the belief that well-orderings of the reals
are provably chaotic in some sense and certainly not definable. For example, the belief would be
that we can prove in ZFC that no well-ordering of the reals arises in the projective hierarchy (that
is, definable in the real field, using a definition quantifying over reals and integers).
This belief is relatively common, but false, if the axioms of set theory are themselves
consistent {{and that can be taken for granted since}} the idea nevertheless has a truth at its
core, which is that although it is consistent that there is a definable well-ordering of the reals (or
the universe), it is also consistent that there is no such definable well-ordering. {{That's what I
like with ZFC. There are no inconsistencies!}}
[Joel David Hamkins (2010), giving in MathOverflow an example of common false beliefs in
mathematics]
http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematicsclosed
§ 305 Interpretation of Limits - Limits of Interpretation?
Consider this sequence
21.
2.1
432.1
43.21
6543.21
654.321
87654.321
8765.4321
1098765.4321
109876.54321
1211109876.54321
121110987.654321
.. .
Its limit in analysis is infinity.
However, we can also interpret this sequence as a supertask of set theory. We put in always two
natural numbers and remove always one natural number. The input is added on the left-hand
side, the the output passes the point for the right-hand side. Then “after having completed” the
supertask, all numbers have been removed from the left-hand side. Therefore the limit in set
theory is less than 1.
After appearing for a short time in MathOverflow and MathStackExchange, this observation got
deleted. But there has been an unmasking comment by the user Michael Greinecker: "There is
no contradiction. Just a somewhat surprising result. And there is no a apriori reason why one
should be able to plug in cardinal numbers in arithmetic formulas for real numbers and get a
sensible result."
Well, we start with only positive integers, i.e., natural numbers or positive finite cardinal
numbers. If however, we decide to interpret their concatenation as a real number, then we get a
different limit. In mathematics limits do not depend on an unspecified “interpretation”. In
matheology, however, facts (or what is taken as facts) are different.
§ 306 The deal with the devil
In the paper http://arxiv.org/PS_cache/math/pdf/0212/0212047v1.pdf J.D. Hamkins describes a
deal with the devil after which you have lost all your money.
Thus, on the first transaction he accepts from you bill number 1, and pays you with bills
numbered 2 and 4. On the next transaction he buys from you bill number 2 (which he had just
paid you) and gives you bills numbered 6 and 8. Next, he buys bill number 3 from you with bills
10 and 12, and so on. When all the exchanges are completed, what do you discover? You have
no money left at all! The reason is that at the nth exchange, the Devil took from you bill number
n, and never subsequently returned it to you.
I think I can contribute an idea that saves you at least one bill. For that sake simply require that
you never hand a bill to the devil unless you are in possession of at least another bill. This
condition, laid down by an additional clause in the contract, must never be violated and remains
valid in eternity. Or put it the other way round: In case you would go bancrupt, simply refuse to
hand him the last bill you have. (There is no last natural number and of course no "time ω"
where everything including the Cantor-diagonal could happen, but before you are in possession
of zero bills, you must have been in possession of at least one - and you never lose bills by
handing out more than a single one.) This will prevent your total bancrupt, won't it?
It will not, however, prevent the total bancrupt of some Cantor-cranks here around. They
cannot defend their ridiculous position that devil breaks contract. He never does. Therefore they
must delete this text as quickly as possible in order to keep the respect of their poor students.
[Devil, 9 July 2013]
Dear Devil, I am voting to close since this is not a real question. Your statement is: if you change
the rules of the game, then you also change the outcome. I don't think anyone disagrees with
that principle, it's just that there is no research level math in your "question". [Vidit Nanda, 9 July
2013]
Sorry, I don't change the rules of the game. This condition is always in every super task implicitly
realised. No natural number is without a successor. Therefore it is nonsense to talk about all
numbers or to enumerate all rationals. [Devil, 9 July 2013]
It seems that most set theorists do not realize that there is no chance to issue a last natural
number. The reason is that there is no last one. But that also implies that you never can issue all
natural numbers. Does this fact restrict mathematics? Some seem to think so:
And also, apparently, we cannot walk from here to there.
en.wikipedia.org/wiki/Zeno's_paradoxes
[Joel David Hamkins, 9 July 2013]
If this supertask has to do anything with the arrival of Zeno’s arrow, i.e., with reality, then the
result of the deal (like anything in reality) must not depend on how the bills are enumerated.
Then the dealer with the devil can withhold an arbitrary amount of numbers and will acquire so
an infinity of wealth.
No, reality is not saved by matheology!
§ 307 Requires set theory accepting the result of a super task?
Littlewood's super-task has been vividly illustrated in this question:
http://mathoverflow.net/questions/7063/a-problem-of-an-infinite-number-of-balls-and-an-urn
But not all mathematicians and very few scientists and philosophers accept that "finally the urn is
empty". One user wrote:
I am a mathematician and I don't accept "0 balls" as "the" answer (because of implicit continuity
assumptions). There is ample evidence at this page that I am not alone in that. [Victor Protsak
26 May 2010]
That is certainly true. However, mathematicians who deny the null-result but accept the notion of
countability are not aware of the fact that every enumeration of a countable set is a super task.
Counting the positive rational numbers, for instance, yields the following super task: In the nth
step fill in all rationals between n-1 and n and, if not yet residing in the urn, also the rational
number qn to be enumerated by the natural number n. Take off the rational number qn. Go to
step n+1. Here we have in every step infinitely many numbers going in and only one going out.
Fraenkel has illustrated this fact by the well known story of Tristram Shandy.
Therefore all mathematicians who don't accept the null-result should not accept countability as a
sensible notion, let alone uncountability.
Those however, who accept the null-result, should try, for a moment, to imagine the possibility to
delay the enumeration a bit by detaining always a natural number until another one is available
http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,42,Folie 42.
Applied to the deal with the devil this method prevents the total bankrupt of the player, applied to
the enumeration of the positive rational numbers, this method prevents their complete
enumeration.
The most natural property of the natural numbers is that it is never possible to use a last one
such that none remains unused. And here "never" means never, and not "in some infinity". But if
things can happen, then:
- The urn: For every step n the contents increases. "At infinity" the urn is empty.
- The Cantor-list: Up to every line n, the diagonal is not contained. "At infinity" the diagonal is
there.
No axioms available that distinguish between these two cases.
My rational question: Is there a way (other than pure belief in the impossible) to circumvent the
fact that the natural numbers never can be exhausted?
This question acquired 7 downvotes in MathOverflow and was deleted soon. Matheologians
must be very afraid of rational thought.
§ 308 Others than all?
On 3 July 2013 Rainbow asked in MathOverflow: Every path in a complete infinite tree
represents a real number of the unit interval between 0 and 1. (Some rational numbers have two
representations.) Assume that you have a can of red paint and that you can colour an infinite
path of the tree with one can. Assume further that you get another can of red paint for every
node that you are colouring for the first time. Then you will first accumulate an infinity of cans of
red paint, but you will nevertheless not be able to colour all paths, since there are uncountably
many paths in the tree (and you can win only countably many cans of red paint). So there
remain uncountably many paths uncoloured in the first run. Start with another colour, say green.
Also with green paint you will not finish. How many different colours will be required?
Andreas Blass replied: "You seem to be ignoring the fact that, after you have colored a
countable family of pathes, say P0, P1, …, Pn, …, there may be other paths Q that are not on this
countable list but have, nevertheless, had all their nodes and edges colored."
I cannot know what Rainbow thinks of this answer. I, for my person, think, that “other” paths
must be different from the paths of the family of coloured paths in order to be “other”. So there
must be at least one node differing from the nodes of a coloured path. But if we leave a coloured
path, we can only do so by using a node of another path that is also a coloured path (since every
path has been coloured). That means we never leave and always remain on a coloured path. It
seems to me that the “fact” Andreas Blass ist talking about refers to a list of numbers. But in the
tree we have not a list. Nevertheless there are provably only countable many paths. And all are
coloured red.
This may be the reason why Rainbow’s answer has been deleted soon. Nobody would know
about the dyslogic applied by Andreas Blass, unless I had, by chance, copied all this stuff.
§ 309 Two identical sequences with very different limits
When constructing the Binary Tree node by node, we need ¡0 steps. Compare for instance
http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT. In the limit the complete Binary Tree
contains 2¡0 paths representing, among others, irrational numbers. Therefore the cardinality of
the set of irrational numbers constructed up to step n can be understood as a sequence f(n) = 0
for every n with limit 2¡0. My question: Has this been observed and discussed in literature
already?
And may I add a second question: If we construct, line by line, a Cantor-list containing all rational
numbers, then the cardinality of the set of irrational numbers contained in that list up to line n is a
sequence g(n) = 0 for every n with limit 0.
Is there any result in research level mathematics why two zero-sequences show such blatantly
different behaviour?
This question got 7 downvotes in MathOverflow but also 3 upvotes. Are there in fact real
mathematicians remaining in this Gomorrah of matheology?
§ 310 How to distinguish between the complete and the incomplete infinite Binary Tree?
How can we distinguish between that infinite Binary Tree that contains only all finite initial
segments of the infinite paths and that complete infinite Binary Tree that in addition also contains
all infinite paths?
Let k denote the kth level of the Binary Tree.
The set of all nodes of the Binary Tree defined by the union of all finite initial segments of the
sequence of levels »0<k<¶ (0, 1, 2, ..., k) contains (as subsets) all finite initial segments of all
infinite paths. Does it contain (as subsets) the infinite paths too?
How could both Binary Trees be distinguished by levels or by nodes?
This question got 7 downvotes in MathOverflow but also 2 upvotes. Are there in fact real
mathematicians remaining in this Gomorrah of matheology?
I note only three further comments of mine:
I cannot understand why some cranks here rate my question negative? I am sure, they have
never pondered about it and cannot answer it. Fortunately some mathematicians seem to have a
larger horizon than their standpoint.
ZF minus infinity is the smaller tree here. But if the existence of the larger tree is claimed and
the Axiom of Infinity is assumed, then it must be in accordance with mathematics. Otherwise it is
as invalid as the axiom stating the existence of 10 different natural numbers with sum 10. And I
am asking whether someone can support the Axiom of Infinity as sensible. A level ω is certainly
not the solution - at least as far as mathematics is concerned.
Here is a simpler explanation: Try to distinguish the set of all terminating fractions and the set
of all real numbers by digits. And if you cannot, try to explain why Cantor's diagonal argument is
said to apply to actually infinite decimal representations only. Perhaps even try to understand,
why I claim that everything in Cantor's list happens exclusively inside of finite initial segments,
such that, in effect, Cantor proves the uncountability of a countable set.
§ 311 How many is ω? Is it larger than ω?
If we count the digits of a real number, we get the result ω. So for every rational approximation
pn we get a digit where it differs from π. On the other hand there are also ω rational
approximations which, up to n, do not differ from π. If we consider this in the finite case, say 3,
then 123 cannot differ from all of its three finite approximations 1, 12, and 123. Any suggestions
how to remedy that gap in case of ω?
Edit: How can a real number be distinguished from all its rational approximations? It can't. Not
by digits. But all rational approximations form a countable set. So the real numbers cannot be
defined other than by finite definitions, which form a countable set (in every speakable language
that can be used in mathematical discourse).
I am afraid the answer to this question is a bit above the "research level" prevailing here. But
who knows?
Questions that cannot be answered without confessing that transfinity is inconsistent are usually
deleted immediately - like the present one.
§ 312 The odd even number ω
The following question
http://mathoverflow.net/questions/21457/are-there-an-odd-number-of-even-numbers-closed
asked on 15 April 2010 at 13:23 by Alex Andronov got a surprising comment:
Yes, the first infinite ordinal ω is even, but other infinite ordinals, such as ω + 1 or ωω + 5 are
odd ordinals. The cardinality of the set of even natural numbers is ω which is even. In fact, under
AC, all infinite cardinals are even. [Joel David Hamkins, 15 April 2010]
Unfortunately, meanwhile not even rudiments of this question are available any longer.
§ 313 A remarkable result of matheology
If we colour every node of some path of the Binary Tree, say the path 0.000…, then we have
coloured this path:
0.
0
0
0
∂
But if we colour in addition to every node all its predecessors, then we have not coloured that
path:
0.
0.0
0.00
0.000
∂
This remarkable observation concerns also the definition of paths: If we define every node of
some path of the Binary Tree, say the path 0.0101010… = 1/3, then we have defined this path.
But if we define in addition to every node all its predecessors, then we have not defined this path
1/3.
The reason of the latter is that none of the finite initial segments of the binary representation of
1/3 is 1/3. The reason of the former is, that, although none of the bits finishes the binary
representation of 1/3, we nevertheless believe, inspired by Cantor, that a binary representation
of 1/3 exists and can be finished. (Of course the “we” in this text has been purest sarcasm.)
Perhaps because of such observations some matheologians are really afraid: “Wolfgang
Mückenheim is probably one of the most dangerous cranks out there. He has a professorship at
the University {{of Applied Sciences}} of Augsburg, Germany, where he is teaching physics and
mathematics!! Currently {{having started in 2003}}, he is teaching a lecture called "History of the
Infinite". This man does real damage.” [Michael Greinecker, 1 May 2012]
http://meta.mathoverflow.net/discussion/1353/nominalist-foundations-of-mathematics/
Thank you, Michael. I hope so because exactly that's my intention. Damaging the nonsense that
you Cantor-cranks call mathematics but that is merely a perversion of mathematics, i.e.,
matheology. (For a definition, see § 001.)
§ 314 Undefinable “reals”?
In order to define uncountably many “real” numbers, infinitely many bits or other letters of a
countable alphabet are required, but nobody can send or receive infinitely many letters. So you
need an uncountable alphabet. Since undefined letters are not suitable to carry information, the
letters of an uncountable alphabet have to be defined first, for sender and receiver. In order to
define uncountably many letters such that they can be used in mathematical discourse, you
need uncountably many sequences of bits. It is no secret that most of these sequences must be
infinite. But infinite sequences cannot be used (written, submitted, received) other than by their
finite names. So the infinite sequences must be defined by finite names constructed from a
countable alphabet . Alas, there are only countably many such finite names available. Therefore,
to commit a gross understatement, not all letters of an uncountable alphabet can have different
definitions. Every single definition has to define uncountably many different letters of the
uncountable alphabet. But letters with identical definitions are scarcely different, and to guess
which of uncountably many letters is meant by a certain definition, is a task that is impossible to
accomplish in mathematics. That is only possible in matheology – because of important nonwritten special information processed and submitted by means that remain unknown outside of
the inner circle of matheologians.
§ 315 Is subcountability as powerful as uncountability in confusing mathematicians?
Andreas Blass stated in MathOverflow (meta) that "eventually, most mathematicians came to
accept that definability should not be required, partly because the axiom of choice leads to nice
results, but mostly because of the difficulties that arise when one tries to make notion of
definability precise." (meanwhile deleted)
That is a real surprise to me. Which mathematicians accepted that and when? Have they been
summoned to a public meeting with voting about abolishing definitions like the astronomers
when Pluto was degraded?
Wouldn't a set with undefined elements contradict the Axiom of Extensionality: “If every element
of X is an element of Y and every element of Y is an element of X, then X = Y.” How could that
be decided for undefined elements?
Wouldn't a set with undefined elements contradict the Axiom of Choice? How could we choose
something that cannot be chosen, in spite of the axiom, because we cannot say or express in
any other way what we intend to choose? Blind choice - the foundation of matheology? It
resembles the attempt of brainless thinking.
But my actual question is this: I have heard of another solution. The set of finite definitions is
countable. That cannot be explained away, can it? But not every finite definition has a meaning.
In fact, if we refrain from using common sense, we cannot even define definability, let alone the
set of meaningful definitions. Therefore this set is not countable but subcountable - and if we
identify subcountability with uncountability, we have won and can continue to enjoy the nice
results of the axiom of choice.
Of course this question has been deleted soon.
§ 316 Does undefinable definability save matheology?
On 13 June 2013 user albino (meanwhile deleted) asked in MathOverflow about set theory
without the axiom of power set.
http://mathoverflow.net/questions/133597/what-would-remain-of-current-mathematics-withoutaxiom-of-power-set
I will report also the interesting subsequent discussion about definability.
The power set of every infinite set is uncountable. An infinite set (as an element of the power
set) cannot be defined by writing the infinite sequence of its elements but only by a finite
formula. By lexical ordering of finite formulas we see that the set of finite formulas is countable.
So it is impossible to define all elements of the uncountable power set. [albino]
You say that the powerset of an infinite set is questionable because it must have some
undefinable elements. You are presuming that the subsets of a set must all be distinguishable by
you, or some entity whose only access to powersets is through formal language. But why is such
an assumption warranted? What makes you think that a thing does not exist unless you can
define it? Is existence a personal belief? [Andrej Bauer]
I think that elements of a set must be distinguishable, as Cantor has put it. [albino]
{{I would say that the existence of personal beliefs is a personal belief. And I think that
undefinable elements cannot be applied. How would you apply an element that you cannot
define? What are elements good for that cannot be applied in mathematics? Matheology. But
unfortunately I came to late to take part in this discussion}}
Also, you are confusing "distinguishable" with "distinguishable by a formula". [Andrej Bauer]
{{He said so but refused to explain the difference. By the way, every distinction in mathematics
occurs by a finite formula.}}
I am afraid I have not got the meaning of your sentence you are confusing "distinguishable" with
"distinguishable by a formula". If you cannot get hold of a notion other than by a finite formula,
how would you distinguish two of them without a finite formula? [albino]
You have focused on this definability issue, but that's just not crucial. I heartily recommend that
you read Joel Hamkin's post that he linked to in the comments to the question. He explained
very well why definability is a deceptive notion. [Andrej Bauer]
{{I think that a person who distinguishes between "distinguishable" and "distinguishable by a
formula" but refuses to explain this, is a deceptive person.}}
… Lastly, concerning your remarks about definability, I refer you as I mentioned in the comments
to an answer I wrote to a similar proposal, which I believe show that naive treatment of the
concept of definability is ultimately flawed. [Joel David Hamkins]
Logicians, I have learned, take some premises and obtain some conclusions. But they do not
judge about truth or practical things. In my opinion "definability" is a practical notion. If I define
something and others are able to understand what I have defined, than that something is
definable (otherwise it may be undefinable or I am not good enough in defining). But with respect
to numbers things are easy. If I say π or e or 1/4, then these numbers are defined. And it is true,
in my opinion, that not more than countably many numbers can be defined by finite strings of
bits. [albino]
Albino, I'm not sure to which logicians you are referring with your first comment. Meanwhile, yes,
your remarks on definability are the usual naive position on definability. If you ever find yourself
inclined to mount a serious analysis of definability, however, then I would suggest that you talk
more with logicians. {{Really? Why are modern logicians despised like parias by all kinds of
scientists?}} In particular, I would point you toward the initial part of my paper on pointwise
definable models of set theory (de.arxiv.org/abs/1105.4597), where we deal with the "Math Tea
argument", which is essentially the argument you are advancing. [Joel David Hamkins]
I am sorry, but since Zermelo's “Beweis, daß jede Menge wohlgeordnet werden kann”, we know
that we have to distinguish between logical proofs and real proofs. If you had proven that all
numbers can be expressed with three digits, I would not believe you. And your proof is rather
similar. So I know it is not a real proof. Nevertheless I read your paper up to Theorem 4. It
reminds me of Zermelo's “If AC, then well-ordering is possible”. But I am not interested in that
kind of logical conclusions but only whether I can do it. [albino]
Oh, I'm very sorry to hear that you aren't interested in logical proof or logical conclusions. I'll
leave you alone, then, to undertake your own kind of proof activity. [Joel David Hamkins]
I do accept logical proof! I accept the logical proof that it is impossible to define more than
countably many objects including all numbers. This stands as solid as the proof that with three
digits you cannot define more than 999 natural numbers. I even accept non-constructive proofs
like Zermelo's, but not as deciding whether something can be constructed - as was Zermelo
original intention. (Compare Fraenkel who said that hitherto nobody could well-order the reals.)
To be short: If your proof is correct, then you have found a contradiction. [albino]
{{I don't think it is a good idea to ask logicians what can be done in reality. Zermelo was the first
to make a fool out of himself, when insisting and "proving" by insisting that every set can be wellordered, Hamkins will not be the last one. To put an axiom may be a good idea in order to find
out what can be thought - but not what can be done.}}
§ 317 Violation of inclusion monotony in infinite applications?
Ex Oriente Lux asked in MathOverflow: Consider the sequence of ordered subsets of natural
numbers, written below each other in form of a triangular matrix:
(1)
(2, 1)
(3, 2, 1)
...
By inclusion monotony we see that every number that is contained in the matrix is in one of its
lines together with all other numbers contained in this matrix. Each vertical row contains ¡0
natural numbers. Since each vertical row is a subset of the matrix, we find that ¡0 numbers are
contained in the matrix. From every horizontal line we see that only a finite set of numbers is in
that matrix, since there is no line with ¡0 natural numbers. If there are ¡0 natural numbers in the
matrix, they cannot be in one and the same line. This fact violates inclusion monotony. Is it
worthwile to accept ¡0 as a number larger than every natural number when it must be payed for
by the disadvantage that inclusion monotony can no longer be trusted?
This very simple question has been deleted soon, as usual if matheologians cannot answer
without unveiling a contradiction between mathematics and the idea of competed infinity. But the
result is without the least doubt: Either there are not "all" natural numbers, i.e., a complete set Ù
such that no further natural can be added, or Ù is contained in all lines of the list without being in
one line of the list although there are "all" lines such that none can be added.
This same effect appears in the Binary Tree, when asking for the tree containing only all finite
initial segments of the infinite paths. Some hold that such a tree contains automatically all infinite
paths nevertheless. But then the list
0.1
0.11
0.111
...
must also contain the number 1/9 (although it is in no single line, it must be there somehow). Or
they try to distinguish between the tree with and without "nodes at level ω". Compare § 198.
§ 318 Does the contents of a set depend on the notation?
Ex Oriente Lux asked in MathOverflow: Does the contents of a set depend on the notation of its
elements? The real numbers of the unit interval [0, 1] can be represented as paths (in a Binary
Tree) or as sequences of bits (in the usual manner).
Examples 1 = 0.111... = 0-1-1-1-..., or 1/3 = 0.010101... = 0-0-1-0-1-0-1-...
The set of all rational numbers of the unit interval can be enumerated. This sequence does not
contain any irrational numbers. But if the same set is written in the form of a Binary Tree, then all
irrational numbers sneak in, since it is impossible to construct the Binary Tree leaving out the
irrational numbers.
My questions: 1) How do all irrational numbers of the unit interval sneak into the Binary Tree that
is constructed only by paths representing rational numbers? Is there an intuitive picture saying
that at the end or slightly before or after the end uncountably many irrationals enter?
And 2) If we define the function f(rational x) = 1 and f(irrational x) = 0, are then the irrationals
also automatically in the set of rationals with f(x) = 1, like they are always in the Binary Tree,
such that the function f is 1 everywhere? Or are they not in this set like in a list of only rationals?
And 3) why is this as it is?
This great question quickly earned 6 downvotes and was deleted after three hours. So it must be
a very dangerous one.
§ 319 Is it possible to find such permutation of natural numbers that it cannot be a limit of finite
permutations? This question has been asked by kakaz on 9 March 2010 in MathOverflow.
http://mathoverflow.net/questions/17653/infinite-permutations
The main answer, provided by J.D. Hamkins on the same day, said no. It got 6 upvotes, only
one less than the question itself.
Another answer, supplied by Luitzen Egbertus Jan on 3 July 2013 got no upvotes but is correct
(well he had much more time to think it over). He said yes:
I think the limit of infinitely many permutations of the set of natural numbers Ù can be non
existent. Consider an enumeration of the rationals –: q1, q2, q3, q4, q5, q6, q7, ..., for instance
the classical enumeration given by Cantor.
Apply operation A: order the rationals pairwise such that q2k+1 and q2k+2 belong to a pair (q1,
q2), (q3, q4), (q5, q6), (q7, ... and order the rationals of the first n parentheses by their magnitude
such that the smaller number comes first. Enumerate them in the new order.
Apply operation B: order the rationals pairwise such that q2k and q2k+1 belong to a pair q1, (q2,
q3), (q4, q5), (q6, q7), ..., and order the rationals of the first n parentheses by their magnitude
such that the smaller number comes first. Enumerate them in the new order.
Repeat operations A and B alternatingly for infinitely incresing but always finite numbers n of
parentheses. The limit would be a well-ordering of all rational numbers by magnitude.
What is the meaning of “all”? Like in Cantor’s enumeration we can continue until we have
included any desired rational number that was fixed in advance. So if we agree that Cantor was
able to enumerate all rational numbers by showing that he can enumerate every desired finite
number of rationals such that the relative position of the enumerated rationals will remain the
same in every following step, we can be sure to have well-ordered by magnitude all rational
numbers because we can well-order every desired finite number of rationals such that the
relative positions of these n rationals will remain the same in every following step (and only after
fixing that we add another rational).
Since a well-ordering by magnitude is impossible, the corresponding limit of the natural numbers
does not exist.
§ 320 Sébastien Palcoux complained in MathOverflow:
I had problems with a question (closed in 9 min, 10 down votes and finally deleted) about
mathematicians, neurosis and all common stereotypes among non-mathematicians on that.
I warn you that I'm not talking about severe neurosis as schizophrenia (John Nash ...), it's not
at all my point.
My point is about mild neurosis, allegedly widespread (by non-mathematicians) among
mathematicians.
§ 321 Prime Mister asked in MathOverflow: Enumerate all rational numbers to construct a
Cantor-list. Replace the diagonal digits ann by dn in the usual way to obtain the anti-diagonal d.
Beyond the n-th line there are f(n) rational numbers the first n digits of which are the same as the
first n digits d1, d2, d3, ..., dn of the anti-diagonal.
f(n) is infinite for every n. So we can safely say that we can find n duplicates of the first n digits
d1, d2, d3, ..., dn of the anti-diagonal.
Define for every n the function g(n) = 1/n. In analysis g(n) has the limit 0. So in the limit there
are infinitely many replica of the diagonal in the list.
This question soon got deleted. The general tenor is: Although every digit of d and all its
predecessors are in a line (in fact in infinitely many lines) of the list, d is not in the list. But can d
be defined by more digits than all numbers in the list can contribute?
Now consider a list of all finite initial segments of Ù
1
1, 2
1, 2, 3
...
Applying the same reasoning as above, we can say that Ù is not in the list. But we know that all
natural numbers are in the list. This means all natural numbers are there but not in one line. So
they must be distributed over more than one line. That means there must be more lines, at least
two lines, that together contain more natural numbers than any of them. Contradiction by the fact
of inclusion monotony of the list.
A similar contradiction results from this list:
0.1
0.11
0.111
...
1/9 is the limit of the listed sequences, but it is not in the list. So if the existence of a decimal
representation of 1/9 is asserted, then it must be distributed over more than one line of the list.
Contradiction by the fact of inclusion monotony of the indices of the sequence of indices.
Result: All that exists of the Cantor-diagonal d, of Ù, and of 1/9's decimal representation, is in
one line of the list. But all that is not all.
§ 322 Breaking circularity
Some days ago, I posted a question about models of arithmetic and incompleteness. I then
made a mixture of too many scattered ideas. Thinking again about such matters, I realize that
what really annoyed me was the assertion by Ken Kunen that the circularity in the informal
definition of natural number (what one gets starting from 0 by iterating the successor operation a
finite number of times) is broken “by formalizing the properties of the order relation on omega”
(page 23 of his “The Foundations of Mathematics”). What does actually “breaking the circularity”
mean? Is there a precise model theoretic statement that expresses this meaning? And what
about proving that statement? Is that possible? [Marc Alcobé García, 14 Jun 2010]
Arithmos answered in MathOverflow in July 2013:
It is impossible to break the circularity immanent in every definition, unless you define some
primitive notions as non-circular. Nevertheless they are. Every single word in "there is a set s,
such that the empty set is an element of s ..." is undefined or defined by other words that are
undefined. This deficiency is not remedied but at most veiled by writing the axiom of infinity or
calling something ω.
Everybody has to begin with some words that he has learned when his mother taught him his
mother-tongue. Therefore it is no surprise that there is much ado about definability, which, again,
is undefinable, at least in first order logic.
Having recognized this, it is irrelevant what primitive notions exactly you start with. Before
defining the natural numbers, students should know how to count and, therefore, how to add the
unit. Then the following three axioms can be used:
- 1 is a natural number.
- If n is a natural number, then n + 1 is a natural number.
- Everything else is not a natural number. (But if you know something else, you know that
anyway.)
To my knowledge, these axioms are the only axioms that define the natural numbers and only
them, namely their sequential character, their inductive property, and their equidistance.
Why use the notion "successor"? Is "successor" more primitive than "addition of the unit"? No.
Knowing what a successor is, is as easy and as difficult as knowing what +1 means - in real life
as well as in formal definitions.
But what is more important, the successor-definition includes infinitely many sequences
isomorphic to the sequence of the natural numbers like:
1, 1/2, 1/3, ...
1, 11, 111, ...
1, 22, 333, ...
1, 1a, 1aa, ...
{}, {{ }}, {{{ }}}, ...
and even such not isomorphic to the sequence of natural numbers:
1, 11, 111, ...
0
00, 000, 000 , ...
1, square, cube, hypercube, ...
1, Fermat, Newton, Leibniz, ...
Andreas Blass commented: You correctly note that the "successor" definition needs additional
information (like the Peano axioms that say the successor function is one-to-one and that 1 is
not a successor) in order to excluse unwanted models. So does the "add a unit" definition; it
might get this additional information as a special case of assumptions about the general notion
of addition rather than in the form of Peano axioms, but, one way or another, the information
must be included in the axiomatic base. Calling something "add a unit" rather than "successor"
doesn't make a substantive difference.
You may certainly believe that "add a unit" is psychologically or epistemologically a more
primitive notion than "successor", but it seems to me at least as reasonable to hold the opposite
opinion. I suspect that the reason "successor" is often (not always) preferred as a primitive
notion is that "add a unit" looks like a special case of a general notion of addition, so authors
would feel obligated to explain that only x Ø x + 1, not x, y Ø x + y is taken as primitive.
Arithmos replied: You incorrectly note that "add a unit" rather than "successor" doesn't make a
substantive difference. The substantial differences are given in my examples.
§ 323 Andreas Blass, in a comment wrote on 1 May 2012
http://meta.mathoverflow.net/discussion/1353/nominalist-foundations-of-mathematics/
There have been a couple of questions recently, from someone with a very long username
abbreviated to user34, attempting to promote his philosophy of mathematics. I'd paraphrase his
attitude as "although I can't say precisely what I mean by definability, every sane person must
agree with me that mathematics is only about definable things."
{{(Paraphrasing is necessary (but not always reliable) if the text has not been fully
understood.) Definability, like set, need not be defined, some even say: cannot be defined.
Definitions have to be given. Mathematics is possible without sets but not without definitions.
Accepting undefinable numbers cannot be excused by refuting to recognize what definability is.}}
Not surprisingly, his questions have been closed and deleted; he's probably lucky that MO's
software doesn't provide buttons for "tar and feather" or "draw and quarter."
{{Yes, some centuries ago this would have been the method of choice.}}
§ 324 In MathOverflow Steven Landburg asked: Consider a country with n families, each of
which continues having children until they have a boy and then stop. In the end, there are G girls
and B = n boys.
Douglas Zare's highly upvoted answer to this question computes the expected fraction of girls
in the population and explains why we shouldn't expect it to equal 1/2. {{This "explanation" is
grossly mistaken. Of course 1/2 is the correct answer, cp. § 302.}} My current question concerns
a different statistic, namely the probability that there are more boys than girls (after all families
have finished reproducing). This probability turns out to be exactly 1/2, and I'm looking for an
intuitive explanation of why.
http://mathoverflow.net/questions/132297/boys-and-girls-revisited
There is a very simple explanation for the lacking intuition, namely: intuition is not lacking but the
expectation E[G/(G+B)] of G/(G+B) is absolutely uncorrelated to the expected answer.
This would have become obvious if the original formulation (the ratio of boys to girls) had been
taken literally by calculating, instead of E[G/(G+B)], the expectation of B/G which is E[B/G] = ¶.
From this result certainly nobody would have concluded that the official answer is false.
[Hilbert7Problem]
... it appears to be the exact opposite of the truth. If the question is "What is B/G?", and if the
official answer is "1/2" {{for B/G it is 1}}, and if the correct answer is "a random variable with
expected value ¶" , then recognizing the correct answer would lead not nobody, but everybody,
to conclude that the official answer is false. [Steven Landsburg]
No, the expactation of fractions E(B/G) is not the fraction of expectations B/G = E(B)/E(G)! The
expectation of the fractions E(B/G) is infinite, since some families have one boy as the first child
and no girl. They contribute 1/0. But this does not influence "the expected fraction of girls in the
population".
I can't believe that nobody in the self proclaimed "elite forum" MathOverflow opposes to
Landsburg's utterings. But I find this is an extremely instructive parallel to Cantorism. Nothing
could show better than this detail what an Overflow of Madness triumphs in Matheology.
§ 325 Importance of priority in mathematics?
The present question {{asked in MathOverflow by Conaeus Traglodythos on 12 July 2013}} does
not ask for priority disputes, like the famous battle between Newton and Leibniz, who was the
first with some invention. This question asks whether the course of mathematics has been
influenced by accidental results of the historical development.
As a famous example take Cantor's diagonal argument. Imagine the case that someone had
discovered prior to Cantor's 1892 publication that the Binary Tree (representing all real numbers
of the unit interval by infinite paths) contains only a countable set of paths that can be
distinguished by nodes. The Binary Tree can be coloured or constructed completely with
countably many infinite paths. Would anybody have given a dime for Cantor's argument? (This is
my question.)
Alas hypothetical questions are not suitable in MathOverflow. It quickly collected five down votes
and then was deleted. However, such questions remain visisble for users with 10000 reputation
points. Their belief in matheology is assumed to be strong enough to read also heretical texts like Catholic cardinals were allowed to read documents of the Vatican's safety cabinet box, like
leading Nazis were allowed to enjoy Jewish music, literature and paintings, like indoctrinating
fanatics always hide the truth only from the commonality (and try to punish whistle-blowers by all
means).
§ 326 An unaccepted method of enumerating the positive rational numbers
On 21 July 2013 Conaeus Traglodythos asked about "A new method enumerating the positive
rational numbers" in MathOverflow: It is well known that Cantor was the first one to succeed in
enumerating all positive rational numbers. He obtained the sequence, the finite initial segments
of which are repeated here - not because they are unknown, but because I wish to apply a new
method to them:
1/1
1/1, 1/2
1/1, 1/2, 2/1
1/1, 1/2, 2/1, 1/3
and so on.
This sequence will never end and for every natural number we will have accomplished much
less than 10-1000000000000000000000000000000000000 of the complete task, but, since every step is
well-defined and absolutely fixed, we conclude from the fact that the enumeration holds up to
every rational number that the enumeration holds for all rational numbers.
My idea is to apply the same method, but, in addition, always to put the finite initial segments
in proper order by size (which is no problem as long as they are finite, i.e., as long as we are
enumerating with finite natural numbers only - and other naturals are not known). This new
method will change the sequence as follows
1/1
1/2, 1/1
1/2, 1/1, 2/1
1/3, 1/2, 1/1, 2/1
and so on.
This sequence will never end, but, since also here every step is well-defined and absolutely
fixed, we can conclude from the fact that enumeration and ordering hold up to every rational
number that they hold for all rational numbers too.
My question: Why don't we accept the second method, or, alternatively, why do we accept the
first one?
Before this question got deleted, there was an interesting comment by Todd Trimble: "Ugh.
When will it stop?"
Even ConTra's answer could be written and published: "Both will stop at the same instant or
never." But nobody except Todd Trimble himself can know whether he could read the answer
and possibly learn from it.
§ 327 On 8 April 2010 J. H. S. asked a question regarding a claim of V. I. Arnold: In his
"Huygens and Barrow, Newton and Hooke", Arnold mentions a notorious teaser that, in his
opinion, modern mathematicians are not capable of solving quickly. Calculate
limxØ0 (sin(tanx) - tan(sinx))/(arcsin(arctanx) - arctan(arcsinx))
The answers given in
http://mathoverflow.net/questions/20696/a-question-regarding-a-claim-of-v-iarnold/132258#132258
including the accepted one (with 31 upvotes) show that professional research mathematicians in
fact cannot solve such problem other than in a very ponderous way.
My answer was shorter: "I think Arnold alludes to the idea that is often used in physics lessons,
for instance when treating mechanical oscillators. For small x we can put
x = sinx = tanx = sintanx = arcsinarctanx etc.
This yields the limit 1 immediately." [WM, 29 May 2013]
For professional research mathematicians this sounds too primitive. In fact many don't
understand at all was has been said. User Misha, for instance, asked: "This gives you
limxØ0 (x - x)/(x - x) = 1. I guess, you are using a system of axioms where 0/0 = 1."
Constantin, a Greek mathematician on sabbatical in Germany, dared to defend my position
asking Misha: "Would you disagree that (x - sinx)/(x - sinx) = 1 for every x including the limit?
That same holds for tan, arctan and so on? In my opinion Arnold cannot have expected that
someone calculates limits. Either you see it - or not."
Few hours later all his contributions were deleted without any announcement and Constantin
had been suspended for a month. Later he has been deleted completely. The impression that
the "great research-level logicians of MathOverflow" are not so great in sober mathematical
thinking must be avoided by all means. Of course also my answer has been deleted. Certainly it
was not "research-level".
§ 328 This paragraph is not related to the present MathOverflow series, but I would like to add
it here because it is so typical and, for the objective reader, so instructive.
My principle teaching, which I considered trivial and well-understood by every mathematician is
this: No infinite sequence can be defined by writing all its terms. We need always give a finite
word as the definition from which every term can be calculated. In case of the infinite sequence
of digits 0.000... this is done by the three 000 in connection with the three points.
Somebody, the name of whom does not matter, wrote with respect to this explanation:
"Everybody understands your trivial crap." However, to the obvious conclusion concerning the
path representing 0 in the Binary Tree: "The path 0.000... like every infinite path cannot be
defined by nodes", he replied: "Of course it can." So he has not understood and probably never
will.
Another reader replied: "Nonsense. If you'd bothered to define a tree properly, it would be clear
that a path is defined exactly by the set of nodes in it's edge." Is it possible that he does not see
that any proper definition together with his sentence "a path is defined exactly by the set of
nodes in it's edge" is not a definition by infinitely many nodes (or terms of the sequences) but a
finite word?
Alas, there are only countably many finite words in all speakable and understandable, in short: in
all usable, languages together.
§ 329 Here is a question rather as simple as the boys-and-girls-question (cp. § 302). It had
existed nearly one year in MathOverflow before I answered it under the pseudonym
Hilbert7Problem (which I considered well-received in MathOverflow).
http://mathoverflow.net/questions/103816/bike-lock-puzzle
I was wondering this when using my bike lock, a combination lock with four dials, each of which
has ten digits (0-9) on it in numerical order.
Suppose a bicyclist decides that, from now on, after putting in his combination on this lock, he
will only give the lock one twist to close it. So, he chooses between 1 and 4 adjacent dials, and
rotates them any number of spaces (other than a multiple of 10, to avoid having the lock end this
procedure in a closed position!)
Unbeknown to the bicyclist, a thief is following him. The thief knows that the bicyclist uses this
procedure to secure his bike. Over a period of days, the thief notes each combination the lock
ends up on. What's the fewest observations that the thief needs to make before she can deduce
the combination with certainty? What's the fewest observations that she needs to make before
she can reduce it to 10 possibilities? How can a shrewd (but stubborn) bicyclist maximize the
number of observations necessary without repeating a combination? [Kaveh, 2 August 2012]
If you are clever, the thief needs 49 different settings of the dials to know the correct setting with
certainty. This is more than half of all 90 = 4ÿ9 + 3ÿ9 + 2ÿ9 + 1ÿ9 possible settings you can
produce (when moving one dial, two adjacent dials, three adjacent dials, and all four dials,
respectively, into their nine possible incorrect positions).
Let the dials be (a, b, c, d). Let the correct position be (0, 0, 0, 0) to have a mental picture. If
you put a in eight different positions, say 1, 2, ..., 8, then the thief does not yet know with
certainty the correct one - although she knows the correct positions 0 of the other three. So if
you decided to turn one or more other dials leaving a at 0, she would quickly know the complete
correct setting. But if you put (a, b), (a, b, c), and (a, b, c, d) also in the eight different positions
avoiding a = 9 , you have 4ÿ8 = 32 positions without revealing the correct information.
You can do even better, if you choose to start with b. Then you can extend your number of
settings by moving (a, b), (b, c), (a, b, c), (b, c, d), and (a, b, c, d) supplying (1 + 2 + 2 + 1)ÿ8 =
48 settings in total. (Of course the order you choose does not matter.) You would get the same
opportunity with c instead of b. d however, like a, would supply only 32 possible positions.
The maximum number of different positions, before the thief has discoverd the correct one, is
for n > 2 digits and an even number m of dials:
(n - 2)(m/2)(m + 2)/2 .
For an odd number m of dials you get
(n - 2)((m + 1)/2)2 .
Addition: Maximality
If no single dial is moved, we have only 48 settings: 3 pairs, 2 triples and 1 quadruple in 8
positions each. But if pair (a, b) has been moved twice, pair (c, d) cannot be moved without
revealing the secret. Hence, we get only 40 settings. That is less than the constructed 48. So, in
order to maximize the number of the secret-maintaining settings, we have to move also at least
one single dial. But having moved it twice, we can no longer move any other single dial or the
pair not containing the first. This subtracts 36 from the 90 possible settings. Since of the
remaining 54 settings 6 are always "the nineth", i.e., revealing the secret, we have at most 48
settings. [Hilbert7Problem, 25 June 2013]
First Hilbert7Problem was praised: "This is very nice (and much better than what I wrote)" [S.
Carnahan] and the answer got six upvotes. Later Hilbert7Problem dared to point out, very
politely though, that the "research-professional's" answer to the boys-and-girls-question (cp. §
302 and §324) shows that at least 100 very stupid users are existing in MathOverflow.
Subsequently he was deleted. But my answer to the bike-lock-puzzle remains; only the author
has been removed. A copy-right violation.
§ 330 Is there an Oort cloud of inaccessible natural numbers?
Hans Peter asked in MathOverflow: When God, at the end of all time, will check what of his
creation has been worthwile, he will also consider the set of natural numbers that ever have
been used by his creatures. And he will find that only a very small subset has been applied.
(This idea goes back to Borel.)
For every usable number we have a finite set of predecessors and an infinite set of ¡0
successors. So there is no usable natural number behind some borderline, although that
borderline cannot be determined yet.
Is it, in principle, possible to find circumstantial evidence for the existence of the ¡0
inaccessible numbers - in order to satisfy platonists like Gödel? Or is postulating them by the
axiom of infinity the only way to lay hold of them?
A first comment by Asaf Karagila: "If time is infinite, then at the end of all time we might have
generated all numbers" showed the belief in the end of the not ending as the foundation of
matherology, since, if time is infinite, there is no end. We might not have generated all natural
numbers at the end of all time, but simply never.
Asaf's second comment showed this again: "Hans, can you imagine any natural number that has
infinitely many predecessors? But there are still infinitely many of them. If time is infinite, it
suffices that everyone just counts the number of days since today in order to ensure that we
used all the natural numbers when time ends."
Hans disproved that there are still infinitely many: "Asaf, I cannot imagine a natural number
having infinitely many predecessors. But that is not surprising because the infinitely many
numbers always remain successors. The end of all times, however, could not be after a finite
number of instants, because then it was finite and was not the end."
§ 331 Hans Peter asked in MathOverflow: Are there undecidable questions that do not depend
on the chosen axioms?
It has been conjectured that the digit sequence of a normal irrational contains all possible finite
digit sequences. Brouwer's famous question after a sequence of nine nines has been answered
in the affirmative, thereby weakening a bit his position of reasoning against tertium non datur.
Nevertheless there seem to be some arguments against tertium non datur. One of them is this:
Will it ever be possible to find out whether the digit sequence of π contains a sequence of
1010000 nines?
Of course this question cannot be decided with the current tools. But my question is this: Can
anybody imagine a way that would promise an answer in the long range? Or should this problem
be considered as undecidable in eternity?
Hans Peter collected 40 reputation points in his short career in MathOverflow, but the ususal
gang of deleters appear not to be interested in too obvious arguments that mathematics is not
performed in heaven but in our physical reality. Although it can be denied, no computer and no
brain and no mathematician can leave it. This situation is comparable to erotics in past times:
Man darf das nicht vor keuschen Ohren nennen, was keusche Herzen nicht entbehren können.
[Goethe, Faust I]
§ 332 Undefinable numbers
Annix asked in MathOverflow on 29 Oct. 2010: Is the analysis as taught in universities in fact the
analysis of definable numbers?
http://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-theanalysis-of-definable-numb
Ten years ago when I studied in the university I had no idea about definable numbers {{No,
such poisonous stuff is not usually taught to normal mathematicians}}, but I came to this concept
myself. My thoughts were as follows:
- All numbers are divided into two classes: those which can be unambiguously defined by a
limited set of their properties (definable) and such that for any limited set of their properties there
is at least one other number which also satisfies all these properties (undefinable).
- It is evident that since the number of properties is countable, the set of definable numbers is
countable. So the set of undefinable numbers forms a continuum.
- It is impossible to give an example of an undefinable number and one researcher cannot
communicate an undefinable number to the other. Whatever number of properties he
communicates there is always another number which satisfies all these properties so the
researchers cannot be confident whether they are speaking about the same number.
- However there are probability based algorithms which give an undefinable number in a limit,
for example, by throwing dice and writing consecutive numbers after the decimal point. {{The
limit is never reached by throwing dice or writing something.}}
But the main question that bothered me was that the analysis course we received heavily
relied on constructs such as 'let's a to be a number that...", "for each s in interval..." etc. These
seemed to heavily exploit the properties of definable numbers and as such one can expect the
theorems of analysis to be correct only on the set of definable numbers. {{Of course!}} Even the
definitions of arithmetic operations over reals assumed the numbers are definable. Unfortunately
one cannot take an undefinable number to bring a counter-example just because there is no
example of undefinable number, but still how to know that all those theorems of analysis are true
for the whole continuum and not just for a countable subset? [Annix, 10:47]
The answer is simple: Analysis is true for all real numbers because there are no undefinable real
numbers. But there are many misunderstandings and quibbles - stuff for several paragraphs.
§ 333 Annix's question "Is the analysis as taught in universities in fact the analysis of definable
numbers?" (cp. § 332) appeared dangerous to many matheologians, so it got closed after less
than 2 hours. It was mainly J.D. Hamkins who pleaded for reopening:
I just wrote a long answer to this question, but it was closed just as I was about to click submit.
Can we re-open please? I think that there are a number of very interesting issues here. [Joel
David Hamkins, 29 Oct. 2010]
But the customary gang of closers did not give up!
I disagree with the continuing votes to close. [Joel David Hamkins, 30 Oct. 2010]
Meanwhile the question has collected 27 upvotes. Obviously the judgement about the quality
of questions is extremely subjective in MathOverflow.
§ 334 Annix's question "Is the analysis as taught in universities in fact the analysis of definable
numbers?" (cp. § 332) has been reopened and answered by J.D. Hamkins:
In recent work (soon to be submitted for publication), Jonas Reitz, David Linetsky and I have
proved the following theorem:
Theorem. Every countable model of ZFC and indeed of GBC has a forcing extension in which
every set and class is definable without parameters.
In these pointwise definable models, every object is uniquely specified as the unique object
satisfying a certain property. {{Of course that is not difficult in a countable model. Alas the only
interesting property of set theory is uncountability.}} Although this is true, the models also believe
{{what models do you allude to, that can believe?}} that the reals are uncountable and so on,
since they satisfy ZFC and this theory proves that. The models are simply not able to assemble
the definability function that maps each definition to the object it defines. [J.D. Hamkins, 29 Oct.
2010]
Then use only the definition as the "object" that it defines. This has always been done in
mathematics with real numbers. None of them is defined by an infinite string of digits or another
"object". Each one is a definition or several equivalent definitions like "divide 1 by 9" or "1/9" or
"0.111..." (which are not infinite decimal representations). If there is more than one definition,
there is no problem, as long as they are equivalent. If you have the choice between several
definitions of π, for instance, that does not hurt. If your model can do that, you have a
contradiction in ZFC because there is nothing uncountable but uncountability is the main result
of ZFC. If not, it is useless.
§ 335 Annix's question "Is the analysis as taught in universities in fact the analysis of definable
numbers?" (cp. § 332) has raised some comments by Andrej Bauer: He {{J.D. Hamkins}} did not
say that it is consistent to "postulate in ZFC that undefinable numbers do not exist". What he
was saying was that ZFC cannot even express the notion "is definable in ZFC". {{Therefore it is
very surprising that ZFC is allegedly able to define sets and numbers, i.e., to do things that
nobody in ZFC can prove to be done correctly.}}
Joel made a very fine answer, please study it carefully. Joel states that there are models of
ZFC such that every element of the model is definable {{although nobody can know precisely
what that means}}. This does not mean that inside the model the statement "every element is
definable" is valid. The statement is valid externally, as a meta-statement about the model.
Internally, inside the model, we cannot even express the statement. {{And externally we cannot
find out whether external statements are meaningful, because "external" is also only some
model - yet a bigger one.}}
Annix answered: I do not say undefinable numbers do not exist. Their existence follows from
axiom of choice {{perhaps it follows, but as a contradiction, because you cannot choose one of
many undefined numbers}} and in theory we can uniquely define each undefinable number by
specifying infinite number of its properties. The problem is that the theorems of analysis as
taught in universities sufficiently rely on the properties of definable numbers. {{That is not a
problem of mathematics since other numbers have no properties. Also it is not a "problem" of
ZFC but a simple contradiction in ZFC. Of course Zermelo would not have been stupid enough
to defended in his 1908 paper the axiom of choice in length as a natural choice if he had been
confronted with undefinable numbers. That nonsense has only become en vogue in the circles of
modern "logicians". Of course nobody can say what real number is undefinable. Why don't the
undefinable-number-cranks believe in undefinable natural numbers? Of course nobody can say
what natural number is undefinable. But then countability-spook would no longer haunt those
poor peolple's mind.}}
§ 336 Annix's question "Is the analysis as taught in universities in fact the analysis of definable
numbers?" (cp. § 332) has raised another comment by Andrej Bauer: This is off-topic, but: it
makes no sense to claim that "constructivist continuum is countable in ZFC sense". What might
be the case is that there is a model of constructive mathematics in ZFC such that the continuum
is interpreted by a countable set. {{Why then not use this as the model of university mathematics
and drop all blather about uncountability?}} Indeed, we can find such a model, but we can also
find a model in which this is not the case {{this is no contradiction, of course}}. Moreover, any
model of ZFC is a model of constructive set theory {{and constructive models contraditct
uncountability because everything constructed, say as a constructed anti-diagonal of a
constructed Cantor-list, is well-defined and therefore definable and therefore belonging to a
countable set}}. You see, constructive mathematics is more general than classical mathematics,
and so in particular anything that is constructively valid is also classically valid {{for instance the
theorem that uncountability does never occur constructively}}.
§ 337 Annix's question "Is the analysis as taught in universities in fact the analysis of definable
numbers?" (cp. § 332) has raised a comment by arsmath: "Definable numbers" are numbers that
are definable in terms of first-order logic over set theory. There are perfectly intelligible numbers
that cannot be defined in your sense. For example, suppose you have a sequence of definable
numbers an that is bounded by a constant. Then b = sup an is a number that is unique and has
an unambiguous meaning, but b is not necessarily definable. {{It is defined by the sequence (an).
Why should another definition be searched? Irrational numbers are never defined in another way
since there is no irrational number that can be defined by giving the value of every digit. In Excel
you are often asked whether you wish to copy only values or also formulas. If you answer "only
values" you wil never get an irrational number.}} Each an is given by a formula {{of course, real
numbers cannot be given in another way}}, but if the formulas are sufficiently different then there
is no way to write down a single formula for b {{then it is already impossible to write down all
formuas for the an. If the sequence should be defined completely, then there must be a first n
from which all following an are defined by the same formula. Otherwise you'd have to write down
infinitely many formulas, which is as impossible. In that case there is not an undefined b but no b
at all.}}
Annix answered in this spirit: Yes, b is not definable. But it is also not unique for any bounded
number of properties we define (i.e. bounded number of sufficiently unique ϕn(x)). Thus for each
limited number (say, N) of ϕn(x) we get infinitely many numbers which satisfy for first n < N.
arsmath replied: I'm not clear what you're after. You said that a researcher cannot give an
example of an undefinable number, and that one researcher cannot communicate an
undefinable number to another. I pointed you towards a counterexample to both claims. {{And I
pointed you to the error you commited.}} You can give a completely explicit family of formulas,
so explicit that they can be generated by a computer program, that gives you a number that's not
definable. We can't say much about that number, but it still have a description that identifies it
uniquely {{and thus defines it}}.
Most theorems of analysis that are false if you only consider definable numbers. For example,
the set of definable numbers does not have the least upper bound property. The intermediate
value theorem is false, etc. {{and undefinable numbers do not in the least improve the situation,
because only definable numbers are looked for in these theorems.}}
Finally, J.D. Hamkins told asrmath the same as I said here: Arsmath, you haven't actually
described a non-definable number, and it is impossible to do so for the reasons expressed in my
answer. {{Yes, that is correct. And therefore the undefinable numbers are without any worth in
mathematics - and elsewhere.}}
Annix took the same position: Either the family of formulas is finite and can be communicated to
the other researcher, then the number b is definable. Or it is infinite, then it cannot be
communicated to the other researcher. You actually did not give an example of b since you say
nothing about the defining formulas. Thus b is not defined so far. {{Correct.}}
arsmath had not yet understood, but this paragraph has become very lengthy already. Therefore
this very instructive discussion will be continued in the next paragraph.
§ 338 The problem of understanding the meaning of definability is basic to modern set theory.
Therefore the discussion triggered by a question of Anixx in MatheOverflow
http://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-theanalysis-of-definable-numb
is continued here:
arsmath: My point is that you can write down an explicit description of a number that
unambigiously defines it to a human being, but that number is not definable. {{On the contrary.
This number is defined.}} I can (in theory) provide a finite computer program that provides the
formulas. (The Wikipedia page sketches a similar construction under "Notion does not
exhaust...".) {{Wikipedia is written by humans. Humans often fail.}} This is a philosophical
question {{not at all}}, but I would say a finite computer program is sufficient. How do I know that
100001000010000 exists? {{Because you have written it here.}} Because I can write a computer
program to compute it. {{No computer could do more than you have done here. In particular
every computer would fail to count in single steps up to 10100.}}
Anixx: In that case your number b is not only definable but even computable.
arsmath: It's not computable because the individual ϕi are not necessarily computable.
Anixx: In that case it is not computable but definable.
arsmath: Why? Definability requires a single formula.
Anixx: If there is a program that can generate such formulas, then there is a single formula for
all. {{Of course.}}
A. Blass: Your last comment is based on the erroneous assumption that one can define how to
pass from a definition to the thing it defines. {{No, the definition of an immaterial thing is the thing
it defines.}} Arsmath described a situation where a sequence of formulas might be definable
(and even computable) but the sequence of real numbers they define is not definable. Joel
explained why your assumption is wrong. {{Joel explained why his assumption is right.}} I
second Andrej's earlier suggestion that you study Joel's answer carefully, and I add my own
suggestion that you assume that Joel meant exactly what he said, not what you think he should
have meant or must have meant. {{What did Joel say? "Arsmath, you haven't actually described
a non-definable number, and it is impossible to do so for the reasons expressed in my answer." I
would recommend that you study this answer very carefully, A. Blass!}}
Anixx answered on the erroneous assumption mentioned by A. Blass: "please tell me where I
did such assumption (and what do you mean under "pass")? Arsmath described a situation
where a sequence of formulas might be definable (and even computable) but the sequence of
real numbers they define is not definable. - It may be not computible but how it can be
undefinable? Can you give an example? {{No, of course he cannot and so he did not since by
definition undefinable numbers are not definable.}}
And, by definition, undefinable number is such number that for any limited set of its properties
there is at least one other number with the same properties. {{So it is!}}
Andres Caicedo: It feels like we are going in circles now. {{No, it feels that some people in fact
have recognized that undefinable numbers are not definable.}}
Finally, some years later, I entered the discussion that I had not been aware of before: It is
completely irrelevant how to pass from a definition to the thing it defines. Important for the
present countability question is only the correspondence of definition and defined object. In
mathematics real numbers and their definitions are in this correspondence: There are many
definitions for some individual real numbers like e, but there are no definitions of individual
numbers that fail to define individual numbers.
§ 339 Annix had asked in MathOverflow on 29 Oct. 2010: Is the analysis as taught in
universities in fact the analysis of definable numbers? I gave the following answer (that of course
has been deleted soon):
A definable real number r is a number that can be defined, i.e., r can be identified and
communicated by a finite sequence of bits in real life, just where mathematics takes place. This
makes the set of simultaneously (in a given language) definable numbers countable. Therefore
all real numbers that can appear in the language of mathematical analysis belong to a countable
set.
Independent of real-life conditions it is impossible to distinguish, in the universe of ZFC or
elsewhere, real numbers by infinite sequences of bits. This claim is proven by the possibility to
construct all infinite sequences of digits by means of a countable set of infinite sequences of
digits as follows:
Enumerate all nodes ai of an infinite binary tree and map them on infinite paths pi such that ai
is in pi. There is no further restriction. The mapping need not be injective. Then construct from
this countable set of paths another binary tree. Mathematical analysis is not able to discern
which paths were used for construction.
This shows that outside of a platonist ZFC-universe there are not uncountably many real
numbers. Real numbers created by Cantor-lists are not defined unless the Cantor-list is welldefined, i.e., every entry of the list is known. That requires a Cantor-list constructed by a finite
definition. But there are only countably many finite definitions of Cantor-lists.
The existing real numbers of analysis cannot be listed. But that does not make their set larger
than any countable set.
§ 340 carl-labande asked in MathOverflow: What is the fade-away-rate of mathematical
induction in practical applications? The unexpected hanging paradox, hangman paradox,
unexpected exam paradox, surprise test paradox ... All these paradoxes and many others, like
the blue-eyed islanders paradox, are mainly based on the unlimited validity of induction. But
perhaps this assumption is incorrect with respect to application of mathematical induction to
practical reality.
If the teacher announces "tomorrow we will write an unexpected exam", then this is clearly a
self-contractory announcement, even in reality.
If the teacher announces that the unexpected exam will be written next week or in any
specified interval of days, then many mathematicians tend to conclude that this is also selfcontradictory in reality because induction shows that the last day of the interval cannot apply,
therefore also the day before the last one cannot apply, and so on. But for a really long interval
induction fails as can be proved. Consider that the teacher announces one or even 100 surprise
tests during the next 3000 days, then induction won't help at all to determine the dates. In order
to prove that, guess 100 dates and compare with a set of 100 random numbers of that interval.
That suggests: in these cases the reasoning based upon induction does not remain valid for
large intervals in reality. (Compare the blue-eyed islanders paradox with 1010 islanders.) The
validity of inductive reasoning is certainly absolute for n = 1, but near to zero for n = 1010 and
has limit 0 for an infinite interval.
The question is of course, whether this problem belongs to mathematics or to reality only. But I
would plead the case that also problems with importance for reality should be scrutinized by
mathematicians.
Therefore my question: What is the fade-away-rate? Can a "function of validity" f(n) be defined
concerning the state of knowledge at time zero?
The question you ask is not a mathematical one. While the question might be research-worthy,
the research in question is not mathematical. [Boris Bukh]
You may be right. But my question is just whether this problem has been considered in
mathematical literature. I guess this is possible. Why should mathematicians refuse to help in
practical questions? Perhaps mathematical methods will be useful to find such a function? [carllabande]
Mathematicians might well love to help in a practical question. They just might not have a clue
how to do so. [Lee Mosher]
Yes, I also made this observation. If a question appears too difficult or if they cannot get to the
interesting nucleus, soon some dwarf-mathematicians gather together and close the disturbing
proof of their inability {{cp. Annix's question, § 332}}. The present question was closed by Boris
Bukh, Gerald Edgar, Steven Landsburg, Felipe Voloch, and Qiaochu Yuan.
As with Anixx's question it was again J.D. Hamkins who discovered some interesting aspects
here: "Suppose you only knew that the exam would be held and would be a surprise with certain
probabilities, perhaps very high. Then one might hope to propagate the inductive reasoning
through this probability, perhaps ultimately giving a probability distribution for when the exam
would occur? Can someone give an answer along these lines?" But in this case he lost against
the dwarfs.
§ 341 Recently Peter Shor's community-wiki-question in MathOverflow: "Can a mathematical
definition be wrong?"
http://mathoverflow.net/questions/31358/can-a-mathematical-definition-be-wrong
has been answered by Hans Peter: Definition: "This is a wrong definition." If definitions cannot
be wrong, then this is truely a wrong definition. If definitions can be wrong, nothing remains to be
shown.
§ 342 gowers asked on 3 April 2012: What was Gödel’s real achievement?
http://mathoverflow.net/questions/20219/what-was-godels-real-achievement
When I first heard of the existence of Gödel's theorem, I was amazed not just at the theorem but
at the fact that the question could be made precise enough to answer: how on earth, even in
principle, could one show that it was impossible to prove something in a given system? That
doesn't bother me now, and that is not my question.
It seems to me that Gödel's theorem is a combination of at least three amazing achievements,
namely these.
1. Formalizing the notions of proof, model, etc. so that the question could be considered
rigorously.
2. Daring to think that there might be true but unprovable statements in Peano arithmetic.
3. Thinking of the idea of Gödel numbering and getting the proof to work.
...
My answer was that none of the three points was Gödel's real achievement: I think Gödel's real
achievement has been overlooked grossly, perhaps because it only appeared in a footnote. He
recognized and wrote in his seminal paper [1] that the true reason for the incompleteness is
caused by the transfinite hierarchy: "Der wahre Grund für die Unvollständigkeit, welche allen
formalen Systemen der Mathematik anhaftet, liegt [...] darin, daß die Bildung immer höherer
Typen sich ins Transfinite fortsetzen läßt [...] während in jedem formalen System höchstens
abzählbar viele vorhanden sind. Man kann nämlich zeigen, daß die hier aufgestellten
unentscheidbaren Sätze durch Adjunktion passender höherer Typen (z. B. des Typus ω zum
System P) immer entscheidbar werden. Analoges gilt auch für das Axiomensystem der
Mengenlehre."
So, if Brouwer is right with his statement excluding the transfinite hierarchy: "De tweede
getalklasse van Cantor bestaat niet" [2], then Hilbert's program [3] gets support from an
unexpected side and can be restarted.
[1] Kurt Gödel: "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter
Systeme I", Monatshefte für Mathematik und Physik 38 (1931) S.173–198, quoted from p. 191.
[2] L.E.J. Brouwer: "Over de grondslagen der wiskunde" (Februari 1907, Dutch) Thesis XIII.
http://www.archive.org/details/overdegrondslag00brougoog
[3] E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar
Univ. Hamburg, Bd. 6, Teubner, Leipzig (1928) 65-85. English translation in J. van Heijenoort:
"From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479.
http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm
[Wolfgang Mueckenheim]
I doubt that gowers has seen my answer because the usual gang deleted it soon. Gödel,
Brouwer, and Hilbert, put together in an unorthodox way, seem to be considered a dangerous
triumvirate.
§ 343 On 11 June 2013 Ari asked in MathOverflow: Can machines generate truly random
sequences?
http://mathoverflow.net/questions/133388/random-infinite-sequence-can-machines-generatetruly-random-sequences
Hans Peter answered: Yes, they can. Couple a computer with a Geiger counter that clicks
between 100 and 1000 times a minute. Count the beeps. If the number is even, let the computer
print 0 if it is odd 1. The sequence is absolutely random. {{Alas, this is not matheology but
MatheRealism. Therefore his answer has been deleted soon.}}
§ 344 Where mathematics has gone wrong
Alekk asked in MathOverflow: I would be interested in knowing examples of results conjectured
by physicists and later proved wrong by mathematicians. Furthermore it would be interesting to
understand why physical heuristics can go wrong, and how wrong they can go (for example,
were the physicists simply missing an important technical assumption or was the conjecture
unsalvagable).
http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrectanswers
I answered: Why do you ask for wrong physicists only?
Being wrong happens to mathematicians as well. In 1833, the year of his dead, Adrien Marie
Legendre presented an overwiev of proofs of the parallel axiom to the French Académie des
Sciences. It included six rigorous proofs, three of which using infinite angular areas. (Here
"rigorous" is to be understood in the meaning of his times as present mathematicians use
"rigorous" in the meaning of our times. But obviously there can never be absolute rigour, neither
then nor today.)
Or take the first proof of the Cantor-Bernstein theorem bei E. Schroeder in 1896. The proof
was wrong, as Schroeder admitted in a letter to A. Korselt (who had improved the proof). Korselt
gives a copy of Schroeder's reply in his paper [A. Korselt: "Über einen Beweis des
Äquivalenzsatzes", Math. Ann. 70 (1911) 294.]
Nevertheless, Korselt's corrected version was not accepted in 1902 by the Annalen. Only 9
years later, he could publish his paper. But that was not widely noticed, so the incorrect proof
survived for a long time.
Cantor wrote to Hilbert on June 28, 1899 that E. Schroeder in 1896 (and Cantor's student F.
Bernstein about Easter 1897) had proved the theorem. So Cantor never noticed Schroeder's
error.
A. Fraenkel mentioned in 1923 (!) that Schroeder's proof was wrong. [A. Fraenkel: "Einleitung
in die Mengenlehre", Springer (1923) p. 58]
E. Zermelo considered Schroeder's proof correct even in 1932. Zermelo remarks in Cantor's
collected works as an editing note: "... wurde erst im Jahre 1896 von E. Schroeder und 1897 von
F. Bernstein bewiesen und seitdem gilt dieser 'Aequivalenzsatz' als einer der wichtigsten Saetze
der gesamten Mengenlehre." [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer (1932) p. 209]
This shows that wrong things can survive in mathematics for about 35 years. Or even longer?
And should that be different now?
§ 345 Where Cantor as a physicist has gone wrong
After this answer had been deleted, I gave a second one: Georg Cantor, the founder of set
theory, also gave lessons in philosophy and in theoretical physics. He devised his set theory in
particular to get a better explanation of physical phenomena. Therefore he can be considered as
a physicist, at least in part.
He stated that "in the universe and on earth and, according to my firm conviction, in every non
vanishing volume of space there are an actually infinite number of created creatures." [Letter to
Cardinal Franzelin of Jan. 22, 1886]
Cantor believed that the material point-like atoms were a countable set and the ether atoms
were an uncountable set (though he did not believe in the existence of atoms for chemical
purposes). Both sets should be dense (in sich dicht) and geometrically homogeneous. [Letter to
Mittag-Leffler, Nov. 16,1884]
These ideas have later been proven incorrect. (By other physicists. Sorry, again not fully met
the topic.)
Of course also this answer of mine has been deleted, as was to be expected in this forum of
pompous Cantor-admirers and -victims.
§ 346 How can undefinable objects be elements of mathematics?
MaBru asked recently, on Friday, 13 Sept. 2013 (obviously not a very lucky date) in
MathOverflow: How can undefinable objects be elements of mathematics?
Consider the complete infinite Binary Tree. It has countably many nodes to be enumerated as
follows:
0
/
\
1
2
/ \
/ \
3 4
5 6
/ \ / \ / \ / \
7 8 9 10 1112 1314
...
Remove the nodes one by one and with each node remove a path containing that node. If the
first nodes of a path are already missing, remove only the remaining tail of the intended path.
First remove node 0 and, for instance, the path (0, 1, 3, 7, ...) Then remove node 2 and, for
instance, the remaining nodes (2, 6, 14, ...) of the path (0, 2, 6, 14, ....) Then remove node 4
and, for instance, the remaining nodes (4, 10, ...) of the path (0, 1, 4, 10, ...). Continue until all
countably many nodes and contably many paths have been removed.
If there are uncountably many paths in the Binary Tree, then they cannot be defined by nodes,
since there are no nodes remaining. With respect to the structure of the Binary Tree, obviously
countably many nodes cannot define more than countably many paths. Those paths can only be
defined by finite definitions. In fact each path is defined by a finite definition like "(0, 1, 3, 7, ...").
Alas there are only countably many finite definitions in all usable languages. How can
undefinable objects be elements of mathematical discourse? [ MaBru]
The problem with the view that "every path has to be defined by a finite definition" is that as soon
as you specify what counts as a definition, you open a possibility to specify (by diagonalization)
a path which is not (by your standards) definable. Why should that path not be an element of
mathematical discourse? [Johan Wästlund]
Of course that path is an element of mathematical discourse, since it belongs to a countable set
of paths. Every defined diagonal of any defined list belongs to the countable set of defined
elements of mathematical discourse. {{In fact a "problem with the view" results only if one insists
that infinity can be finished, that there is a list of all natural numbers. But who would tolerate
such an Overflow of Madness?}}
A further question. Perhaps the closers can answer, but probably they cannot and therefore
will prefer to delete this question: What views exist in mathematics besides "the view that 'every
path has to be defined by a finite definition'"? [MaBru]
MaBru appears clairvoyant. Instead of an answer the question got deleted.
§ 347 Countability and super tasks
Recently I have been convinced that enumerating the rational numbers is a super task:
Counting the positive rational numbers can be interpreted as a super task: In the nth step fill
into an urn all rationals between n-1 and n and, if not yet residing in the urn, also the rational
number qn to be enumerated by the natural number n. Take off the rational number qn. Go to
step n+1.
But, in spite of ardent prayers, I cannot believe that such super tasks can be finished with the
result "urn empty". Further I don't know a suitable axiom. Therefore my question: Can this result
be circumvented? Can someone supply a convincing argument that the enumeration of the
rational numbers is not a super task? [superintendent]
Comments after some hours:
Nobody wiling to help me? [superintendent]
Nobody able to help me? [superintendent]
You should find another web site that is more appropriate for your question. It might help if you
understood the difference between defining a map on infinite sets and doing a supertask. [S.
Carnahan]
What is the difference between defining a map on infinite sets and doing a supertask? That's
what I wish to know! But you are not willing to say in two words what the difference is? Can you
point me to someone who would know it??? [superintendent]
Of course this dangerous question could not persist in MathOverflow; it was deleted
immediately; no traces remain. Perhaps someone is able to figure out the difference?
§ 348 Three bawdy points
When discussing the super task of § 347 in MathOverflow it is generally regarded as
unmathematical and unprofessional and punished by deletion to mention one of the following
three points. It appears like mentioning genitals in the presence of ladies in Queen Victoria's
times!
1) Do you properly distinguish every and all? Every number has finitely many predecessors and
infinitely many successors. In mathematics you can easily conclude that every natural number
belongs to a set that contains less than half of all natural numbers because "infinitely many" is
larger than "finitely many".
2) When the urn is empty in the end of the unending, it contains no rational number. Do you
agree that a stringent property of the super task requires that in every step only one number
leaves the urn? If there is a state with none, then there must have been a preceding state with
one. Or do you simply neglect this property?
3) Can you tell me why Cantor's bijection is far from being a super task? The above delusions
are simply nonsense because ............................................ (please fill in your argument).
§ 349 Pete L. Clark, addressing me, wrote in MetaMathOverflow in July 2010
http://tea.mathoverflow.net/discussion/484/physicists-can-be-wrong/
what I will comment here:
P.L. Clark: I am not sure that you understand the technical items of set theory. ... You use terms
like completed versus potential infinity, which are not part of the modern vernacular.
WM: You are not well informed. But partially you are right. Many set theorists do not like to talk
about finished infinity. These words have been omited from the official vocabulary with good
reason: An intelligent newbie would get somewhat confused. But what you formalize in modern
set theory is just completed infinity - even if you don't know that.
P.L. Clark: ... try out your argument on the following simpler case: consider the infinite graph on
the integers where for all n, n is adjacent precisely to n-1 and to n+1. There are countably many
nodes on this graph but there are uncountably many random walks: they again correspond to
infinite sequences from a two-element set. This is disturbing to you because...?
WM: This is disturbing because there is no chance to have an infinite random walk executed,
described, or used in mathematics other than by a finite description. Alas, there are only
countably many finite descriptions in all usable languages, i.e., in all languages that can be
learned, written and read. (An uncountable alphabet would be useless, because an alphabet is
something to look up. But an uncountable alphabet cannot fit into a list - not even into the whole
universe. Therefore it is a contradictio in adjecto. I would never get in touch with people who
seriously consider uncountable alphabets because they must be so much confused that they are
probably even dangerous.)
P.L. Clark: Let's not be disingenuous: you are notorious on the internet for your writings about
set theory and especially Cantor's uncountability arguments. But Cantor's work on set theory has
been explored and vetted with extreme care by mathematicians for more than a hundred years.
WM: Same has been done by (even more) astrologians with astrology.
P.L. Clark: Nowadays our attitude to allegations of flaws in Cantor's work is similar to that of
many biologists when presented with attacks to evolution from "creation scientists":
WM: Crazy! It is just the other way round. Cantor's first proofs of finished infinity were based on
God and the Holy Bible. He coined his expression cardinal number when he tried to convince
cardinal Franzelin of his absured ideas in a letter of 22 Jan. 1886. And he was a hard core
creationist with respect to nature as well as to mathematics. He even tried in several cases, by
intrigues, to keep Darwinists from university chairs. Always without success, fortunately.
P.L. Clark: ... it is not a debate we are eager to have, and we feel that we are at least entitled to
restrict ourselves to discussants who show an understanding and technical mastery of the
relevant material (which is, for mathematics, not that technical: for instance, many bright high
school students know it well).
WM: Sorry, due to my professional responsibilities I often have to meet bright high school
students but I never met a high school student who was informed that modern matheology
requires to accept undefinable objects as real numbers. But when I told some bright students
about that "fact", they spontaneously asked what a Cantor-list of undefinable numbers and its
diagonal number would look like.
P.L. Clark: There's certainly room for philosophical doubts about uncountable (or even
countably infinite) sets, but this is not the appropriate forum for that.
WM: In particular since my arguments are not philosophical ones but simply mathematics. (A
summary will be given in § 350.)
P.L. Clark: From our point of view, your criticisms are simply not valid.
WM: I know. That will never change. But perhaps I will save some young students from getting
matheologians. I'll do my best.
§ 350 The Four Most Simple Contradictions of Transfinite Set Theory
I started to learn about large cardinals a while ago, and I read that the existence, and even the
consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally
regular, is unprovable in ZFC. Nevertheless large cardinals were studied extensively in the last
century and (apart from attempts that went too far as the Reinhardt-Cardinals) nobody ever
found a contradiction to ZFC. [erinna, 29 Oct. 2010]
http://mathoverflow.net/questions/44095/arguments-against-large-cardinals
The reason seems to be that all contradictions are eagerly and quickly deleted. Today I found
this post by WM in sci.math:
The actually infinite is self-contradictory as has been shown by many proofs. I will sketch here
only four of them:
1) If actual infinity exists, then you can in ¡0 steps well-order the rational numbers by
magnitude.
2) A Cantor-list that is complete with respect to the rationals contains every finite initial
sequence of the anti-diagonal infintely often. If a list contains every finite initial sequence of π,
then it contains π because there is not more than every finite initial sequence. Why? See here:
3) The Binary Tree that contains only all rational numbers of the unit interval contains also all
irrational numbers of the unit interval although none of them has been added explicitly. Only
countably many rational numbers have been added. But when ready, abracadabra, the Binary
Tree contains uncountably many irrational numbers.
4) The sequence
1
2, 1
3, 2, 1
...
contains Ù in every column. So the whole matrix contains Ù. But none of the lines contains Ù.
On the other hand, the matrix is constructed such that a set of finite lines never contains more
than one of them. And the matrix contains only finite lines. (How many of them does not matter
and does not change the facts.)
If only one of these ideas is correct {{in fact all are}} then there is no cardinal number larger
than infinity. Isn't this an argument against large cardinals? [Hans Peter, MathOverflow, 10 June
2013]
§ 351 An air of unreality
When modern set theory is applied to conventional mathematical problems, it has a
disconcerting tendency to produce independence results rather than theorems in the usual
sense. The resulting preoccupation with "consistency" rather than "truth" may be felt to give the
subject an air of unreality. [Saharon Shelah: "Cardinal arithmetic for skeptics",
arXiv:math/9201251 (1992)]
An air of unreality? Very solid air. Rather ore.
§ 352 Colours
Theorem. The set Ù of all natural numbers does not exist.
Proof. Let us assume that the actual infinity ¡0 or the set Ù of all natural numbers can exist.
Also assume that there exists a set of ¡0 different colours as {red, yellow, green, blue, ...} (the
wavelength-range may be as large and differences in wavelength may be as small as desired).
We represent the set of all natural numbers by use of the initial segments of colours encoded by
r, y, g, b, ...
r
ry
ryg
rygb
...
The first horizontal line shows number one, the second number two, and so on.
It is obvious that the whole set of ¡0 different colours (or the greatest initial segment) will not
be used in any horizontal line (because of the finiteness of every horizontal line). In other words
the number qayn {{the 22th letter of the Persian alphabet, the first letter of the word
"impossible"}} of different colours used is less than ¡0. On the other hand, the number qayn of all
different colors used is larger than every finite number. So the following relation holds:
Every finite number n < qayn < ¡0
Since every number less than ¡0 is a finite number, we have qayn is a finite number and qayn is
larger than every finite number:
qayn < qayn
This deduction is a contradiction. And now the proof is completed. Therefore the existence of the
set Ù of all natural numbers is impossible.
[S.S. Mirahmadi (Sept. 2013), Qom seminary, Qom, Iran]
[email protected]
§ 353 Hessenberg's proof is this: If a mapping from Ù to all its subsets would exist, then there
must one natural number n be mapped on that subset Sn that contains only natural numbers that
are mapped on subsets which do not contain the natural numbers which are mapped on them.
But does this subset Sn exist?
If the set of 2¡0 subsets of the natural numbers would exist, i.e., if Hessenbergs proof would be
valid, then one should expect that also all permutations of the natural numbers would exist and
(by the bijection of Ù and –) one should further expect that also all permutations of the rational
numbers, each rational number indexed by a natural number, should exist. Each permutation is
a well-ordering. One of them would be the well-ordering of – that is simultaneously the wellordering by size.
This is a contradiction. Like Hessenberg's assumption.
§ 354 The Continuum C (or the field R) appears as a numerical approximation to a complex
reality of observations. - It is not a set in the original sense of traditional set theory. In particular,
the power set axiom cannot be applied to it. [...] the entaglement phenomenon involving light (or
electrons), discovered by Albert Einstein, 1935, John Stewart Bell, 1964, and Alain Aspect,
1982, demonstrates that the Continuum of light is interiorly somehow “tightly interlaced”, so that
the distance, even enormous, between its “entangled” points becomes unimportant; (3) this
signifies that the Continuum is not a set, a “bag of points”, but that the points on it appear as the
consequence of our activities.
[Edouard Belaga: "From Traditional Set Theory – that of Cantor, Hilbert, Gödel, Cohen – to Its
Necessary Quantum Extension", Institut des Hautes Études Scientifiques (2011)]
http://preprints.ihes.fr/2011/M/M-11-18.pdf
§ 355 The preface {{of my book "Mathematik für die ersten Semester"
https://portal.dnb.de/opac.htm;jsessionid=CBA7F11D6BADF59FACD934DA2F83CE82.prodworker4?method=showFullRecord&currentResultId=Wolfgang+M%C3%BCckenheim%26any&c
urrentPosition=1
}} suggests that the book lays a solid foundation for, amongst other things, computer science.
{{That is correct.}} Presumably in WMaths, the thing that is not exactly a set of real numbers is
countable, as is the thing that is not quite a set of computable numbers. {{Meanwhile it should be
clear that "countable" with respect to infinite is a self-contradictory notion. Why not simply say
infinite?}} Apart from simply being wrong {{How that? Today everybody can be enlightened
enough to know that a set with more than countably many elements is simply the delusion of a
small minority of lunatics who adhere to Cantor's transfinity which simply has no place in any
rational mind. Even if God had created an uncountable set, no human could apply more than
countably many of its elements. No mathematical result, no question or equation could apply or
ask for an undefinable real number. But mathematics is what mathematicians do - not what
matheologians counterfactually believe.}}, it raises the question of whether or not they are the
same. Only WM knows. {{The book says that Ù is a proper subset of —.}} [Ben Bacarrise, "§ 350
The Four Most Simple Contradictions of Transfinite Set Theory", sci.logic (26 Sept. 2013)]
§ 356 It seems likely that only in the realm of pure mathematics can the idea of infinity be
entertained. In the context of actual, manifest, realisable quantities things seem much more like
the situation in a computer where all phenomena have definite resolution and size. One can
never create an infinitely large file because that would require an infinite amount of time and
infinite computational resources such as memory.
In my own work, which uses computational concepts to model reality I take the position that
the phenomena can be arbitrarily large and detailed but they always have a definite finite value.
So this allows for potential infinity but totally disallows actual infinity. Given that any set must be
actually represented using data (e.g. binary data), then no set can be infinitely large and if one
removes any members of the set then the cardinality (size) of the set is reduced. So any
representable set cannot be an infinite set and any infinite set cannot be actually represented.
Furthermore, in the context of computational metaphysics, representation is equivalent to
existence. If something is represented and it takes part in the overall simulation of the universe
then it exists in that universe but if it cannot be represented then it cannot exist. So if actual
infinities exist then there cannot be any discrete computational foundation to reality but so far no
actual infinites have ever been discovered.
Even with the domain of pure mathematics, infinities can only exist because they are
symbolically represented and never actually represented. No one has ever written out an infinite
number of integers thereby actually representing the set of integers. It is only ever referred to but
never fully represented. If one required sets to be fully represented then mathematics could not
operate on actual infinite sets; it could only operate on potentially infinite sets which always have
finite representations (e.g. {1, 2, 3}) but which are unlimited in their length. Such sets are
arbitrarily large but always have a definite finite size.
[John Ringland: Does Infinity Exist?]
http://www.anandavala.info/TASTMOTNOR/Infinity.html
§ 357 Nonstandard mathematics shows new ways of making mathematical discourse more
intuitive without losing logical rigor and giving more flexible ways of constructing mathematical
objects. We may say that by discriminating between "actual finiteness" and "ideal finiteness", we
obtain a better system of handling infinity than the "actual infinity" offers.
The followings are some of the features of our approach radically different from the usual
mathematics.
Sets are finite. The usual "infinite sets" such as Ù and – are considered as proper classes so
that the totality is not considered as a definite object.
Sorites Axiom. A number x is called accessible if there is a certain concrete method of
obtaining it. We postulate the existence of inaccessible numbers as the most basic axiom of our
framework. The accessible numbers form an nonending number series which is closed under
the operation x Ø x+1 but differes from the total number series. Accordingly, fundamental
notions such as transitivity, equivalence relation, provability, compatibility, etc. become relative
to the number series chosen.
The overspill axiom. If an objective condition holds for all accessible numbers, then it holds
also for an inaccessible number. Here a condition is called objective if it can be specified without
the notion of accessibility.
Vague conditions. The vaguness of the accessibility prohibits us to regard the collection of
accessible numbers as a set. It is a proper class contained in a finite set, called semiset in
Alternative Set Theory of Vopenka.
Continua are not infinite sets. The real line is considered as the "quotient" of the proper class
– by the indistinguishability relation defined by r º r' if and only if k|r - r'| < 1 for every accessible
number k.
[Toru Tsujishita: "Alternative Mathematics without Actual Infinity", arXiv (2012)]
http://arxiv.org/pdf/1204.2193v2.pdf
§ 358 Das Unendliche und die Theologie / Infinity and Theology (1)
Für die breite Masse gilt Adam Ries(e) als größter (und häufig auch als einziger) deutscher
Mathematiker. In bildungsnäheren Schichten nimmt Carl-Friedrich Gauß diese Position ein. Die
Mathematiker selbst aber verehren ihren größten Kollegen in Georg Cantor. Die höchste
Auszeichnung der DMV trägt sein Konterfei und seinen Namen, denn er hat die Mathematik
unendlich erweitert und bereichert --- so glauben jedenfalls die meisten. Über Georg Cantor,
Schöpfer der Mengenlehre und Gründer und erster Vorsitzender der Deutschen MathematikerVereinigung, wurden mehr biografische Notizen gesammelt und veröffentlicht als über jeden
anderen Mathematiker des 19. Jahrhunderts. Aus diesen Mosaiksteinen lässt sich ein
plastisches Bild seiner Weltsicht zusammensetzen. Einige Aspekte, vor allem theologischer
Natur, die zu seinem Verständnis des Unendlichen führten und dies scheinbar untermauerten,
sollen in den folgenden Beiträgen nachgezeichnet werden.
The educationally disadvantaged populace admires Adam Ries(e) as the greatest (and often as
the only) German mathematician. For the educated class Carl-Friedrich Gauss assumes this
position. The mathematicians themselves however admire Georg Cantor as their greatest
colleague. The highest award of the German Mathematical Union (DMV) carries Cantor's
likeness and name, because he extended and enriched mathematics infinitely - at least many
believe that. More biographical material has been collected about Georg Cantor, inventor of set
theory and first president of the DMV, than about any other mathematician of the 19th century.
From these tesserae we can obtain a vivid picture of his world view. In the following paragraphs
some of the theological aspects which lead to and seemingly supported his understanding of the
infinite will be reproduced. The translations into English are mine. The German originals can be
seen in de.sci.mathematik.
§ 359 Das Unendliche und die Theologie / Infinity and Theology (2)
Gestatten Sie mir aber dazu zu bemerken, daß mir die Realität und absolute Gesetzmäßigkeit
der ganzen Zahlen eine viel stärkere zu sein scheint als die der Sinnenwelt. Und daß es sich so
verhält, hat einen einzigen, sehr einfachen Grund, nämlich diesen, daß die ganzen Zahlen
sowohl getrennt wie auch in ihrer actual unendlichen Totalität als ewige Ideen in intellectu Divino
im höchsten Grade der Realität existiren. [Cantor an Hermite, 30. Nov. 1895]
Allow me to remark that the reality and the absolute principles of the integers appear to be much
stronger than those of the world of sensations. And this fact has precisely one very simple
reason, namely that the integers separately as well as in their actually infinite totality exist as
eternal ideas in intellectu Divino in the highest degree of reality.
§ 360 Das Unendliche und die Theologie / Infinity and Theology (3)
Hochverehrter Pater Ign. Jeiler.
Es freut mich sehr, aus Ihrem freundlichen Schreiben vom 20ten Oct. zu ersehen, daß jetzt Ihre
Bedenken gegen das "Transfinitum" geschwunden sind. Gelegentlich will ich Ihnen aber einen
kleinen Aufsatz verfaßen und zuschicken, in welchem ich in scholastischer Form detaillirt zeigen
möchte, wie sich meine Resultate gegen die bekannten Argumente vertheidigen lassen und vor
Allem, wie durch mein System die Grundlagen der christlichen Philosophie in allem
Wesentlichen unverändert bleiben, nicht erschüttert, sondern vielmehr eher gefestigt werden,
und wie sogar damit ihre Ausbildung nach verschiedenen Seiten gefördert werden kann.
[Cantor an P. Ignatius Jeiler, 27. Okt. 1895]
Highly esteemed Pater Ign. Jeiler,
I am very glad to see from your friendly letter of 20 Oct. that meanwhile your qualms against
the "transfinitum" have disappeared. Betimes I will write and send you a little essay where I want
to show you in scholastic form in detail how my results can be defended against the well-known
arguments and, above all, how by my system the foundations of christian philosophy in all
essentials remain unchanged, they are not shaken but rather become fixed, and how even their
development in different directions can be promoted.
§ 361 Das Unendliche und die Theologie / Infinity and Theology (4)
Da ich nun aber auch keine Stütze von früheren wissenschaftlichen Autoritäten für meine
Ansichten habe finden können, so weit ich auch in die beiden letzten Jahrhunderte, in das
Mittelalter und selbst in das griechische Alterthum mich zurückversetzte, war es mir, so
sonderbar es Ihnen vielleicht vorkommen wird, eine gewisse Befriedigung in Exodus, cap. XV, v.
18 wenigstens eine Art von Anklang an die transfiniten Zahlen zu finden, indem es dort heisst:
"Dominus regnabit in infinitum (aeternum) et ultra". Ich meine dieses "et ultra" ist eine
Andeutung dafür, dass es mit dem ω nicht sein Bewenden hat, sondern dass es auch darüber
hinaus noch was giebt. [Cantor an Lipschitz, 19. Nov. 1883]
I have not been able to find support for my opinions from ancient scientific authorities, how far I
went back into the two last centuries, into the Middle Ages and even into Greek antiquity.
Therefore it was a certain satisfaction for me, how strange this may appear to you, to find in
Exodus XV verse18 at least something reminiscent of transfinite numbers, namely the text: "God
is king in eternity and beyond". I think this "and beyond" is a hint to the fact that ω is not the end
but that something is existing beyond.
§ 362 Das Unendliche und die Theologie / Infinity and Theology (5)
Unter einem Actual Unendlichen ist dagegen ein Quantum zu verstehen, das einerseits nicht
veränderlich, sondern vielmehr in allen seinen Teilen fest und bestimmt, eine richtige Konstante
ist, zugleich aber andrerseits jede endliche Größe derselben Art an Größe übertrifft. Als Beispiel
führe ich die Gesamtheit, den Inbegriff aller endlichen ganzen positiven Zahlen an; diese Menge
ist ein Ding für sich und bildet, ganz abgesehen von der natürlichen Folge der dazu gehörigen
Zahlen, ein in allen Teilen festes, bestimmtes Quantum, ein aphorismenon das offenbar größer
zu nennen ist als jede endliche Anzahl.
Fußnote: Man vgl. die hiermit übereinstimmende Auffassung der ganzen Zahlenreihe als
aktual-unendliches Quantum bei S. Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui
dicunt ea, quae infinita sunt, nec Dei posse scientia comprehendi. Wegen der großen
Bedeutung, welche diese Stelle für meinen Standpunkt hat, will ich sie wörtlich hier aufnehmen
[...] Indem nun der h. Augustin die totale, intuitive Perzeption der Menge (nü), "quodam ineffabili
modo", a parte Dei behauptet, erkennt er zugleich diese Menge formaliter als ein aktualunendliches Ganzes, als ein Transfinitum an, und wir sind gezwungen, ihm darin zu folgen.
[Cantor an Prof. Dr. med. A. Eulenburg, Berlin, 28. Feb. 1886]
By the actual infinite we have to understand a quantity that is not variable but fixed and defined
in all its parts, really a constant, but also exceeding every finite size of the same kind by size. As
an example I mention the set of all finite positive integers; this set is a self-contained thing and
forms, apart from the natural sequence of its numbers, a fixed and defined quantity, an
aphorismenon, which is obviously larger than every finite number.
Footnote: Compare the concurring perception of the whole sequence of number as an actually
infinite quantum by S. Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui dicunt ea,
quae infinita sunt, nec Dei posse scientia comprehendi. Because of its great importance for my
position I will quote it here in full [...] While now the h. Augustin claims the total, intuitive
perception of the set (nue), "quodam ineffabili modo", a parte Dei, he acknowledges this set
formally as an actual infinite entity, as a transfinitum, and we are forced to follow him in this
matter.
§ 363 Das Unendliche und die Theologie / Infinity and Theology (6)
Sie haben, soviel ich weiß, in Spanien 10 Universitäten, von denen aber die Theologie
ausgeschloßen ist, welche bei Ihnen nur in Priesterseminaren gelehrt wird. Die frühere
Einrichtung der Universitäten, wo die Theologie mit einbegriffen war, halte ich für die beßere,
sowohl für Spanien, wie auch für Frankreich, wo ja derselbe Ausschluß eingeführt worden ist.
Nicht nur lege ich Werth darauf, daß die übrigen Wissenschaften sich nicht principiell feindlich
gegen die angestammte Theologie verhalten, sondern ich glaube, daß auch die Theologie durch
eine engere Anlehnung an die übrigen Facultäten nur Nutzen für sich ziehen kann.
[Cantor an Don Zoel Garcia de Gáldeano, 1893, zitiert in einem Brief von Cantor an
Baumgartner, 15. Dez. 1893]
You have, as far as I know 10 universities in Spain. Theology however is excluded and is only
taught in seminaries. The former constitution of universities, which included theology, is better in
my opinion. That holds for Spain as well as for France, where the same exclusion has been
introduced. I do not only care about a non-hostile attitude of the other sciences towards the
ancestral theology, but I believe that also theology can only stand to gain from a close relation to
the other faculties.
§ 364 Das Unendliche und die Theologie / Infinity and Theology (7)
Die Speculation, zumal die mathematische nimmt schon seit zehn Jahren ungefähr, nachdem
ich die eigentlichen Hauptschwierigkeiten in Bezug auf das Transfinite überwunden hatte, nur
einen geringen Theil meiner Zeit in Anspruch, die ich vielmehr der Theologie und "guten
Werken" widme.
[...] Sie können also sicher sein, daß ich an dem, Don Zoel Garcia de Galdeano gegenüber
eingenommenen Standpuncte fest halten und, soweit es in meinen Kräften steht, das meinige
dazu beitragen werde, in Spanien gesündere Zustände anzubahnen, wie ich dies auch für Italien
und Frankreich durch meine Verbindungen mit den dortigen Mathematikern seit vielen Jahren
erstrebe.
[...] Zum Begriffe einer Universität gehört das einigermaaßen friedliche Zusammenleben und
Wirken der vier Facultäten; und wenn dieses Verhältniß seit der Reformation in vielen
catholischen Ländern zuerst in‘s Schwanken gekommen und nachher ganz aufgelöst worden ist,
so muß Alles geschehen (mit Vorsicht und Klugheit selbstverständlich) um jenen, allein
naturgemäßen Zustand mit der Zeit wieder herbeizuführen.
[Cantor an P. Alexander Baumgartner SJ, 27. Dez. 1893]
The speculation, in particular the mathematical one, occupies only a small part of my time, after
having overcome the original main difficulties with respect to the tranfinite. I devote the
speculation to theology and "good works".
[...] You can be sure that I will adhere to the standpoint, mentioned to Don Zoel Garcia de
Galdeano, and contribute according to my power to initiate healthier states in Spain, as I have
been trying for years for Italy and France by means of my relations with the mathematicians
there.
[...] The institution university requires the rather peaceful collaboration of the four faculties.
Caused by the reformation this relation in many catholic countries first has been shaken and
then completely deleted. Everything has to be done (with care and cleverness, of course) to
reestablish step by step this only natural state.
§ 365 Das Unendliche und die Theologie / Infinity and Theology (8)
Sie würden auch ein "gutes Werk“ thun, wenn Sie darauf hinwirken wollten, daß gelegentlich,
wenn auch nur auf kurze Zeit, einige Ihrer jüngeren, der Metaphysik sich widmenden Patres
hierher zu mir geschickt würden, um mit mir über das actuale Unendliche, (diese „quaestio
multis molestißima de infinita multitudine“ wie sich Card. Franzelin in seinem Tr. de Deo uno
sec. nat. Thes. XLI ausdrückt) privatißime zu disputiren. Denn Sie können sich darauf verlaßen,
daß, der Standpunct, den in dieser Frage die Mehrzahl Ihrer Patres (aber auch die Mehrzahl der
kathol. Theologen) einnimmt, auf die Dauer völlig unhaltbar ist. [...] bemerke ich, daß größere
Vorkenntniße in der Mathematik zum Verständniß meiner Lehre nicht nöthig sind, sondern nur
eine gründliche philosophische Vorbildung, wie sie ja bei Ihnen auf‘s Beste und Schönste erlangt
wird.
[...] Das einzige, wofür ich diesem Langbehn {{deutscher Erfolgsautor des Buches
"Rembrandt"}} dankbar bin, ist, daß er [...] mich auf eine angebliche Aehnlichkeit meines Kopfes
und Gesichts mit dem heiligen Ignatius von Loyola aufmerksam machte. Deßen Exercitien
kenne und lese ich seit vielen Jahren. Möglicherweise ist aber auch dieser Vergleich ein
ebensolcher Quatsch und ebenso verrückt, wie die meisten Vergleiche seines "Rembrandt“.
[...] Soll übrigens meine Wirksamkeit in Spanien erfolgreich werden, so würde ich für eine
geschickte Cooperation Ihrer Patres in verwandtem Sinne sehr dankbar sein.
[Cantor an P. Alexander Baumgartner SJ, 27. Dez. 1893]
You would also do a "good work" if you would push some of your younger, metaphysically
interested patres to occasionally visit me for a short time and discuss privatissime with me about
the actual infinite, (this "quaestio multis molestissima de infinita multitudine“ as Card. Franzelin
calls it in his Tr. de Deo uno sec. nat. Thes. XLI). You can be sure that the point of view of the
majority of patres (but also of the catholic theologians) is in the long term completely untainable.
[...] I mention that great previous knowledge of mathematics is not required to understand my
techings, but only extensive philosophical knowledge, as it is learned best and most beautifully
at your institution.
[...] The only reason I have to be grateful to Langbehn {{German author of the best seller
"Rembrandt"}} is his hint to his observation that my head allegedly resembles the face of the
holy Ignatius of Loyola. I know his spiritual exercises and have been reading them for many
years. Perhaps this has had an influence on my looks. But perhaps this comparison is as much
nonsense and silly as most comparisons in his "Rembrandt".
[...] Should my agitation in Spain become successful, I would be very grateful for a clever
cooperation of your patres in kindred spirit.
§ 366 Das Unendliche und die Theologie / Infinity and Theology (9)
Was die dritte auf das A. U bezügliche Frage, nämlich nach dem A. U in Deo aeterno
omnipotenti seu in natura naturante betrifft, (den letzteren Ausdruck habe ich einigen grossen
Scholastikern entnommen) so zweifle ich nicht, dass wir hier wieder hinsichtlich der Bejahung
ganz einer Ansicht sind. Das letztere A.U, d.h. das A.U in Deo, nenne ich, wie Sie in meinem
Schriftchen "Grundlagen" bemerkt haben werden, das Absolute und es fällt dasselbe ganz
ausserhalb der Zahlentheorie. Dagegen sind das A. U in abstracto und in concreto, wo ich es
Transfinitum nenne, nicht nur Gegenstand einer erweiterten Zahlentheorie, sondern auch, wie
ich noch zu zeigen hoffe, einer avancirten Naturwissenschaft und Physik.
[Cantor an P. Ignace Carbonnelle SJ, 28. Nov. 1885, aus C. Tapp: "Kardinalität und Kardinäle",
Franz Steiner Verlag (2005)]
With respect to the third question concerning the A. I {{actual infinite}}, namely the A. I in Deo
aeterno omnipotenti seu in natura naturante (the last expression I have adopted from some
great scholastics) I have no doubt that we agree again in its approval. The last A. I, i.e., the A. I
in Deo, I call the Absolute, as you will have noted in my little essay "Grundlagen", and this falls
completely out of number theory. The A. I in abstracto and in concreto, however, where I call it
transfinitum, are not only subject of an extended number theory but also, as I hope to show, of
an advanced natural science and physics.
§ 367 Das Unendliche und die Theologie / Infinity and Theology (10)
Mit großem Interesse habe ich Ihre Schrift: "Die Lehre des hl. Thomas von Aquino über die
Möglichkeit einer anfangslosen Schöpfung“ studiert. Es war mir eine innige Befriedigung, von so
berufener Seite die Stellung des heil. Thomas zur Frage des actualen Unendlichen erörtert zu
sehen und mich zu überzeugen, daß ich den heil. Thomas in diesem Puncte und den damit
zusammenhängenden Fragen richtig verstanden habe, daß vor Allem seine Argumentation
gegen das actuale Unendliche in creatis resp, gegen die Möglichkeit act. unendl. großer Zahlen
für ihn selbst nicht die Bedeutung einer demonstratio, quae usquequaque de necessitate
concludit und metaphysische Gewissheit liefert, gehabt hat; sondern sie war in seinen eigenen
Augen nur in gewissem Grade probabel. [...] Ich stehe überall auf demselben Boden wie Sie und
freue mich daher umsomehr, daß aus Ihrem Schreiben Ihre Absicht hervorzugehen scheint,
meine Lehre vom Transfiniten einer gründlichen Prüfung zu unterwerfen.
[Cantor an P. Thomas Esser 0P, 5. Dez. 1895, aus C. Tapp: "Kardinalität und Kardinäle", Franz
Steiner Verlag (2005)]
With great interest I have studied your essay: The teachings of holy Thomas of Aquino about the
possibility of a creation without beginning. It was very satisfying for me to see the position of holy
Thomas concerning actual infinity be discussed from such a profound expert and to learn that I
had correctly understood holy Thomas in this point and related questions, in particular that his
arguments against the actual infinite in creatis or against the possibility of actually infinitely great
numbers has, for himself, not the meaning of a demonstratio, quae usquequaque de necessitate
concludit leading to metaphysical certainty, but was in his own eyes only probably in a certain
degree. [...] I stand everywhere on the same ground as you and I am the more happy that your
letter appears to contain your intention to thoroughly examine my teachings of the transfinite.
§ 368 Das Unendliche und die Theologie / Infinity and Theology (11)
Die allgemeine Mengenlehre [...] gehört durchaus zur Metaphysik. Sie überzeugen sich hiervon
leicht, wenn sie die Kategorieen der Kardinalzahl und des Ordnungstypus, dieser Grundbegriffe
der Mengenlehre, auf den Grad ihrer Allgemeinheit prüfen.
[...] und auch der Umstand, dass die unter meiner Feder noch stehende Arbeit in
mathematischen Journalen herausgegeben wird, modificirt nicht den metaphysischen Inhalt und
Charakter derselben.
[...] Von mir wird der christlichen Philosophie zum ersten Mal die wahre Lehre vom Unendlichen
in ihren Anfängen dargeboten. Ich weiß ganz sicher und bestimmt, dass sie diese Lehre
annehmen wird, es fragt sich nur, ob schon jetzt oder erst nach meinem Tode. Dieser Alternative
stehe ich vollkommen gleichmüthig gegenüber, sie berührt nicht meine arme Seele, die ich
vielmehr, verehrter Pater, in Ihr und der Ihrigen frommes Gebet empfehle.
[Cantor an P. Thomas Esser OP, 15. Feb. 1896, aus C. Tapp: "Kardinalität und Kardinäle",
Franz Steiner Verlag (2005)]
The general set theory [...] definitely belongs to metaphysics. You can easily convince yourself
when examining the categories of cardinal numbers and the order type, these basic notions of
set theory, on the degree of their generality.
[...] and the fact that my presently written work is issued in mathematical journals does not
modify the metaphysical contents of this work.
[...] By me christian philosophy is for the first time confronted with the true teachings of the
infinite in its beginnings. I know quite firmly and certainly, that my teachings will be accepted.
The question is only, whether this will happen before or after my death. But I am completely
calm about this alternative. It does no touch my poor soul wich, however, dear Pater, I
recommend to your and yours pious prayer.
§ 369 Das Unendliche und die Theologie / Infinity and Theology (12)
Am Meisten würde es mich aber freuen, wenn meine Arbeiten {{über transfinite Cardinalzahlen
und transfinite Ordnungstypen}} der meinem Herzen am Nächsten stehenden christlichen
Philosophie, der "philosophia perennis“, zugute kämen, was nur dann denkbar und möglich
wäre, wenn sie von der alten, nun durch S. Heiligkeit Leo XIII. so herrlich erneuerten,
wiedererstandenen Schule genau und eingehend untersucht und geprüft würden.
[Cantor an R. P. Thomas Esser 0P, 19. Dez. 1895]
I would be most happy if my works {{on transfinite cardinal numbers and transfinite order types}}
would be for the benefit of the christian philosophy which is next to my heart, namely the
"philosophia perennis". This would onyl then be thinkable and possible, if they would be
scrutinized by the old, meanwhile by His Holiness Leo XIII so beautifully restored, revived
school.
§ 370 Das Unendliche und die Theologie / Infinity and Theology (13)
Ein Wesen existiert, das alle positiven Eigenschaften in sich vereint. Das bewies der legendäre
Mathematiker Kurt Gödel mit einem komplizierten Formelgebilde. Zwei Wissenschaftler haben
diesen Gottesbeweis nun überprüft - und für gültig befunden.
[...] Die Existenz Gottes kann fortan als gesichertes logisches Theorem gelten.
{{So wie viele Sätze in ZFC. Dies ist zweifellos ein wichtiges Beispiel für die durch automatische
Beweisprüfer gewonnene Sicherheit.}}
http://www.spiegel.de/wissenschaft/mensch/formel-von-kurt-goedel-mathematiker-bestaetigengottesbeweis-a-920455.html
A being exist which reconciles all positive properties in itself. That has been proven by the
legendary mathematician Kurt Goedel by means of a complicated formula. Two scientists have
scrutinized this proof of God - and have approved it.
[...] The existence of God can in future be assumed to be a proven logical theorem. {{Like many
theorems of ZFC.}}
[Translated from the German text of SPIEGEL-ONLINE
http://www.spiegel.de/wissenschaft/mensch/formel-von-kurt-goedel-mathematiker-bestaetigengottesbeweis-a-920455.html
Goedel's ontological proof has been analysed for the first-time with an unprecedent degree of
detail and formality with the help of higher-order theorem provers. {{An important example for the
advantage of formalizing and the safety gained by checking theorems by means of theorem
provers}}
Christoph Benzmüller, Bruno Woltzenlogel Paleo: "Formalization, Mechanization and
Automation of Gödel's Proof of God's Existence", Arxiv (2013)
http://arxiv.org/abs/1308.4526
§ 371 Das Unendliche und die Theologie / Infinity and Theology (14)
Gestatten Sie, Monsignore, dass ich Ihnen beifolgend einen kleinen Aufsatz (in Correctur)
überreiche, von dem ich mir erlauben werde, Ihnen einige Exemplare unter Kreuzband zu
senden, sobald die Abzüge vollendet sein werden.
Es würde mich freuen, wenn der darin enthaltene Versuch, die drei Hauptfragen mit Bezug auf
das actuale Unendliche gehörig abzugrenzen, einer Prüfung auch seitens der christlichcatholischen Philosophen unterzogen würde.
[Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 17. Dez. 1885]
Ich kann mich gegenwärtig mit metaphysischen Erörterungen wenig beschäftigen; gestehe
jedoch, dass nach meiner Meinung das, was der Herr Verfasser das „Transfinitum in natura
naturata“ nennt, sich nicht vertheidigen lässt, und in einem gewissen Sinne, den ihm der Herr
Verfasser jedoch nicht zu geben scheint, den Irrthum des Pantheismus enthalten würde.
[Antwort von Franzelin an Cantor, 25. Dez. 1885]
Versuche, die ich schon vor vielen Jahren und neuerdings wiederholt gemacht habe, Mitglieder
der deutschen Provinz der S. J. zu einer solchen vertraulichen wissenschaftlichen
Correspondenz über das Actual-unendliche zu veranlassen, sind, obgleich viele von ihnen
meine Arbeiten seit mindestens zehn Jahren kennen und in Händen haben, ohne jeden Erfolg
geblieben, während doch der hochselige Cardinal J. B. Franzelin in seinen gerade vor 10 Jahren
an mich gerichteten Briefen auf die Bedeutung der Frage für Theologie u. Philosophie deutlich
genug hingewiesen hat.
[Cantor an R. P. Thomas Esser 0P, 19. Dez. 1895]
Monsignore, may I present you the included galley proofs of the little essay, of which I will send
you some copies as soon as it has been completed.
I would be glad if my attempt to properly distinguish between the three main questions with
respect to the actual infinite could be scrutinized thoroughly by christian-catholic philosophers.
[Cantor to Cardinal Joannis Baptistae Franzelin SJ, 17 Dec. 1885]
Presently I am rather unable to consider metaphysical arguments. But I confess that in my
opinion that which is called by the author the "Transfinitum in natura naturata“, cannot be
defended and in a certain sense, which however the author does not seem to claim, would
include the error of pantheism.
[Answer from Franzelin to Cantor, 25 Dec. 1885]
Attempts that I have made many years ago and repeatedly recently, to win members of the
German province of S. J. {{the Jesuites}} for a confidential scientific correspondence about the
actual infinite, have been without success although many of them have been knowing and
possessing my works for more than ten years, whereas the late Cardinal J.B. Franzelin very
plainly has been pointing to the importance of this question for theology and philosophy in his
letters directed to me.
[Cantor to R. P. Thomas Esser 0P, 19 Dec. 1895]
§ 372 Das Unendliche und die Theologie / Infinity and Theology (15)
Dementsprechend unterscheide ich ein "Infinitum aeternum sive Absolutum", das sich auf Gott
und seine Attribute bezieht, und ein "Infinitum creatum sive Transfinitum", das überall da
ausgesagt wird, wo in der Natura creata ein Actualunendliches constatirt werden muss, wie
beispielsweise in Beziehung auf die, meiner festen Ueberzeugung nach actual unendliche Zahl
der geschaffenen Einzelwesen, sowohl im Weltall, wie auch schon auf unsrer Erde und, aller
Wahrscheinlichkeit nach, selbst in jedem noch so kleinen ausgedehnten Theil des Raumes,
worin ich mit Leibniz ganz übereinstimme.
[Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 22. Jan. 1886]
Accordingly I distinguish an "Infinitum aeternum sive Absolutum" that refers to God and his
attributes, and an "Infinitum creatum sive Transfinitum" that has to be applied wherever in the
Natura creata an actual infinite is observed, like, for example, with respect to the, according to
my firm conviction, actually infinite number of created individuals, in the universe as already on
our earth and, most probably, even in each extended part of the space, however small it may be.
Here I agree completely with Leibniz.
§ 373 Das Unendliche und die Theologie / Infinity and Theology (16)
Obwohl ich weiss, dass diese Lehre vom "Infinitum creatum", wenn auch nicht von allen, doch
von den meisten Kirchenlehrern bekämpft wird und im Besonderen auch vom grossen St
Thomas Aquinatus [...] so sind doch die Gründe, welche in dieser Frage im Verlauf
zwanzigjähriger Forschung [...] sich mir aufgedrängt und mich gewissermaaßen gefangen
genommen haben, stärker, als Alles, was ich bisher dagegen gesagt fand, obgleich ich es in
weitem Umfange geprüft habe. Auch glaube ich, dass die Worte der heiligen Schrift, wie z. B.
Sap. c. 11, v. 21: "Omnia in pondere, numero et mensura disposuisti" in denen ein Widerspruch
gegen die actual unendlichen Zahlen vermuthet wurde, diesen Sinn nicht haben; denn gesetzt
den Fall, es gäbe, wie ich bewiesen zu haben glaube, actual unendliche "Mächtigkeiten“ d. h.
Cardinalzahlen und actualunendliche "Anzahlen wohlgeordneter Mengen“ d. h. Ordinalzahlen
[...] die ebenso wie die endlichen Zahlen feste, von Gott gegebene Gesetze befolgen, so würden
ganz sicherlich auch diese transfiniten Zahlen in jenem heiligen Ausspruche mitgemeint sein
und es darf daher, meines Erachtens, derselbe nicht als Argument gegen die actual unendlichen
Zahlen genommen werden, wenn ein Cirkelschluss vermieden werden soll.
Dass aber ein „Infinitum creatum“ als existent angenommen werden muß, läßt sich mehrfach
beweisen. [...]
Ein Beweis geht vom Gottesbegriff aus und schliesst zunächst aus der höchsten
Vollkommenheit Gottes Wesens auf die Möglichkeit der Schöpfung eines Transfinitum
ordinatum, sodann aus seiner Allgüte und Herrlichkeit auf die Nothwendigkeit einer thatsächlich
erfolgten Schöpfung des Transfinitum.
Ein andrer Beweis zeigt a posteriori, dass die Annahme eines "Transfinitum in natura naturata“
eine bessere, weil vollkommenere Erklärung der Phänomene, im Besondern der Organismen
und psychischen Erscheinungen ermöglicht, als die entgegengesetzte Hypothese. {{Letzteres
wurde von Cantor niemals weiter ausgeführt.}}
[Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 22. Jan. 1886]
Although I know that this teaching of the "Infinitum creatum", is objected, if not by all, yet by most
doctors of the church, and in particular by the great St Thomas Aquinatus [...] the reasons that
have imposed themselves on me and rather captivated me during 20 years of reasearch [...] are
stronger than everything contrary I have heard, although I have checked that very carefully.
Further I believe that the words of the Holy Bible like Sap. c. 11, v. 21: "Omnia in pondere,
numero et mensura disposuisti“ which have been assumed to contradict infinite numbers, do not
have that meaning. Given the case, actually infinite "powers", i.e., cardinal numbers and actually
infinite "numbers of well-ordered sets", i.e., ordinal numbers [...] existed, as I think to have
proved, which like finite numbers obey firm laws given by God, so clearly also these transfinite
numbers would be covered by that holy remark - and it cannot be used against actually infinite
numbers if a circular argument shall be avoided.
It can be proved in different ways that an "Infinitum creatum“ has to be assumed. [...]
One of the proofs starts from the notion of God and concludes first from the highest perfection
of the Supreme Being on the possibility of the creation of a Transfinitum ordinatum. Then from
God's loving kindness and glory on the necessity of an actually created Transfinitum.
Another proof shows a posteriori that the assumption of a "Transfinitum in natura naturata“
delivers a better, more complete, explanation of the phenomena, in particular of the organisms
and physical phenomena than the contrary hypothesis. {{This has never been further elaborated
by Cantor.}}
§ 374 Das Unendliche und die Theologie / Infinity and Theology (17)
Vom Pantheismus glaube ich jedoch, dass er, und vielleicht nur durch meine Auffassung der
Dinge, mit der Zeit ganz überwunden werden könnte. [...] Was aber den Materialismus und die
damit zusammenhängenden Richtungen betrifft, so scheinen sie mir, gerade weil sie die
wissenschaftlich unhaltbarsten und am Leichtesten widerlegbaren sind, zu den Uebeln zu
gehören, von welchen das menschliche Geschlecht in dem zeitlichen Dasein nie ganz zu
befreien sein wird.
[Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 22. Jan. 1886]
In Ihrem werthen Schreiben an mich sagen Sie nämlich erstens ganz richtig (vorausgesetzt daß
Ihr Begriff des Transfinitum nicht blos religiös unverfänglich, sondern auch wahr ist, worüber ich
nicht urtheile), "ein Beweis geht vom Gottesbegriffe aus und schließt zunächst aus der höchsten
Vollkommenheit Gottes Wesens auf die Möglichkeit der Schöpfung eines Transfinitum
ordinatum.“ In der Voraussetzung, daß Ihr Transfinitum actuale in sich keinen Widerspruch
enthält, ist Ihr Schluß auf die Möglichkeit der Schöpfung eines Transfinitum aus dem Begriffe
von Gottes Allmacht ganz richtig. Allein zu meinem Bedauren gehen Sie weiter und schließen
"aus seiner Allgüte und Herrlichkeit auf die Nothwendigkeit einer thatsächlich erfolgten
Schöpfung des Transfinitum". Gerade weil Gott an sich das absolute unendliche Gut und die
absolute Herrlichkeit ist, welchem Gute und welcher Herrlichkeit nichts zuwachsen und nichts
abgehen kann, ist die Nothwendigkeit einer Schöpfung, welche immer diese sein mag, ein
Widerspruch.
[Franzelin an Cantor, 26. Jan. 1886]
I believe that pantheism, perhaps only by means of my theory of the things, can be overcome
completely. [...] Materialism and related ideas seem to me to belong to the evils of which, just
because they belong to the scientifically most untenable and easiest refutable, the human race
in its temporal existence will never be completely be released of.
[Cantor to Cardinal Joannis Baptistae Franzelin SJ, 22 Jan. 1886]
In your valued letter to me you say first quite right (provided that your notion of the transfinitum is
not only compatible with religion but also true, what I do not judge), "one of the proofs starts from
the notion of God and concludes first from the highest perfection of the Supreme Being on the
possibility of the creation of a transfinitum ordinatum." Assuming that your transfinitum actuale is
free of contradictions your conclusion on the possibility of the creation of a transfinitum out of the
notion of God's omnipotence is quite right. But to my regret you go on and conclude from his
"loving kindness and glory on the necessity of an actually created transfinitum". Just because
God himself is the absolute infinite good and the absolute glory, which good and which glory
nothing can be added and nothing can be missing, the necessity of some creation, whatever it
might be, is a contradiction.
[Franzelin to Cantor, 26 Jan. 1886]
§ 375 Das Unendliche und die Theologie / Infinity and Theology (18)
Auf eine weitere Korrespondenz über Ihre philosophischen Ansichten kann ich bei meinen vielen
Beschäftigungen, durch welche ich auf ein ganz anderes Feld angewiesen bin, mich ferner nicht
einlaßen; Sie mögen mich also entschuldigen, wenn ich auf Ihre etwaigen Repliken, welche ich
jedoch insoweit sie sich auf Ihr System beziehen zu unterlaßen bitte, nicht werde antworten
können.
[Franzelin an Cantor, 26. Jan. 1886]
{{Wie jeder von seiner Sache Überzeugte konnte es aber auch Cantor nicht unterlassen, zu
antworten. Er richtete noch zwei Briefe an den Kardinal, nämlich am 29. Jan. 1886 und in
anderem Zusammenhang am 18. Feb. 1886, worauf jedoch keine Antwort erfolgte. Eine
vollständige Sammlung aller bekannten Korrespondenz Cantors im geistlichen Umfeld findet
sich in C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der
Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol.
53, Franz Steiner Verlag (2005).}}
Ewr. Eminenz sage ich meinen herzlichsten Dank für die Ausführungen des gütigen Schreibens
vom 26ten dss., denen ich mit voller Ueberzeugung zustimme; denn in der kurzen Andeutung
meines Briefes v. 22. dss. war es an der betreffenden Stelle nicht meine Meinung, von einer
objectiven, metaphysischen Nothwendigkeit zum Schöpfungsact, welcher Gott, der absolut Freie
unterworfen gewesen wäre, zu sprechen, sondern ich wollte nur auf eine gewisse subjective
Notwendigkeit für uns hindeuten, aus Gottes Allgüte und Herrlichkeit auf eine thatsächlich
erfolgte (nicht a parte Dei zu erfolgende) Schöpfung, nicht bloss eines Finitum ordinatum,
sondern auch eines Transfinitum ordinatum zu folgern.
[Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 29. Jan. 1886]
I am not able to continue the correspondence about your philosophical opinions with you
because of my many occupations which direct me to quite another field. You might excuse if I
will not react on your possible replies, which however, as far as they will be related to your
system, I beg you to refrain from.
[Franzelin to Cantor, 26 Jan. 1886]
{{Like everyone who is sure of his ground Cantor could not refrain from answering. He directed
two further letters to the Cardinal, on 29 Jan. 1886 and, with another topic, on 18 Feb. 1886,
which however remained without reply. A complete collection of Cantor's known correspondence
with clerics is supplied by C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische
Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner
Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005).}}
Your Eminence, I thank you very much indeed for the clarification given in your kind letter of 26
January which I agree to with full conviction, because in the short hint in my letter of 22 january I
did not opine to talk about an objective, metaphysical necessity of the act of creation, which
God, the absolutely Free had been subject to, but I only wanted to point to a certain subjective
necessity for us, to conclude from God's loving kindness and glory on an actually done (not a
parte Dei to be done) creation, not only of a Finitum ordinatum but also of a Transfinitum
ordinatum.
[Cantor to Cardinal Joannis Baptistae Franzelin SJ, 29 Jan. 1886]
§ 376 Here are some arguments of the orthodox internet matheologian Hancher alias Virgil,
standing in place of many others, concerning the matrices of FISONs (Finite Initial Sequences
Of Naturals)
1
1, 2
1, 2, 3
...
or
1
2, 1
3, 2, 1
...
that cannot have more lines than columns.
He said:
- no row ends without a last n and no column starts without a first n, so where does an unpaired
row or column fit into that diagram?
- there must be precisely the same number of rows as columns
- the completion of that diagram has exactly as many rows as columns
- Nor more columns than lines, whether truncated finitely or carried to infinite completion.
- There are precisely the same number of rows as columns, and both are the same as the
number of naturals.
- So how can the number of rows differ from the number of columns if they both are in one-toone correspondence with the set of naturals?
- No one has opposed that no FISON has ¡0 elements.
- there are ¡0 different FISONs
At least one of these statements is wrong, namely in contradiction with the others.
§ 377 Das Unendliche und die Theologie / Infinity and Theology (19)
[...] weit gewichtigere Gründe hinzufügen lassen, die aus der absoluten Omnipotenz Gottes
fließen und denen gegenüber jede Negation der Möglichkeit eines "Transfinitum seu Infinitum
actuale creatum" wie eine Verletzung jenes Attributes der Gottheit erscheint.
[Cantor an Prof. Dr. Constantin Gutberlet, 24. Jan. 1886]
[...] far more Important reasons can be added which result from the absolute omnipotence of
God and with respect to which every negation of the possibility of a "Transfinitum seu Infinitum
actual creatum" appears like a violation of that attribute of God.
§ 378 Das Unendliche und die Theologie / Infinity and Theology (20)
Soll ich auch einen Punct erwähnen, worin ich mit Ihnen nicht ganz einverstanden bin, so ist es
das unbedingte Vertrauen, welches von Ihnen dem modernen sogenannten Gesetz von der
Erhaltung der Energie entgegengebracht wird. Ich will durchaus nicht die Lehre von der
Aequivalenz der verschiedenen sich ineinander umsetzenden natürlichen Kraftformen in Zweifel
ziehen, soweit sie experimentell hinreichend begründet ist. Das wogegen ich ernste Bedenken
hege, ist sowohl die Erhebung des angeblichen Gesetzes zu einem metaphysischen Prinzip,
von dem die Erkenntniss so gewichtiger Sätze, wie die Unsterblichkeit der Seele abhängig sein
soll wie auch die von den Herren Thomson, v. Helmholtz, Clausius und Genossen beliebte und
durch nichts gerechtfertigte Ausdehnung und Anwendung des Satzes von der Erhaltung der
Energie auf das Weltganze, woran phantastische Speculationen geknüpft werden, die ich für
ganz werthlos halte. {{Tja, wie man sich (und andere) täuschen kann.}}
[Cantor an Prof. Dr. Constantin Gutberlet, 1. Mai 1888]
Should I also mention a point where I do not quite agree with you, so it is your unreserved
confidence in the modern so-called law of energy conservation. I do not wish to doubt the
teaching of the equivalence of the different natural forces transforming into each other as far as
this has been experimentally verified. That wich I have serious reservations against is the
elevation of the asserted law into the rank of a metaphysical priciple which governs the
recognition of so important theorems as the immortality of the soul as well as its completely
unjustified extension and application onto the whole world system as the gentlemen Thomson, v.
Helmholtz, Clausius, and comrades, like to do, who add phantastic speculations which in my
opinion are without any value. {{Amazing, how ignorant one can be (and make others)}}
§ 379 Das Unendliche und die Theologie / Infinity and Theology (21)
Der Aufsatz von Pohle über die objective Bedeutung des unendlich Kleinen enthält sehr schöne
und gehaltreiche Betrachtungen. Nur irrt er mit der Annahme, daß das unendlich Kleine als
actuelles integrirendes oder constitutives Element zur Erklärung des Continuums resp. zur
Begründung der Infinitesimalrechnung nothwendig sei. Ich trete mit ihm für die objective
Bedeutung des unendliche Kleinen ein, doch nicht des unendlich Kleinen, sofern es ein
actuelles, unendlich klein seiendes wäre, vielmehr nur sofern es ein potenzielles, unendlich klein
werdendes ist. Als Element des Continuums ist das Unendlichkleine nicht bloß unbrauchbar,
sondern auch an sich undenkbar resp. unmöglich, wie ich streng beweisen kann.
[Cantor an Prof. Dr. Constantin Gutberlet, 1. Mai 1888]
The paper by Pohle about the objective meaning of the infinitely small contains quite nice and
comprehensive reflections. But he errs in the assumption that the infinitely small be necessary
as an actually integrating or constituent element for the explanation of the continuum or as a
foundation of the infinitesimal calculus. I agree with him concerning the objective importance of
the infinitely small, but not the infinitely small as far as it is something actual, being infinitely
small, rather only something potential, becoming infinitely small. As an element of the continuum
the infinitely small is not only unusable but even unthinkable or impossible as I can strictly prove.
§ 380 Resolution of the dissent between mathematics and matheology. Hancher alias Virgil,
one of the defenders of matheology, explains the difference:
WM: How can you dare to say that ¡0 columns are spanned by FISONs in the table
1
1, 2
1, 2, 3
...
Virgil: Because I dare speak truth.
WM: If not every natural is in one and the same FISON, then at least two FISONs are required to
contain every natural - two or more or infinitely many. But everybody who claims that this is true
should be able to name the first FISON of the required set of FISONs.
Virgil: If that were so, then somewhere in wikipedia or elsewhere on the net somebody other
than WM must have posted that "fundamental law of arithmetic".
WM: Every set of FISONs has a first element.
Virgil: That would only apply if there were only one set of FISONs whose union was Ù.
From these few sentences everybody can recognize the basic requirements of matheology - in
addition to the fact that enumerating infinite sets is a super task (cp. § 347) that cannot be
accepted as part of mathematics and that the resulting idea of uncountable sets implies the
"existence" of undefinable "real" numbers.
§ 381 Das Unendliche und die Theologie / Infinity and Theology (22)
Da er {{Cantor}} sich wegen dieses kühnen Unternehmens von allen Seiten angegriffen sah,
suchte er Sukkurs bei mir, dem einzigen, der, wie er glaubte, mit seiner Auffassung
übereinstimmte. Da er von edler Gesinnung war, teilte er nicht die Verachtung, mit welcher die
ungläubige Wissenschaft die christlichen Philosophen behandelt. Es war auch nicht die bloße
Not, welche ihn zu mir führte, sondern, wie er sagte, habe er darum eine katholikenfreundliche
Gesinnung, weil seine Mutter katholisch war. Er befragte mich über die Lehre der Scholastiker in
betreff dieser Frage. Ich konnte ihn besonders auf den hl. Augustin und auf den P. Franzelin,
den späteren Kardinal, hinweisen. Dieser mein hochverehrter Lehrer verteidigte die aktual
unendliche Menge in der Erkenntnis Gottes, gestützt auf die ausdrückliche Lehre des hl.
Augustin, und er war es, der mir den Anstoß zu jener Schrift gegeben, und mich bei den heftigen
Angriffen damit beruhigte, daß ich nur die Lehre des hl. Augustin vortrage. An den Kardinal
wandte sich Cantor selbst, und Äußerungen desselben teilt er, ohne ihn zu nennen, in einem
Aufsatze der "Zeitschrift für Philosophie und philosophische Kritik" mit.
[C. Gutberlet: Philos. Jahrbuch der Görres-Ges. 32 (1919) 364]
Since he {{Cantor}}, because of his bold endeavor, had been attacked from all sides, he tried to
get support from me, the only one who, as he believed, agreed with his opinions. Since he was
noble minded, he did not share the contempt of the disbelieving science against the christian
philosophers. And it was not only pure poverty which lead him to me, but, as he said, he had a
catholic-friendly attidude because his mother was catholic. He inquired with me about the
teachings of the scholastics with respect to this question. I could point him in particular to St.
Augustin and to P. Franzelin, the later cardinal. This highly esteemed teacher of mine defended
the actually infinite set in the cognition of God, supported by the explicit teaching of St. Augustin,
and it has been he, who induced that writing of mine and who calmed me during the violent
attacks with the argument that I only had repeated the teaching of St. Augustin. Cantor himself
then addressed the cardinal and reported his statements, without revealing his name, in an
essay of the "Zeitschrift für Philosophie und philosophische Kritik".
§ 382 Das Unendliche und die Theologie / Infinity and Theology (23)
Ich bin niemals von einem "Genus supremum" des actualen Unendlichen ausgegangen. Ganz
im Gegentheil habe ich streng bewiesen, daß es ein "Genus supremum" des actualen
Unendlichen garnicht giebt. Was über allem Finiten und Transfiniten liegt, ist kein "Genus"; es ist
die einzige, völlig individuelle Einheit, in der Alles ist, die Alles umfasst, das "Absolute", für den
menschlichen Verstand Unfassbare, also der Mathematik gar nicht unterworfene, Unmessbare,
das "ens simplicissimum", der "Actus purissimus", der von Vielen "Gott" genannt wird.
[Cantor an Mrs. Chisholm-Young, 20. Juni 1908]
I have never been assuming a "Genus supremum" of the actual infinite. On the contrary I have
proven strictly that a "Genus supremum" of the actual infinite does not exist. That which is higher
than all finite and transfinite is not a "Genus", it is the only absolutely individual unit, which
contains all, which comprehends all, the "Absolute", for the human intellect incomprehensible,
therefore not being subject to mathematics, unmeasurable, the "ens simplicissimum", the Actus
purissimus, which by many is called "God".
§ 383 Das Unendliche und die Theologie (24)
Nun hat Herr Bernstein die neue Unvorsichtigkeit begangen, in den mathem. Annalen zeigen zu
wollen, daß es "Mengen giebt, die nicht wohlgeordnet werden können". Ich habe keine Zeit nach
dem Fehler in seinem Beweise zu suchen, bin aber fest überzeugt, daß ein solcher vorhanden
ist. Hoffentlich kommt bald die Zeit und Gelegenheit, wo ich meine volle Meinung über alle
derartigen prämaturirten Versuche aussprechen kann.
[...] Das Fundament für meine Auffassung des Erlösungswerkes ist, daß Jesus der
vorausgesagte Messias der Juden und als solcher seiner Menschheit nach ein richtiger
Nachkomme Davids ist. Dies wissen wir auf's Sicherste von ihm selbst und als solcher gilt er
nach seiner Auferstehung allen seinen Aposteln. Von hier aus komme ich, wie Sie gesehen
haben, auf Grund des neuen Testaments dazu, zwei Josephs zu unterscheiden, den Königl.
Joseph und leiblichen Vater Christi und den Nährvater Joseph.
[...] Was die Auferstehung Christi betrifft (zu welcher Sie meine Stellung wissen wollen), so ist
sie durch die Schriften des neuen Testaments aufs Beste und Umfassendste bezeugt; ich
glaube fest daran, als an eine Thatsache und grüble nicht über das "Wie" derselben.
[Cantor an Jourdain, 3. Mai 1905]
Recently Mr. Bernstein has committed the new carelessness, to try to show in the mathem.
Annalen that "there are sets existing which cannot be well-ordered". I have not the time to look
for the error in his proof but I am firmly convinced that such an error exists. Hopefully time and
opportunity will come soon to frankly express my full opinion about all those immature attempts.
[...] The fundament of my opinion about redemption is that Jesus is the predicted Messiah of
the Jews and as such in his human nature a real descendant of David. This we know absolutely
sure from himself and as such he has been considered by all his apostles after his resurrection.
From this point I arrive, as you have seen, based on the New Testament, at the distinction of two
Josephs, the royal Joseph and physical father of Christ and the breadwinner Joseph.
[...] Concerning the resurrection of Christ (about which you have inquired me), this has been
attested best and most comprehensively by the writings of the New Testament. I firmly believe it
as a fact and do not brood over the "how" of it.
§ 384 Das Unendliche und die Theologie / Infinity and Theology (25)
Wie mir ein mir persönlich bekannter Ordensbruder des Autors erzählte, ist P. Esser momentan
in Rom als Mitarbeiter an einer hochwichtigen vom heil. Vater berufenen Commißion zur
Revision des Index [...] {{Diese "hochwichtige" Kommission erneuerte den Index der verbotenen
Bücher, d.h. der Bücher, die ein gläubiger Katholik nicht lesen durfte.}}
[Cantor an Prof. Dr. F.X. Heiner, Hochwürden, 31. Dez. 1895]
As I have been told by a personally known brother monk of the author, P. Esser is momentarily
in Rome as a co-worker of an extremely important commission, appointed by the Holy Father in
order to revise the Index [...] {{This "extremely important" commission renewed the index of
prohibited books, i.e., those books which a devout Catholic was not allowed to read.}}
§ 385 Das Unendliche und die Theologie / Infinity and Theology (26)
Vom katholischen Standpuncte muß man froh sein, daß Sie den Herrn Prof. Riehl [...] los
werden und man kann nur wünschen, daß kein Gesinnungsgenoße von ihm an seine Stelle tritt.
Denn diese Sorte ist im Stande, viel Unheil anzurichten, wie Sie es ja an Riehl selbst durch
lange Jahre hin erfahren haben. [...] Die Kieler Theologen mögen selbst sich überzeugen, was
sie an ihm haben und mögen zusehen, wie sie mit ihm fertig werden. Außerdem können wir
nicht wißen, ob nicht etwa die Göttliche Vorsehung gerade solche radikalen Leute den
protestantischen Universitäten aus dem Grunde zuweisen läßt, damit der Zersetzungs- und
Auflösungsprozeil des Protestantismus dadurch beschleunigt werde. Hätten wir wohl ein
Intereße daran, dies zu verhindern? Mit Nichten!
[...] Die Thatsache, daß ein Schüler und Freund des Herrn Prof. Riehl (D. Förster) wegen
Majestätsbeleidigung verurtheilt worden ist, müßte der Großherzoglich Badischen Regierung auf
privatem Wege (durch den Ihnen befreundeten Abgeordneten) als ein Grund vorgehalten
werden, den von Prof. Riehl empfohlenen Candidaten mit dem grössten Misstrauen zu
begegnen. Im Senat dürfte es besser sein, diesen Gesichtspunct nicht zu berühren.
[Cantor an Prof. Dr. F.X. Heiner, Hochwürden, 31. Dez. 1895]
From the catholic point of view we have to be happy that you got rid of Prof. Riehl [...] and one
can only wish that he will not be replaced by a like-minded person. Because this sort of men is
able to cause much damage, as you have experienced with Riehl over many years. [...] The
theologians of Kiel may convince themselves what they have got and may look how they can live
with him. Further we cannot know whether divine Providence places just such radical in
Protestant universities in order to accelerate the undermining and decay of Protestantism. Would
we be interested to hinder that? Not at all!
[...] The government of the grand duke of Baden should be informed in a private way (by your
friend, the member of parliament) of the fact that a pupil and friend of Prof. Riehl (D. Foerster)
has been sentenced because of lèse-majesté. This should be a reason to meet the candidate
recommended by Prof. Riehl with greatest suspiciousness. In the senate it may be preferable
not to touch this point.
§ 386 Das Unendliche und die Theologie / Infinity and Theology (27)
Sollten Sie Zeit haben, meine Arbeiten zu lesen; so werden Sie vielleicht finden, dass sehr
wenig mathematische Vorkentniße zum Verständniß derselben erforderlich sind.
[Cantor an Hemann, 28. Juli 1887]
Mit Bezug auf die Frage des actualen Unendlichen in creatis wiederhole ich zunächst, was ich
Ihnen vor einem Jahre schrieb, daß eine gelehrte Vorbereitung in der Mathematik zum
Verständniß meiner betreffenden Arbeiten ganz und gar nicht nöthig ist, so daß ein genaues
Studium der letzteren dazu ausreicht. Jeder, und vor allen der geschulte Philosoph ist in der
Lage, die Lehre vom Transfiniten zu prüfen und sich von ihrer Richtigkeit und Wahrheit zu
überzeugen.
[Cantor an Hemann, 2. Juni 1888]
Should you have time to read my papers, you might find that very little previous mathematical
knowledge is required for the understanding.
[Cantor to Hemann, 28 July 1887]
With respect to the question of the actual infinite in creatis I repeat first of all, what I wrote you
one year ago. For the understanding of my relevant papers a scholarly preparation in
mathematics is not at all necessary but a careful study is sufficient. Everyone and in particular
the trained philosopher is able to scrutinize the teaching of the infinite and to convince himself
from its correctness and truth.
[Cantor to Hemann, 2 June 1888]
§ 387 Das Unendliche und die Theologie / Infinity and Theology (28)
Daß S. Thomas der auf Aristoteles zurückführenden Schulmeinung in Bezug auf die actual
unendlichen Zahlen nur mit grossen Zweifeln und halben Herzens anhing, läßt sich mit
Sicherheit feststellen [...] Denn die Thomassche Doctrin "mundum incepisse sola fide tenetur
nec demonstrative probari posse" [Daß die Welt angefangen hat, wird nur im Glauben
festgehalten, und es ist nicht möglich, dies durch einen Beweis zu begründen.] findet sich
bekanntlich nicht bloß in jenem opusculo, sondern auch [...] noch an vielen anderen Stellen.
Diese Doctrin wäre aber unmöglich, wenn der Aquinate den Satz "es gibt keine actual
unendlichen Zahlen" für erwiesen gehalten hätte. Denn aus diesem Satz (wenn er wahr wäre)
würde demonstrative mit größter Evidenz folgen, daß eine unendliche Zahl von Stunden vor
diesem Augenblick nicht verflossen sein könnte; Es würde also das Dogma vom Weltanfang (vor
endlicher Zeit) nicht als bloßer Glaubenssatz haben vertheidigt werden können.
[Cantor an Hemann, 2. Juni 1888, entnommen einschließlich der Übersetzung der lateinischen
Passage aus C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der
Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol.
53, Franz Steiner Verlag (2005) p. 380f]
It can be absolutely ascertained that St Thomas only with great doubts and half-heartedly
adhered to the received opinion concerning the actually infinite numbers, going back to Aristotle.
Thomas' doctrine "It can only be believed but it is not possible to be proven that the world has
begun" is known to appear not only in that opusculo but also [...] in many other places.
This doctrine however would be impossible if the Aquinatus had thought that the theorem
"there are no actually infinite numbers" was proven. Because from this theorem, it would
immediately follow with greatest evidence that an infinite number of hours could not have passed
before the present moment. The dogma of a beginning of the world could not have been
defended as a pure doctrine of belief.
§ 388 Das Unendliche und die Theologie / Infinity and Theology (29)
Es hat der paganistisch falsche Satz {{Es gibt keine actual unendlichen Zahlen}} auch ohne die
Eigenschaft eines von der Kirche anerkannten Dogmas zu besitzen oder je besessen zu haben,
vermöge seiner dogmenähnlichen Verbreitung unermeßlichen Schaden der christlichen Religion
und Philosophie verursacht und man kann es daher dem heiligen Thomas von Aquino, meines
Erachtens, nicht hoch genug zu Dank anrechnen, daß er diesen Satz als durchaus zweifelhaft
auf‘s Deutlichste gekennzeichnet hat.
[Cantor an Hemann, 21. Juni 1888, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 384]
Zum Vergleich: Thomas von Aquin schreibt in der Summa Theologica I, Q. 7, A. 4: Nulla autem
species numeri est infinita, quia quilibet numerus est multitudo mensurata per unum. Unde
impossibile est esse multitudinem infinitam actu, sive per se, sive per accidens. Item, multitudo
in rerum natura existens est creata, et omne creatum sub aliqua certa intentione creantis
comprehenditur, non enim in vanum agens aliquod operatur. Unde necesse est quod sub certo
numero omnia creata comprehendantur. Impossibile est ergo esse multitudinem infinitam in
actu, etiam per accidens. Sed esse multitudinem infinitam in potentia, possibile est.
http://www.corpusthomisticum.org/sth1003.html
Keine Sorte von Zahlen ist unendlich; denn jede Zahl ist eine durch die Eins zu messende
Vielheit. Also kann es unmöglich, an sich oder zufällig, eine aktual unendliche Vielheit geben.
Desgleichen ist jede Vielheit, die in der Natur der Dinge existiert, geschaffen; und jedes
Geschaffene unterliegt einer bestimmten Absicht des Schaffenden; denn kein Wirkender wirkt
ziellos. Also ist es notwendig, daß alles Geschaffene unter eine ganz bestimmte Zahl fällt. Daher
kann es unmöglich eine aktual unendliche Vielheit geben auch nicht per Zufall. Aber eine
potentiell unendliche Vielheit ist möglich.
Ja, wir sind dem Aquinaten für diese klaren Worte zu Dank verpflichtet!
The pagan, wrong proposition {{There is no actually infinite number}}, even without possessing
the property of being a dogma acknowledged by the church or ever having been in that
possession, has, because of its dogma-like popularity, done unmeasurable damage to Christian
religion and philosophy, and one cannot, in my opinion, thank holy Thomas of Aquino too
effusively that he has clearly marked this proposition as definitely doubtful.
[Cantor to Hemann, 21 June 1888, quoted from C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 384]
For comparison: Thomas Aquinatus writes in his Summa Theologica I, Q. 7, A. 4: But no species
of number is infinite; for every number is multitude measured by one. Hence it is impossible for
there to be an actually infinite multitude, either absolute or accidental. Likewise multitude in
nature is created; and everything created is comprehended under some clear intention of the
Creator; for no agent acts aimlessly. Hence everything created must be comprehended in a
certain number. Therefore it is impossible for an actually infinite multitude to exist, even
accidentally. But a potentially infinite multitude is possible.
http://www.sacred-texts.com/chr/aquinas/summa/sum010.htm
Yes we have to be grateful for those clear words!
§ 389 Das Unendliche und die Theologie / Infinity and Theology (30)
Es wird von Ihnen das Verhältniß der beiden Sätze
I. "Die Welt hat sammt der Zeit vor einem endlichen Zeitabschnitt angefangen oder was
dasselbe sagt: die bisher verflossene Zeitdauer der Welt ist (mit dem Maaß etwa einer Stunde
gemeßen) eine endliche."
welcher wahr und christlicher Glaubenssatz ist und:
II. "Es giebt keine actual unendlichen Zahlen."
welcher falsch und heidnisch ist und daher kein christlicher Glaubenssatz sein kann,
ich sage, es wird von Ihnen das Verhältniß dieser beiden Sätze nicht richtig gedacht. [...]
Aus der Wahrheit des Satzes I folgt aber mit Nichten, wie Sie in Ihrem Briefe anzunehmen
scheinen, die Wahrheit des Satzes II. Denn der Satz I bezieht sich auf die concrete creatürliche
Welt; Satz II aber auf das ideale Gebiet der Zahlen; im letzteren könnte das actual Unendliche
vertreten sein, ohne deshalb in jener nothwendig vorkommen zu müßen. {{Das ist falsch. Jede
Zahl besitzt wie jeder Gedanke eine physikalische Darstellung als Elektronenkonfiguration in
einem Gehirn.}}
[Cantor an Hemann, 21. Juni 1888, entnommen aus C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 383]
Your understanding of the relation of the two propositions
I. "The world including the time has begun before a finite time interval or, what is the same, the
duration of the world elapsed until now (e.g., measured by hours) is finite."
which is true and a Christian dogma and
II. "Actually infinite numbers do not exist."
which is false and pagan and therefore cannot be a Christian dogma,
I say you have not the correct idea about the relation of these two propositions. [...]
The truth of proposition I does not at all imply, as you seem to assume in your letter, the truth
of proposition II. Because proposition I concerns the concrete world of creation; proposition II
concerns the ideal realm of numbers; the latter could include the actually infinite without its
necessarily being included in the former. {{That is wrong. Every number has, like every thought,
a physical representation as an electron configuration in a brain.}}
§ 390 Das Unendliche und die Theologie / Infinity and Theology (31)
Die Lehre vom Transfiniten ist weit davon entfernt, die Thomassche Doctrin in ihren
Fundamentenzu erschüttern. Dagegen wird meine Lehre in gar nicht so ferner Zeit als eine
geradezu vernichtende Waffe gegen allen Pantheismus, Positivismus und Materialismus sich
erweisen! {{Cantor als Terminator.}}
[Cantor an P. Joseph Hontheim S. J., 21. Dez. 1893]
Es besteht wohl volle Übereinstimmung darüber, dass Matheologie und Materialismus absolut
unvereinbar sind. Matheologie erfordert den Glauben an eine platonistische immaterielle
Existenz aller Mengen (freilich ohne die Menge aller Mengen und auch ohne die Menge aller
Mengen ohne die Menge aller Mengen und auch ohne die Menge aller Mengen ohne die Menge
aller Mengen ohne die Menge aller Mengen und auch ohne ...). Wie Engel im Himmel und
Seelen im Irgendwo müssen Zahlen ohne jedes rationale Fundament existieren - die meisten
ohnehin undefinierbar, unidentifizierbar, unerkennbar und natürlich unbenutzbar, also
unbrauchbar. Der Materialismus dagegen lehnt die Existenz immaterieller, unphysikalischer
Objekte ab. Es handelt sich um den allgegenwärtigen Widerstreit zwischen Glühen und Wissen,
zwischen Religion und Wissenschaft.
The teaching of the transfinite is far from shaking the fundaments of the doctrin of Thomas. The
time is not far, however, that my teaching will turn out to be a really exterminating weapon
against all pantheism, positivism and materialism. {{Cantor, the terminator!}}
I think we all can agree that matheology and materialism are absolutely incompatible.
Matheology is impossible without the credo in a platonist immaterial existence of all numbers
and all sets (without the set of all sets though and without the set of all sets without the set of all
sets and without the set of all sets without the set of all sets without the set of all sets and
without ...). Like angels in the heaven and souls in the somewhere numbers have to exist without
any rational foundation - most of them even remaining undefinable, unidentifyable,
unrekognizable, and unusable, i.e., useless. Materialism does not accept the existence of
immaterial, unphysical objects. We see the eternal conflict between burning and knowing,
between religion and science.
§ 391 Das Unendliche und die Theologie / Infinity and Theology (32)
Besonders kühn für seine Zeit, in dieser Sache, erscheint Rod. Arriaga S. J. [Rodrigo de
ARRIAGA SJ (1592 - 1667)]. (Ich bemerke hierbei, daß die Lehre vom creatürlichen actualen
Unendlichen (was ich Transfinitum nenne) bei Rod. Arriaga keineswegs widerspruchsfrei
begründet ist; dasselbe gilt von dem Minimen Em. Maignan [Emanual MAIGNAN OMin (1601 1676)]. Beide habe ich erst kennen gelernt lange nachdem ich meine Theorie innerlich fertig und
in‘s Klare gebracht hatte. Es fehlt auch beiden die richtige Begriffsbildung der transfiniten
Cardinalzahlen (Mächtigkeiten) und der transfiniten Ordnungstypen und Ordnungszahlen, also
gerade dasjenige Instrument, mit dessen Hülfe die ganze Lehre einwandsfrei wird.); aber auch
Suarez S. J. [Francisco SUAREZ SJ (1548 - 1617)] steht mir nicht so fern, wie es vielleicht den
Anschein hat. [...] Bei meiner Hochschätzung und Verehrung Ihres religiösen Ordens könnte ich
von keiner Seite mehr Ermuthigung ziehen, in meiner Arbeit fortzufahren, als von Ihnen und den
Ihrigen!
[Cantor an P. Joseph Hontheim S. J., 21. Dez. 1893, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 396f]
{{Die Wurzeln der Mengenlehre reichen in Zeiten zurück, in denen Giordano Bruno und Galileo
Galilei wegen Ketzerei verurteilt wurden.}}
Especially bold in this matter, with regard to his time, appears Rod. Arriaga S. J. [Rodrigo de
ARRIAGA SJ (1592 - 1667)]. (I mention here that the teaching of the creational actual infinite
(what I call transfinitum) by Rod. Arriaga has not at all been founded free of contradiction; same
is true for the Minime Em. Maignan [Emanual MAIGNAN OMin (1601 - 1676)]. Both I have only
become acquainted with a long time after I had completed my theory internally and had cleared
it. Both of them are lacking the correct notions of transfinite cardinal numbers (Maechtigkeiten)
and the transfinite order types and ordinal numbers, just that tool which helps to make the whole
theory faultless.); but also Suarez S. J. [Francisco SUAREZ SJ (1548 - 1617)] is not so
disconnected from my position as it might appear. [...] With respect to my high esteem and
admiration of your religious order I could not win more encouragement from any party to
continue in my work than from you and yours.
{{The roots of set theory reach back into times which saw Giordano Bruno and Galileo Galilei
sentenced as heretics.}}
§ 392 Das Unendliche und die Theologie / Infinity and Theology (33)
Von Leibniz beispielsw. ist es sicher, daß er e. creatürl. Unendl. i. verschied. Beziehungen als
wirklich existirend angenommen hat. [...] Dagegen hat Leibniz sowenig wie seine Vorgänger u.
Nachfolger die act. unendl. d. h. transf. Zahlen u. Ordnungstypen erkannt; er bestreitet sogar
ihre Möglichkeit.
[Cantor an P. Ignatius Jeiler, OFM, 20. Mai 1888, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 412]
It is certain that for instance Leibniz has assumed the creational infinite in different relations as
really existing. [...] On the other hand Leibniz has as little as his predecessors and successors
recognized the actually infinite transfinite numbers and order types; he even refutes their
possibility.
{{For further reading see § 20 and § 21.}}
§ 393 Das Unendliche und die Theologie / Infinity and Theology (34)
Während das hier Hervorgehobene (daß der Satz "totum majus est sua parte" in gewissem
Sinne falsch ist) in Bezug auf die Substanzialformen allgemein anerkannt ist (beispielweise
bleibt die Seele eines lebenden Organismus beim Wachsen oder Abnehmen des Körpers ihrem
wesentlichen Sein nach stets dieselbe {{ein wichtiger Aspekt im Reiche des aktual
Unendlichen}}), scheint man zu glauben, daß es für die accidentalen Formen nicht auch zutreffe.
Dieses Vorurtheil ist eben aus der Wahrnehmung entstanden, daß, wie ich soeben hervorhob,
bei endlichen Mengen, auf die man allein seine Betrachtungen beschränkt hatte, der Satz "tot. e.
majus sua parte" in Bezug auf die diesen Mengen zukommenden Cardinalzahlformen stets
richtig ist; ohne weitere Untersuchung, aber auch ohne jegliche Berechtigung, wurde seine
Gültigkeit im bezeichneten Sinne auch auf unendliche Mengen übertragen, und man darf sich
daher über die Widersprüche nicht wundern, die aus einer so grundfalschen Voraussetzung sich
ergaben.
[Cantor an P. Ignatius Jeiler, OFM, 20. Mai 1888, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 418]
Classical logic was abstracted from the mathematics of finite sets and their subsets [....]
Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to
all mathematics, and finally applied it, without justification, to the mathematics of infinite sets.
[Hermann Weyl: "Mathematics and logic: A brief survey serving as a preface to a review of The
Philosophy of Bertrand Russell", American Mathematical Monthly 53 (1946) 2-13.]
Warum ist es falsch, das Prinzip "totum majus est sua parte" auf unendliche Mengen zu
übertragen, nicht aber die Prinzipen der klassischen Logik, z.B. die Implikation, die u.a. auf
diesem Prinzip basiert?
Whereas the emphasized (that the principle "totum majus est sua parte" is wrong in a certain
sense) with respect to substantial forms is acknowledged in general (the soul of a living
organism, for instance, in its essential being remains always the same during the growing or
decreasing of the body {{an important aspect in the realm of the actually infinite}}) one seems to
believe that this does not refer to accidencial forms. This prejudice originates from the
observation that, as I just stressed, observation has been restricted to only finite sets which
always obey the principle "totum majus est sua parte" with respect to the cardinal number forms
belonging to them; without further investigation, but also without any justification, its validity has
been carried over to infinite sets, and there is no reason to be surprized about the contradictions
resulting from such an utterly wrong premise. [Cantor]
Classical logic was abstracted from the mathematics of finite sets and their subsets [....]
Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to
all mathematics, and finally applied it, without justification, to the mathematics of infinite sets.
[Weyl]
Why is it wrong to carry over to infinite sets the principle "totum majus est sua parte" but not the
principles of classical logic, for instance the implication that, among others, is based upon this
principle?
§ 394 Das Unendliche und die Theologie / Infinity and Theology (35)
Sie sagen [...] daß Ihnen d. Begriff des Transfiniten Schwierigkeiten verursache, weil Sie den
Satz nicht aufgeben könnten, daß, wo additio möglich, ein Potenzielles vorhanden sein muß. Es
ist aber von mir nicht behauptet. worden, daß ein Transfinitum nur Act. sei, vielmehr ist das
Transf. in dem Sinne Potenz, in welchem es vermehrbar ist; nur das Absolute ist actus purus
oder vielmehr actus purissimus.
[Cantor an P. Ignatius Jeiler, OFM, 20. Mai 1888, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 417]
Die Menge aller Zeilen einer Cantor-Liste ist vermehrbar. Also ist auch die Menge der sie
nummerierenden natürlichen Zahlen vermehrbar. Also ist eine nicht in der Liste vorkommende
Diagonalzahl kein Indiz für eine größere Mächtigkeit der reellen Zahlen sondern lediglich dafür,
dass die Menge der natürlichen Zahlen abermals und beliebig oft verdoppelt werden kann.
Tatsächlich könnte sie bei ausreichender Geduld und dem Vorhandensein des aktual
Unendlichen auch aktual unendlich oft verdoppelt werden, womit sich |Ù| ¥ ¡0ÿ2¡0 ergäbe.
You say [...] that you have problems with the notion of the transfinite because you cannot give
up the theorem that the possibility of additio implies the presence of a potential. But it has not
been asserted by me that a transfinitum be only act rather the transfinite in this sense is potency
in which sense it is multiplyable; only the absolute is actus purus or rather actus purissimus.
The set of all entries of a Cantor-list can be multiplied. Therefore also the set of numbers
enumerating the entries can be multiplied. Therefore the diagonal number resulting from the list
does not indicate a greater cardinality of the real numbers but only that the set of natural
numbers can be doubled again and again, as often as desired. In fact, given sufficient patience
and given that there are ¡0 steps possible, the natural numbers could be multiplied infinitely
often to get as many as |Ù| ¥ ¡0ÿ2¡0.
§ 395 Das Unendliche und die Theologie / Infinity and Theology (36)
Um scholastisch zu reden: was weiterer Vermehrung fähig ist, ist in potentia zu diesem weiteren
actus, also ein potentielles; fällt also unter diesen Begriff. Ihr transfinitum könnte also hiernach
nur eine Unterabtheilung des gewöhnlich gelehrten potentiellen Infiniten sein.
[Jeiler an Cantor, 22. Juni 1890]
Betrachte die folgende Liste: 1
2
3
...
1
11
111
...
Die natürlichen Zahlen 1, 2, 3, ... sind natürlich keiner Vermehrung fähig. Die natürlichen Zeilen
1, 11, 111, ... sind natürlich einer Vermehrung fähig. Man muss das nur richtig verstehen.
Alle Elemente aller abzählbaren Mengen können (durch natürliche Zahlen) nummeriert werden
und können durch die sie nummerierenden natürlichen Zahlen bezeichnet werden. Die
natürlichen Zahlen bilden demnach die umfangreichste Menge von allen Mengen, die
ausschließlich definierbare Elemente enthalten. Und andere kommen in der Mathematik nicht
vor.
To use scholastic terms: Something that can be multiplied is in potentia to this further actus,
hence something potential; it belongs to that notion. Your transfintum could be some subsection
of the usually taught potential infinite.
Consider the following list:
1
2
3
...
1
11
111
...
The natural numbers 1, 2, 3, ... are not capable of natural multiplication. The natural rows 1, 11,
111, ... are naturally capable of multiplication. You have to understand that only.
All elements of all countable sets can be enumerated by (natural numbers) and can be
denoted by the natural numbers enumerating them. The natural numbers therefore are the most
extensive set of all sets that contain exclusively definable elements. And others do not appear in
mathematics.
§ 396 Das Unendliche und die Theologie / Infinity and Theology (37)
Die Resultate, zu denen ich gelangt bin, sind diese: Ein solches Transfinitum, sowohl wenn es in
concreto, wie auch in abstracto gedacht wird, ist widerspruchsfrei, also möglich und von Gott
erschaffbar, so gut wie ein Finitum. [...] Alle diese besonderen Modi des Transfiniten existiren
von Ewigkeit her als Ideen in intellectu divino. [...] Wenn Sie diese Thatsache so ausdrücken,
daß Sie sagen: "jedes Transfinite ist in potentia zu einem weiteren actus und in sofern ein
potenzielles", so ist nichts dagegen einzuwenden. Denn actus purus ist nur Gott; dagegen jedes
Creatürliche, in dem von Ihnen gebrauchten Sinne, "in potentia zu einem weiteren actus sich
befindet."
Dennoch kann das Transfinite nicht als eine Unterabtheilung dessen angesehen werden, was
man gewöhnlich "potentielles Unendliches" nennt. Denn letzteres ist nicht (wie jedes individuelle
Transfinite und allgemein wie jedes Ding, das einer "Idea divina" entspricht) in sich bestimmt,
fest und unveränderlich, sondern ein in Veränderung Begriffenes Endliches, das also in jedem
seiner actuellen Zustände eine endliche Größe hat; Wie beispielsweise die vom Weltanfang
verflossene Zeitdauer, welche, wenn man sie auf irgend eine Zeiteinheit, z. B. ein Jahr, bezieht,
in jedem Augenblicke endlich ist, aber immerzu über alle endlichen Grenzen hinaus wächst,
ohne jemals wirklich imendlich groß zu werden.
[Cantor an P. Ignatius Jeiler, Ord. S. Franc., 13. Okt. 1895, nach H. Meschkowski: "Georg
Cantor: Leben, Werk und Wirkung" 2. Aufl. BI, Mannheim (1981) p. 271f]
Hier ist ein Beipiel für beide Formen des Unendlichen: Betrachte die Folge oder Liste
0.0
0.1
0.11
0.111
...
Als Diagonalzahl kann (dkk) = 0.111... erzeugt werden. Wäre das Unendliche nur potentiell (in
Veränderung Begriffenes Endliches - zu jeder Zeile gibt es eine nachfolgende) dann wäre die
Diagonalzahl selbst in jedem ihrer "actuellen Zustände" in der Liste. Um eine Diagonalzahl zu
erhalten, die sich von jeder Zahl der Folge unterscheidet, muss die Liste aktual unendlich viele
Zeilen besitzen (die Zeilenzahl muss größer als jede natürliche Zahl sein). Da aber die Zeilen
über die Diagonale mit den Spalten gekoppelt sind, ergibt sich ein Widerspruch. Die ersten
Indizes k liefern eine aktual unendliche Diagonalzahl, die zweiten Indizes k eine nur potentiell
unendliche.
The results, which I have arrived at, are as follows: Such a transfinite is free of contradiction,
therefore possible and creatable by God, as well as a finite. [...] All special modes of the
transfinite exist forever as ideas in intellectu divino. [...] If you express this fact by saying: "every
transfinite is in potentia to another actus and thus is a potential", so I have no objection.
Because actus purus is only God; but every creational, in the mentioned sense, is in potentia to
another actus.
Nevertheless the transfinite cannot be considered a subsection of that which is usually called
"potentially infinite". Because the latter is not (like every individual transfinite and in general
everything due to an idea divina) determined in itself, fixed, and unchangeable, but a finite in the
process of change, having in each of its actual states a finite size; like, for instance, the time
elapsed after the biginning of the world, which, measuered in some time-unit, for instance a
year, is finite in every moment, but always growing beyond all finite limits, without ever becoming
really infinitely large.
[Cantor to P. Ignatius Jeiler, Ord. S. Franc., 13 Oct. 1895, fromH. Meschkowski: "Georg Cantor:
Leben, Werk und Wirkung" 2nd ed. BI, Mannheim (1981) p. 271f]
Here we have an example for both kinds of the infinite: Consider the list or sequence
0.0
0.1
0.11
0.111
...
The diagonal number can be made (dkk) = 0.111... If infinity was only potential (always growing after every row there follows another row), then the diagonal number was in the list "in each of
its actual states". In order to get a diagonal number that is not in the list, the list need to have
actually infinitely many terms (the number of rows must be larger than every natural number).
Since horizontal and vertical rows are coupled by the diagonal, we get a contradiction. The first
indices k supply an actually infinite diagonal number, the second indices k an only potentially
infinite.
§ 397 Das Unendliche und die Theologie / Infinity and Theology (38)
{{An mindestens vier verschiedenen Stellen seiner erhaltenen Korrespondenz (und das ist leider
nur ein kleiner Bruchteil der gesamten) betont Cantor:}} Zur Auffassung des Grundgedankens
der Lehre des Transfiniten bedarf es keiner gelehrten Vorbildung in der neueren Mathematik;
dieselbe kann dazu sogar hinderlich sein, weil in der sogenannten Infinitesimalanalysis das
potenziale Unendliche sich in den Vordergrund gedrängt und selbst bei den Heroen die Meinung
gezeitigt hat, als beherrschten sie mit ihren "Differentialen“ und "Integralen“ die Höhen des
Wissens und Könnens. Strenggenommen ist aber überall das potenz. Unendl. ohne ein zu
Grunde liegendes A. U. (über das sich nur die meisten jener Herren keine Rechenschaft geben
mögen oder können) undenkbar. Wenn Sie also etwa in diesen Kreisen auf "fachmännisch"
competentes Urtheil zu der vorliegenden Frage zählen sollten, so könnten Sie sich vielleicht
getäuscht sehen; das alleinige Forum ist hier die höchstgebietende Vernunft, welche kein
Ansehen der privilegirten, gelehrten, akademischen Zünfte anerkennt; sie bleibt und herrscht,
wir Menschen kommen und gehen.
[Cantor an Prof. Aloys Schmid, 18.? April 1887 nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 504]
{{At least in four of his preserved letters (and that is merely a small fraction of his complete
correspondence) Cantor emphasizes:}} To understand the teaching of the transfinite no
scholarly education in newer mathematics is required; this could even be a nuisance because in
the so-called infinitesimal analysis the potential infinite has pushed to the fore and lead to the
opinion, even of the heroes, as if they with their "differentials" and "integrals" mastered the
hights of knowledge and skill. Strictly speaking, the potential infinite is always unthinkable
without the foundational A. I. (which only most of those gentlemen cannot account for). So, if you
expect to get an "expertly" competent judgement from those circles you may find your
expectations disappointed. The only forum here is the Empress Reason which does not
acknowledge any reputation of priviledged, scholarly, academical guilds. She persists and rules we humans come and go.
§ 398 Das Unendliche und die Theologie / Infinity and Theology (39)
Alle sogenannten Beweise (und es dürfte mir wohl keiner verborgen geblieben sein) gegen das
geschöpfliche A. U. beweisen nichts, weil sie sich nicht auf die richtige Definition des
Transfiniten beziehen. Die beiden für seine Zeit und auch heute noch kräftigsten und
tiefsinnigsten Argumente des S. Thomas Aquinatus [...] werden hinfällig, sobald ein Princip der
Individuation, Intention und Ordination actual unendlicher Zahlen und Mengen gefunden ist.
{{Für undefinierbare Zahlen ist Individuation eine unerfüllbare Forderung.}}
[Cantor an Aloys Schmid, 26. März 1887]
All so-called proofs (and I hardly may have missed anyone) against the creational Actual Infinite
do not prove anything because they do not refer to the correct definition of the transfinite. The
two, for their time and even today, strongest and profoundest arguments of St Thomas
Aquinatus [...] are invalid as soon as a principle of individuation and ordination of actually infinite
numbers and sets has been found. {{With respect to undefinable numbers individuation is an
unattainable demand.}}
§ 399 Das Unendliche und die Theologie / Infinity and Theology (40)
Der R. P Ign. Carbonelle hat in seiner schönen Schrift "Les confins de la science et de la
philosophie, 3e ed. t. I cap. 4" den Versuch gemacht, den Gerdilschen Beweis für den zeitlichen
Weltanfang dadurch zu retten, dass er zwar sehr scharfsinnig und kenntnissreich den Satz
vertheidigt: "Le nombre actuellement infini n'est pas absurd", aber demselben den harten,
unbarmherzigen und dissonirenden Nachsatz giebt: "mais il est essentiellement indéterminé".
Auf diesen Nachsatz würde er vielleicht verzichtet haben, wenn er schon damals meine Arbeiten
gekannt hätte, die sich von Anfang an, seit bald zwanzig Jahren, fast ausschliesslich mit dem
Beweis der Individuations-, Specifications- und Ordinationsmöglichkeit des actualen
Unendlichen in natura creata beschäftigen. Mit jenem Nachsatz steht und fällt aber der vom R.
P. Carbonelle unternommene mathematische Beweis für den zeitlichen Anfang der Schöpfung.
Was endlich die dritte These Ihres geschätzten Schreibens betrifft, so bin ich ganz auf Ihrer
Seite, wenn Sie mit Nic. v. Cusa sagen, daß "in Gott Alles Gott ist", wie auch, dass "die
Erkenntnis Gottes objectiver Seits das Incommensurable nicht als commensurabel, das
Irrationale nicht als rational zu erkennen vermag, weil die göttliche Allerkenntnis, wie die
göttliche Allmacht nicht auf Unmögliches gehen kann." {{Dagegen verlangt die heutige
Matheologie Unmögliches, nämlich die Individuation undefinierbarer Zahlen.}}
[Cantor an Aloys Schmid, 26. März 1887]
The R. P. Ign. Carbonelle, in his beautiful essay "Les confins de la science et de la philosophie,
3e ed. t. I cap. 4", has tried to save Gerdil's proof for a temporal beginning of the world by very
astutely and scholarly defending the proposition: "Le nombre actuellement infini n'est pas
absurd", but adding the hard, merciless, and dissonant afterthought: "mais il est essentiellement
indéterminé". Perhaps he would have refrained from that afterthought if he had known at his
time my works already, which from its beginning, for meanwhile nearly twenty years, has been
concerned with ways of individuation, specification and ordination of the actual infinite in natura
creata. But the proof for a beginning of the world in finite time, undertaken by R. P. Carbonelle,
stands or falls with just this afterthought .
Finally, with respect to the third thesis of your esteemed letter I fully agree that you with Nic. of
Cusa say that "in God all is God" as well as that "the cognition of God objectively cannot
recognize the incommensurable as commensurable, cannot recognize the irrational as rational,
because the divine omniscience as well as the divine omnipotence cannot give rise to the
impossible". {{Alas, present matheology can give rise to the the impossible, namely die
individuation of undefinable numbers.}}
§ 400
Eine Zahl ist definierbar durch ihre Ziffern oder durch ihr Bildungsgesetz. Die Menge aller
definierbaren Zahlen ist abgeschlossen unter allen mathematischen Operationen. Die Menge
aller definierbaren Zahlen ist eine Untermenge der abzählbaren Menge aller endlichen
Definitionen.
Die hierfür vorausgesetzte Sprache ist Deutsch. Jede endliche Definition in jeder beliebigen
Sprache kann in die deutsche Sprache übersetzt werden.
Schluss: Die Mathematik, sofern sie in Deutsch betrieben werden kann, enthält nur abzählbar
viele Zahlen.
A number is definable by its digits or by its construction rule. The set of definable numbers is
closed under all mathematical operations, i.e., every definable number leads always to other
definable numbers. The set of all definable numbers is a subset of the countable set of all finite
definitions.
The basic language is English. Every definition in whatever language can be translated into
English.
Conclusion: Mathematics, as far as it can be expressed in English, contains only countably
many numbers.
§ 401 Das Unendliche und die Theologie / Infinity and Theology (41)
Wenn hier gesagt wird, daß ein mathematischer Beweis für den zeitlichen Weltanfang nicht
geführt werden könne, so liegt der Nachdruck auf dem Wort "mathematischer" und nur soweit
stimmt meine Ansicht mit der des St. Thomas überein. Dagegen dürfte gerade auf Grund der
wahren Lehre vom Transfiniten ein gemischter mathem. metaphysischer Beweis des Satzes
wohl zu erbringen sein und insofern weiche ich allerdings von St. Thomas ab, der die Ansicht
vertritt: S. th. q. 46, a. 2 concl. "Mundum non semper fuisse, sola fide tenetur, et demonstrative
probari non potest."
[Cantor an Aloys Schmid, 26. März 1887, nach H. Meschkowski, W. Nilson: "Georg Cantor
Briefe", Springer, Berlin (1991) p. 285]
If it is said here that a mathematical proof of the beginning of the world in finite time cannot be
given, the stress is on the word "mathematical" and only in that respect my opinion is in
agreement with St. Thomas. On the other hand, based upon the true teaching of the transfinite,
a mixed mathematical-metaphysical proof of the theorem might well be possible. In so far I differ
from St. Thomas, who holds the opinion: "We know by belief only that the universe did not
always exist, and that cannot be checked by proof on its genuineness."
§ 402 Das Unendliche und die Theologie / Infinity and Theology (42)
Ich halte es für sehr werthvoll, daß den schamlosen Angriffen Haeckels gegen das Christentum
der angemasste Schein der Wissenschaftlichkeit nunmehr vor dem weitesten Kreisen entrissen
wird. Die vornehme Scheu vor herzhafter Polemik (in anderen Kreisen so verbreitet!) musste
gegenüber solchen Nichtswürdigkeiten weichen.
[Cantor an Friedrich Loofs, 24. Feb. 1900]
I do highly appreciate that the pretended scientific appearance has been snatched away from
Haeckel's shameless attacks against Christianity in front of the widest audience. The noble
shyness toward hearty polemics (in other circles so usual!) had to give way with respect to such
wretchedness.
§ 403 Das Unendliche und die Theologie / Infinity and Theology (43)
Als Philosoph thut man meines Erachtens überhaupt gut daran, sich in allen mathematischphilosophischen Fragen mathematischen Autoritäten gegenüber möglichst skeptisch zu
verhalten, eingedenk des wahren Pascalschen Ausspruchs: "Il est rare, que les géomètres
soient fins, et que les fins soient géomètres." [Es ist selten, dass die Mathematiker Scharfsinnige
sind und dass die Scharfsinnigen Mathematiker sind.]
[Cantor an Aloys Schmid, 8. Mai 1887, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 509]
As a philosopher you do well, in my opinion, to be very sceptical against mathematical
authorities in all mathematical-philosophical questions, in memory of Pascal's true saying: "Il est
rare, que les géomètres soient fins, et que les fins soient géomètres." {{It is rare that
mathematicians are sharp-witted and that the sharp-witted are mathematicians.}}
§ 404 Das Unendliche und die Theologie / Infinity and Theology (44)
Die Thatsache der act. unendl. grossen Zahlen ist sowenig ein Grund für die Möglichkeit einer a
parte ante unendlichen Dauer der Welt, dass vielmehr mit Hülfe der Theorie der transfiniten
Zahlen die Nothwendigkeit eines von der Gegenwart in endlicher Ferne gelegenen Anfangs der
Bewegung und Zeit bewiesen werden kann.
Die ausführliche Begründung dieses Satzes verschiebe ich auf eine andre Gelegenheit, da ich
Ihnen den frohen Ferienanfang nicht mit mathematisch-metaphysischen Erwägungen
beschweren möchte.
[Cantor an Aloys Schmid, 5. Aug. 1887, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 517]
The fact of the act. infinitely large numbers is so little a reason for the possibility of an a parte
ante infinite duration of the world that, on the contrary, by means of the theory of transfinite
numbers the necessity of a beginning of motion and time in finite distance from the present can
be proven.
The detailled grounds of this theorem I will postpone to another opportunity because I would
not like to weigh down the merry beginning of your holidays with mathematical and metaphysical
considerations.
§ 405 Das Unendliche und die Theologie / Infinity and Theology (45)
Dagegen ist [...] in der ersten Hälfte des vorigen Jahrhunderts ein merkwürdiger Versuch von
dem berühmten Franzosen Fontenelle gemacht worden (in dem Buche "Eléments de la
Géometrie de l'infini", Paris 1727), actual unendliche Zahlen einzuführen; dieser Versuch ist
jedoch gescheitert und hat dem Verfasser nicht ganz unverdienten Spott seitens der
Mathematiker eingetragen, welche im 18ten Jahrh. und im ersten Viertel dieses Jahrh. gewirkt
haben; die heutige Generation weiss nichts mehr davon. Fontenelle's Versuch musste scheitern,
weil seine unendlichen Zahlen einen flagranten inneren Widerspruch mit sich auf die Welt
brachten; es war leicht, diesen Widerspruch aufzudecken und ist dies von dem R. P. Gerdil
bestens geschehen. Wenn aber d'Alembert, Lagrange und Cauchy geglaubt haben, dass damit
die schlummernde Idee des Transfiniten tödtlich für alle Zeiten getroffen worden sei, so
erscheint mir dieser Irrthum weit grösser, als der des Fontenelle und umso gravirender, als
Fontenelle in der bescheidensten Weise sich als Laie in der Mathematik bekennt, während jene
drei nicht nur Mathematiker von Fach, sondern wahrhaft grosse Mathematiker waren.
[Cantor an Prof. Aloys Schmid, 26. März 1887, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 500f]
In the first half of the last century a curious attempt has been made by the famous French
Fontenelle (in the book "Eléments de la Géometrie de l'infini", Paris 1727), to introduce actually
infinite numbers; this attempt however has failed and has brought him some mockery, not quite
undeserved, from the mathematicians who were active in the 18th century and in the first quarter
of this century; the present generation does not know about that. Fontenelle's attempt was
doomed to failure because his infinite numbers brought with them a flagrant contradiction; it has
been easy to show this contradiction, and that has been done best by R. P. Gerdil. But if
d'Alembert, Lagrange and Cauchy have believed that the dormant idea of the transfinite has
been stroken deadly by that for all times, then this error appears by far greater than that of
Fontanelle and the more grave because Fontenelle in the most humble way confesses to be a
layman in mathematics whereas those three have not only been professionals but really great
mathematicians.
§ 406 Das Unendliche und die Theologie / Infinity and Theology (46)
Wenn man aber irgendetwas in der Wahrheit erkannt hat, so weiss man auch, dass man diese
Wahrheit besitzt und man findet (auch wenn man, wie ich, sagen kann "non quaero ab
hominibus gloriam" [von den Menschen verlange ich keinen Ruhm]) eine Art von Verpflichtung,
soweit und solange die Kräfte dazu einem geschenkt werden, das Gewußte anderen
mitzutheilen. Von diesem Gesichtspuncte aus wollen Ew. Hochwürden es freundlich
entschuldigen, wenn ich im Folgenden ausführlicher das schon in meinen bisherigen
Mittheilungen Gesagte vertreten und ergänzen werde.
[Cantor an Prof. Aloys Schmid, 18.? April 1887, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 504]
If one has recognized the truth of something, then one knows to be in possession of the truth
and one feels (even if saying like me "non quaero ab hominibus gloriam" {{I do not want glory
from mankind}} sort of duty, as far and as long as power reaches, to tell it to others. Under this
aspect you, Reverend Father, will kindly forgive that I in the following will in greater detail
support and amplify what I said in my recent messages.
§ 407 Das Unendliche und die Theologie / Infinity and Theology (47)
Nachdem ich erst kürzlich Ihr Werk "Institutiones philosophicae“ durchgesehen habe, bin ich zu
der Ueberzeugung gelangt, dass ich in den allerwichtigsten metaphysischen Fragen von der
Philosophie des heiligen Thomas Aquinatus, welche durch Ew. Hochwürden so meisterhaft und
lichtvoll vertreten wird, nicht sehr abweiche und dass diejenigen Puncte, in welchen eine
Differenz zu constatiren wäre, solche sind, welche eine Modification der Lehre des grossen
Philosophen gestatten und vielleicht selbst wünschenswerth erscheinen lassen.
[Cantor an Révérend Père Matth. Liberatore S. J., 7. Feb. 1886, nach C. Tapp: "Kardinalität und
Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor
und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 453]
After recently skimming through your paper "Institutiones philosophicae“ I got the impression
that I do not much deviate in the most important metaphysical questions concerning the
philosophy of Saint Thomas Aquinatus, which you, Referend Father, support so masterly and
enlightening, and that those points, where a difference could be stated, are such in which
modifying the teaching of the great philosopher might be allowed and perhaps even be
desirable.
§ 408 Das Unendliche und die Theologie / Infinity and Theology (48)
Hinsichtlich der Frage, die du in deinen Arbeiten behandelst, bin ich der Ansicht, daß eine
unendliche Vielheit als in Wirklichkeit existierende unmöglich, als im Denken existierende nicht
nur möglich, sondern aktual im göttlichen Intellekt gegeben ist; denn sicher nimmt Gott alle
möglichen Dinge unterschieden wahr, und es gibt unendlich viele mögliche Dinge. Ich glaube,
daß dies die Lehrmeinung des heil. Thomas ist.
lch stimme dir nicht zu im Hinblick auf die Theorie von den einfachen Seienden, die als
konstitutive Prinzipien der Körper zugelassen werden. Die wesensmäßige Zusammensetzung
der Körper kann meiner Meinung nach nur durch Materie und Form im Sinne des Aristoteles und
der Scholastiker erklärt werden.
Ich glaube, daß die Lehre des heiligen Thomas keine Modifikation in Bezug auf ihre
fundamentalen Teile vertragen kann. Wenn irgendeiner ihrer Teile weggenonmnmen wird, stürzt
das ganze Gebäude zusammen; so sehr ist jene Lehre in sich zusammenhängend.
[Liberatore an Cantor, 24. Feb. 1886 {{Antwort auf § 407}}, Originaltext Latein, Übersetzung aus
C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz
zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz
Steiner Verlag (2005) p. 454}]
With respect to the question that thou are treating in thy works, I mean that an infinite multitude
as in reality existing is impossible, as in thinking existing is not merely possible but actually is
being given in the divine intellect; surely God perceives all possible things as distinct, and there
are infinitely many possible things. I believe that this is the doctrine of holy Thomas.
I do not agree with you concerning the theory of the simple beings which are admissible as the
constituent priciples of bodies. The essential constitution of bodies can, in my opinion, be
explained only by means of matter and form in the sense of Aristotle and the scholastics.
I believe that the doctrine of holy Thomas cannot bear any modification with respect to the
fundamental parts. When any of its parts is removed, the whole edifice will collapse; so closely
this teaching is interconnected.
§ 409 Das Unendliche und die Theologie / Infinity and Theology (49)
Zur Sache möchte ich noch hinzufügen, dass Sie in der bisherigen Mathematik, im Besonderen
in der Differential- und Integralrechnung wenig oder gar keine Auskunft über das Transfinitum
erhalten können, weil hier das potenziale Unendliche, ich sage nicht die alleinige, aber die an
die Oberfläche (mit welcher sich die meisten Herren Mathematiker gern begnügen)
hervortretende Rolle spielt. Selbst Leibniz, mit dem ich auch sonst in vielen Beziehungen nicht
harmonire, hat sich [...] in Bezug auf das A. U. in die auffallendsten Widersprüche verwickelt,
[Cantor an Aloys Schmid, 26. März 1887, nach C. Tapp: "Kardinalität und Kardinäle:
Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und
katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 500]
On account of the matter I would like to add that in conventional mathematics, in particular in
differential- and integral calculus, you can gain little or no information about the transfinite
because here the potential infinite plays the important role, I don't say the only role but the role
that is visible next to the surface (which most colleagues are readily satisfied with). Even Leibniz
[...] from whom I deviate in many other respects too has fallen into most eclatant contradictions
with respect to the actual infinite.
§ 410 Das Unendliche und die Theologie / Infinity and Theology (50)
Sanctissimo Domino Nostro
Papae LEONI XIII
Ad Epistolas Apostolicas Sanctitatis Tuae Henoticas cum spectarem imprimis ad illam, 14.
Apr. anni 1895 datam, quam ad Populum Anglicum misisti, confessionem fidei Francisci Baconi
«Seculi et gentis suae decoris, ornatoris et ornamenti literarum» Christianis Omnibus et
praecipue Anglicanae Ecclesiae Sectatoribus in memoriam revocare opportunum existimavi.
Permitte, Pontifex Maxime, ut septem exemplaria novae editionis hujus opusculi Sanctitati
Tuae dedecem et ut tria volumina operum Francisci Baconi addam. Oro rogoque Te, Beatissime
Pater, ut accepta habere velis haec decem munera, que offere audeo, ut signa sint meae
reverentiae meique Amoris Tuae Sanctitatis et Ecclesiae S. Catholicae Romanae.
Tuae Sanctitatis humillimus et addictissimus servus
Georgius Cantor
Mathematicus.
Unserem heiligsten Herrn Papst Leo XIII.
In Betracht der wohlbekannten Briefe Ihrer Apostolischen Heiligkeit, besonders jenes, unter
dem 14. April 1895 gegebenen, den Du an das englische Volk geschickt hast, habe ich es für
nötig gehalten, das Glaubensbekenntnis des Francis Bacon, "seines Jahrhunderts und seines
Volkes Zier, Schmücker und Schmuck der Gelehrsamkeit" allen Christen und insbesondere den
Anhängern der Anglikanischen Kirche ins Gedächtnis zu rufen.
Erlaube, Größter Brückenbauer, daß ich sieben Exemplare einer neuen Ausgabe jenes
kleinen Werkes Dir widme, und daß ich drei Bände der Werke des Francis Bacon beifüge.
Ich bete und bitte Dich, Seeligster Vater, daß Du annehmen wollest jene 10 kleinen Gaben,
die ich anzubieten wage, die Zeichen sein sollen meiner Verehrung und meiner Liebe zu Deiner
Heiligkeit und der Heiligen Katholischen Römischen Kirche.
Deiner Heiligkeit demütigster und höchst zugetaner Diener
Georg Cantor
Mathematiker.
Eine Antwort des Papstes ist nicht bekannt.
[Cantor an Papst Leo XIII, 13. Feb. 1896, H. Meschkowski, W. Nilson: "Georg Cantor Briefe",
Springer, Berlin (1991) p. 383]
Our Holiest Father, Pope LEO XIII
With regard to the well-known Letters of your Apostolic Holiness. in particular that published on
April 14, 1895 that you have sent to the English people, I have held it necessary to remind all
Christians, in particular the adherents of the Anglican Church, of the creed of Francis Bacon "the
fine specimen of his century and his nation, adorning literature and being its adornment".
Permit, Greatest Pontifex, that I dedicate to you seven specimen of a new edition of that little
work and that I include three volumes of the works of Francis Bacon.
I further pray and ask you, Beatissime Pater, to accept those 10 litttle gifts, which I dare to
offer to you and which shall be a token of my admiration and of my love to your Holiness and to
the Holy Catholic Roman Church.
Your Holiness most humble and devoted servant
Georg Cantor
mathematician.
An answer of the Pope is not known.
§ 411 Ein neuer Überabzählbarkeitsbeweis / A fine proof of uncountability
Wir betrachten eine Abzählung aller rationalen Zahlen und bilden die Antidiagonalzahl
d = 0,d1d2d3 ...dn... Die Menge der Dezimalstellen dn (Ziffer samt Index) steht in Bijektion mit der
Menge der natürlichen Zahlen.
Jede endliche Ziffernfolge 0,d1d2d3...dn besitzt unendlich viele Fortsetzungen, zum Beispiel
0,d1d2d3...dn000... oder 0,d1d2d3...dn111... Also existieren in der Abzählung bis zu jeder
Dezimalstelle dn von d unendlich viele Zahlen mit derselben Dezimalstellenfolge.
Da die Antidiagonalzahl aber von allen rationalen Zahlen verschieden ist, muss diese
Verschiedenheit durch die Verschiedenheit von Dezimalstellen bewirkt werden. Und weil die
Dezimalstellen mit endlichen Indizes dazu nicht taugen, muss mindestens eine weitere
Dezimalstelle zu d gehören. Damit ist die Überabzählbarkeit der Folge von Dezimalstellen von d
bewiesen. Der Beweis sollte leicht auf jede andere Folge übertragbar sein.
Damit ist gezeigt, dass jede unendliche Menge überabzählbar ist.
The digits dn of the antidiagonal d are in bijection with Ù.
The entries of a rationals-complete list do not all differ at a finite index n from the antidiagonal
d. For every finite index n there are infinitely many duplicates of 0.d1d2d3...dn. So if d differs by
its digits from all rationals of the rationals-complete list, then the digits of d must be uncountable.
There must be at least one digit more than those which are in bijection with Ù.
A fine proof of uncountability of sequences.
§ 412 What is the meaning of "to exist" in mathematics?
There are three alternatives with respect to numbers:
1) A number exists if it can be individualized in mathematical discourse such that its numerical
value can be calculated by everybody without any error.
2) A number exists if it can be individualized in mathematical discourse such that its numerical
value can be calculated by everybody with error less than any given epsilon.
3) A number exists if it cannot be individualized but if there is some dubious proof saying that
some numbers of some sort should exist.
Are there any other definitions possible?
§ 413 Irrational numbers have no decimal (or binary or whatever integer-positive-base)
expansion. It is impossible for an infinite list of decimals to appear in mathematical discourse,
dialogue, or monologue other than as the finite rule how to calculate every decimal at a finite
place but never all decimals, since beyond every finite index there are infinitely many further
indices. Every decimal that appears in mathematical discourse, dialogue, or monologue belongs
to a rational number.
By the way that is also the reason why Cantor's uncountability proof must fail.
A matheologian (for the definition see § 001) answered: "Both your assertions above are
incorrect."
Wouldn't that claim oblige him to support his opinion by listing all decimals of a nonterminating
decimal representation of a real number of his choice?
§ 414 The adaptation of strong axioms of infinity is thus a theological venture, involving basic
questions of belief concerning what is true about the universe.
[A. Kanamori, M. Magidor: "The evolution of large cardinal axioms in set theory" in: Higher Set
Theory, Lecture notes in mathematics 669, G.H. Müller und D.S. Scott (eds.), Springer, Berlin
(1978) p. 104]
§ 415 I came to the conclusion some years ago that CH is an inherently vague problem [...].
This was based partly on the results from the metatheory of set theory showing that CH is
independent of all remotely plausible axioms extending ZFC, including all large cardinal axioms
that have been proposed so far. In fact it is consistent with all such axioms (if they are consistent
at all) that the cardinal number of the continuum can be “anything it ought to be”, i.e. anything
which is not excluded by König’s theorem. The other basis for my view is philosophical: I believe
there is no independent platonic reality that gives determinate meaning to the language of set
theory in general, and to the supposed totality of arbitrary subsets of the natural numbers in
particular, and hence not to its cardinal number. Incidentally, the mathematical community
seems implicitly to have come to the same conclusion: it is not among the seven Millennium
Prize Problems established in the year 2000 by the Clay Mathematics Institute, for which the
awards are $1,000,000 each; and this despite the fact that it was the lead challenge in the
famous list of unsolved mathematical problems proposed by Hilbert in the year 1900, and one of
the few that still remains open.
[Solomon Feferman: "Philosophy of mathematics: 5 questions" p. 12]
http://www.academia.edu/160395/Philosophy_of_mathematics_5_questions
By continuum hypothesis, CH, we understand the assumption that the set of real numbers
(falsely identified by Cantor with the geometric continuum) has cardinal number ¡1 = 2¡0. It is
absolutely meaningless, first because the continuum is inherently different from the set of real
numbers, second because all alephs, countability and uncountability, are ill-defined super tasks,
and third because there is not the least application of CH in science or mathematics.
§ 416 Alien mathematics: is π universal?
Aliens evolving elsewhere would probably not be humanoid.
There is one area, though, where aliens are generally expected to be much like us:
mathematics. It is often suggested that a good way to contact extraterrestrials is to send signals
with the prime numbers, or digits of π. Each Voyager spacecraft carried a golden phonograph
record of sounds and images of Earth, including a description of our number system. [...]
It seems silly to imagine that intelligent creatures would think that 2 + 2 is different from 4. But
I’m not so sure that they would necessarily understand 2, 4, or +. Let alone π. History shows that
our mathematics depends not just on logical universals, but on what sort of creatures we are,
where we live, and what we think is important.
[Ian Stewart: "Alien mathematics: is Pi universal?" (2010)]
http://www.telegraph.co.uk/travel/7954877/Alien-mathematics-is-Pi-universal.html
Aliens would know that two aliens and two aliens give four aliens. Aliens would probably know
that the circumference of the circle isn't a rational multiple of its diameter so that π cannot be
expanded by digits. But I wish I could know what aliens think about transfinite set theory - after
they were informed. I am quite sure that this kind of "mathematics" is unique in the universe.
§ 417 Eine Konsequenz des aktual Unendlichen / An implication of actual infinity
Jede Menge Sn der Folge
S1 = {1}
S2 = {1} » {2}
S3 = {1} » {2} » {3}
...
ergibt sich als Vereinigung von {n} und {1, 2, 3, ..., n-1}. Unendlich viele Vereinigungen führen zu
unendlich vielen Mengen, doch keine Menge enthält unendlich viele (alle) natürlichen Zahlen.
Vielmehr fehlen in jeder der Mengen unendlich viele.
Nochmals: Die Folge enthält unendlich viele Vereinigungen. Mit jeder wächst die Menge der
darin enthaltenen natürlichen Zahlen. Doch die Menge Ù aller natürlichen Zahlen ist als
Grenzwert einer streng monoton wachsenden Folge nicht in der Folge enthalten.
Vereinigen wir aber alle Folgenglieder Sn (also alle Fehlversuche, Ù zu erzeugen) ohne
irgendetwas hinzuzufügen, dann erhalten wir die Menge Ù aller natürlichen Zahlen.
Das ist eine Konsequenz des aktual Unendlichen. Ist sie akzeptabel?
Every set Sn of the sequence
S1 = {1}
S2 = {1} » {2}
S3 = {1} » {2} » {3}
...
emerges from the union of {n} and {1, 2, 3, ..., n-1}. There are infinitely many unions causing
infinitely many sets but no set contains infinitely many (all) natural numbers. In every set
infinitely many of all natural numbers are missing.
I repeat, there are infinitely many unions, each one adds another natural number n, but the set
Ù, as the limit of a strictly monotonic increasing sequence of sets, is not a term of this sequence.
But if we union all the terms Sn of the sequence (i.e., all the successless attempts to establish
Ù) without adding anything further, then we get the complete set Ù of all natural numbers.
An implication of actual infinity. Are you happy to accept it?
§ 418 Would you trust in such a theory?
Consider a geometry that contains the axiom: "For every triple of points there exists a straight
line containing them." When you ask for the straight line that contains the points (0,1), (0,2),
(1,0) the masters of the theory reply that some straight lines cannot be constructed but that they
certainly "exist". If you ask what in this case existence would mean, you are called a crank.
Would you trust in such a theory and its masters?
You do already. The axiom of choice says that every set can be well-ordered, i.e., all its
elements can be indexed such that every non-empty subset has an element with smallest index.
There are only countably many indices, but what about uncountable sets? Who cares!
But even if you are not outwitted by this case, you certainly trust in set theory, don't you?
The common interpretation of the notion "set" is that all its elements "exist". The axiom of
infinity then says that every element of an inductive set is preceded by finitely many elements
but followed by ¡0 elements.
Nobody has ever succeeded to show one of the trailing ¡0 elements. All that could be done is
to show elements belonging to first the 0% set.
In fact the situation is not very different from the geometry with the straight lines through every
triple of points. It is rather the same.
If you are in despair now then you show that you are able to follow mathematical arguments.
And I can comfort you:
Set theory does not require that sets have to be completed, neither does the axiom of infinity.
This axiom has the same wording in potential infinity (from where it originally has been taken)
and does not require ¡0 but only infinitely many successors to every element of an inductive set.
That is a big difference.
And the axiom of choice is a very natural one and is quite right because there are no
uncountable sets.
§ 419 Warum versagt das Cantorsche Diagonalisierungsverfahren?
Jede abbrechende Dezimalzahl gehört zur potentiell unendlichen Menge der abbrechenden
Dezimalzahlen. Sie sind durch Auflistung aller ihrer Ziffern identifizierbar (weil nach der letzen
von Null verschiedenen nur noch Nullen folgen, die den Zahlenwert unverändert lassen).
Jede unendliche Dezimalzahl gehört zur aktual unendlichen Menge aller nicht abbrechenden
Dezimalzahlen. Sie sind im mathematischen Diskurs nicht durch Auflistung ihrer Ziffern
identifizierbar, sondern erfordern ein endliches Bildungsgesetz.
Eine Cantor-Liste, die alle durch Ziffern identifizierbaren Zahlen enthält (was nach Cantor
möglich ist), kann keine durch Ziffern identifizierbare Antidiagonalzahl enthalten, die sich von
allen Zahlen der Liste unterscheidet.
Ebenso kann der Binäre Baum keinen durch Auflistung seiner Knoten identifizierbaren Pfad
enthalten, der sich von allen Pfaden des Binären Baums unterscheidet.
In beiden Fällen erfordert die erfolgreiche Diagonalisierung die Bildung eines durch Auflistung
von Ziffern bzw. Knoten nicht identifizierbaren Grenzwertes.
Zum Ende des 19. Jahrhunderts, als Cantor sein Diagonalverfahren veröffentlichte, war noch
nicht bekannt, dass nur abzählbar viele endliche Definitionen für Bildungsgesetze existieren
können. (Tatsächlich existieren zu jedem Zeitpunkt sogar nur endlich viele.) Dies wurde erst zu
Beginn des 20. Jahrhunderts einigen wenigen Mathematikern bewusst und erst zu Beginn des
21. Jahrhunderts ins allgemeine Bewusstsein gerückt. Damit ist die Existenz von
überabzählbaren Zahlenmengen (für andere Mengen gelten ähnliche Überlegungen) im
mathematischen Diskurs ausgeschlossen.
Die Bezeichnungen für diese im mathematischen Diskurs nichtexistenten aber von vielen
Mitgliedern der intellektuellen Arrièregarde noch immer für wahr genommenen Mengen
existieren als endliche Definitionen weiterhin, haben aber keinen Bezug zur Mathematik.
Fußnote: Die Verfechter des Cantorschen Diagonalverfahrens behaupten, es sei ausreichend,
dass für jede Listenzahl eine Ziffer existiert, die sich von der entsprechenden Ziffer der
Antidiagonalzahl unterscheidet. Hier wird das Unendliche als potentiell behandelt. Das genügt
aber nicht (es genügt nur scheinbar, wenn die Bedeutung des Unendlichen geflissentlich
vertauscht und es nun als aktual behandelt wird - oder wenn man die Unterscheidung gar nicht
kennt), wenn man die Antidiagonalzahl mit Hilfe ihrer Ziffern von allen Listenzahlen
unterscheiden möchte, denn bis zu jeder (und also auch der gerade betrachteten) Ziffer der
Antidiagonalzahl gibt es unendlich viele übereinstimmende Zahlen in einer Liste aller
abbrechenden Dezimalzahlen. Um die Antidiagonalzahl von allen Listenzahlen zu unterscheiden
(wie bereits bemerkt wird nur damit bewiesen, dass sie nicht in der aktual unendlichen Liste
enthalten ist), genügen Ziffern allein also nicht. Dazu bedarf es einer endlichen Definition, aus
der alle Ziffern der Antidiagonalzahl hervorgehen.
§ 419' Why does Cantor's diagonal argument fail?
Terminating decimal numbers can be identified by listing all their digits (because the last one
different from zero is followed by zeros only which do not change the numerical value). Infinite
decimal numbers cannot be identified in mathematical discourse by listing all their digits digit by
digit. Each one requires a finite construction rule.
A Cantor list containing all terminating decimal numbers cannot yield an antidiagonal that
differs from all numbers of the list and can be identified by its digits. (Similary, the Binary Tree
cannot be diagonalized such that the antidiagonal path differs from all paths of the Binary Tree
and can be identified by its nodes.) In both cases the diagonal argument requires an infinite
string, a limit that cannot be identified by listing all digits or nodes. A limit always requires a finite
construction rule.
At the end of the 19th century, when Cantor published his diagonal argument, it had not been
known that only countably many finite words for expressing rules can exist*. It became known to
few mathematicians only at the beginning of the 20th century and has become popular
knowledge only at the beginning of the 21st century. This fact excludes the existence of
uncountable sets of numbers in mathematical discourse, dialogue, or monologue.
The names for such sets, which nevertheless are yet asserted to exist somewhere (in a never
defined place) by the intellectual arrièregarde, are persisting but without any relation to
mathematics.
*) In order to eliminate Koenig's paradox of a first undefinable real number (which by these very
words has been defined), it is usually argued that no list of all finite rules exists that could be
followed by this number. However, that does not contradict the fact that the number of definable
numbers is countable as is every subset of a countable set.
§ 420 Why mathematics contradicts set theory
In set theory there exists the list of all finite subsets of Ù. They can be noted in binary where the
infinite sequence 000... is the empty set, 11000... is the set {1, 2}, and 111... is Ù (but the latter,
as an infinite set, is not in the list). In set theory this list can be diagonalized.
In mathematics the antidiagonal cannot be defined by its digits because all finite digit
sequences are in the list and an infinite digit sequence cannot be communicated in mathematical
discourse, dialogue, or monologue. Therefore the antidiagonal must be addressed by a finite
definition like "antidiagonal of this particular (insert definition) list".
The list of all infinite subsets of Ù, defined by finite definitions, e.g., in Latin letters (like "set of
all even numbers") and noted as infinite binary sequences (like 010101...) is a sublist of the list
of all finite definitions. But this sublist is subcountable. It has an antidiagonal, that is also defined
by the finite definition "antidiagonal of this particular (insert definition) list".
However, in set theory there is nothing subcountable. Every subset of any set exists and can
be addressed. According to set theory there is no undefined subset of Ù, hence no
subcountability as above. This means that in set theory, based upon mathematics, the axioms of
extensionality and of power set are in contradiction with the list of all finite definitions of infinite
subsets of Ù. In mathematics based upon the actual infinity of set theory there is
subcountability.
In potential infinity, this contradiction cannot occur. "Antidiagonal of this particular (insert
definition number n) list" can be inserted infinitely often since no list is ever completed.
§ 421 Für jede natürliche Zahl n gilt fraglos |{1, 2, 3, ..., n}| = n und falls ein Grenzwert X
existiert, so gilt auch
lim |{1, 2,3,..., n}| = X
n→ X
sowie
lim n = |{1, 2,3,..., X } |
n→ X
also
|{1, 2, 3, ..., X}| = X
Die naheliegenden Konsequenzen
|{1, 2, 3, ..., ¡0}| = ¡0
|{1, 2, 3, ..., ω}| = ω
|{1, 2, 3, ..., ¶}| = ¶
gelten aber nicht. Jeder zünftige und vermutlich schon jeder zukünftige Adept der Mengenlehre
hat gelernt, dass allein
|{1, 2, 3, ...}| = ¡0
zulässig und richtig ist.
§ 422 What is a definition of a number?
The definition of a number must allow transmitter and receiver in mathematical discourse,
dialogue, and monologue to identify this one (1) number uniquely.
If the question was: "How can we define a number?, then the answer could only be: "A number
can be identified by a finite string of symbols taken from a countable alphabet". There are many
ways to do so. Every definable number has infinitely many finite definitions. Most are known for
the number zero or 0 or 0.000..., because in addition to the three finite definitions just given
there are lots of sequences with improper limit oo, each of which has a sequence of reciprocals
with limit 0.
The set of finite strings, however, is countable. In order to get a set of uncountably many
numbers, infinite strings of symbols are required (because uncountable countable lists are
impossible, but unlistable alphabets cannot be learned or applied, i.e., uncountable alphabets
are not alphabets). Such an infinite string must be capable of uniquely defining a number. It is
not enough to distinguish the number from all its finite approximations like 1/9 = 0.111... can be
distinguished from all its finite approximation 0.1, 0.11 and so on. That would require that 1/9 is
already "given", of course by a finite definition. Infinite definitions cannot "be given". A distinction
by digits is impossible because all digits belong to the set of finite approximations. It is only the
property of being "non-terminating" that distinguishes 1/9 uniquely from all its approximations,
but this property cannot be obtained from checking any digits but only from the finite definition.
(An always negative result with always infinitely many further digits remaining to check can only
be accepted as a final exclusion in an experimental science like physics. Mathematics requires
final proofs!)
Although it is clear from the above argument, that a number cannot be defined by an infinite
sequence of digits, it can be proved in addition that the set of all infinite sequences of digits is
countable. For this proof consider the set of all infinite sequences of symbols, or, without loss of
generality, the Binary Tree which contains as paths the allegedly uncountable set of real
numbers in the unit interval. All infinite bit sequences that in a unique way define a real number
of this interval (some numbers having even two such sequences) are paths in the Binary Tree.
All possible paths that are defined by nodes only will be covered when each node is covered by
at least one infinite path containing it. Since the number of nodes is countable, this is
accomplished by countably many paths. More paths cannot be defined by nodes. But even
covering every node by countably many paths would not result in using more than countably
many paths.
This excludes the acceptance of an uncountable set of numbers in any mathematical theory that
is free of contradictions.
§ 423 The purposes of the Binary Tree proof is to show first that there are not more than
countably many infinite paths and second that there is no actually infinite path.
The universe of ZFC set theory contains uncountably many real numbers each of which has a
unique place in both the natural order by size (trichotomy) and all well-orderings, at least one of
which is existing. This implies that every real number has a unique definition such that using this
definition will call one and only one real number like a magic spell. Since there are at most
countably many finite definitions, every real number is said to have an infinite definition. Such a
definition can be tfansformed into bits. A very convenient method to do so is to identify each
definition of a real number of the unit interval with an infinite path in the complete infinite Binary
Tree
0.
/ \
0 1
/ \ / \
0 101
...
This Binary Tree contains a countable set of nodes n and infinite path p which are infinite sets of
nodes, as we will assume for the outset.
(1) The complete structure of the Binary Tree can be constructed by a countable set of infinite
path p(n).
Proof: Every path p(n) is constructed such that it begins at the root node "0." and contains the
node n and then continues in a way that can be arbitrarily chosen and does not matter for the
proof, since every following node m will be constructed for another time by the path p(m)
containing it. Note that every node can be connected with the root node in one and only one
way. This way, called the common finite initial segment f(n) of all those paths that contain node
n, is uniquely defined by node n.
If all nodes of a path p (like 0.111...) have been constructed by paths, then the path p itself
may have been used or may not have been used. We must say either that p has been
constructed since p as a set of nodes cannot be different from all its nodes. Or we must say that
it is impossible to discern the path p by the set of its nodes but only by a finite definition (like
"0.111..."). However finite definitions are not under investigation here. Therefore all possible
infinite paths that can be distinguished by nodes belong to a countable set of paths.
(2) In fact it is impossible to distinguish a path p (like 0.111...) by its nodes because all its nodes
belong to other paths (like 0.1000..., 0.110000..., 0.111000...) too. The sequence of these other
paths (here those with a finite number of nodes 1) is infinite and contributes all infinitely many
nodes of p. The limit of this strictly increasing sequence does not belong to the sequence. But it
would, if it was a path in the Binary Tree that could be identified by its nodes.
In addition it is clear that in mathematical discourse never an infinite set can be identified by
listing all its elements.
Conclusion: All paths in the Binary Tree are potentially infinite sequences of finite initial
segments of infinite paths. Down to every level j there are only countably many such sequences
consisting of finite initial segments of less than j + 2 nodes. Since one sequence cannot have
many limits, there cannot be more than countably many limits. This result is in agreement with
the number of not more than countably many definable real numbers.
§ 424 Actual Infinity: We never get it - but we get it!
In actual infinity the set Ù of all natural numbers exists as a union such that no natural number is
missing. Considering the union stepwise
{1}
{2, 1}
{3, 2, 1}
...
{..., 3, 2, 1} = Ù
we see that Ù does not appear in any enumerated line, i.e., it appears never, but it appears
according to the motto: It doesn't matter that we never get it - if only we get it. So the sequence
somehow has to reach, create, or complete its limit.
This case can be translated into analysis. If actual infinity applies here, then the sequence
0.000
0.1000...
0.11000...
0.111000...
...
0.111...
reaches, creates, or completes its limit with an actual infinity of digits too. However, this implies
that Cantor's diagonal argument fails in case the antidiagonal d of the sequence is chosen to be
d = 0.111... Of course d is not completed in any enumerated line but only in the infinite - alas
there it is already welcomed by itself.
Usually set theorists deny that d belongs to the infinite list. Therefore the projection of d on the
horizontal axis is never completed (that would require a completed line). But its projection on the
vertical axis is completed. And from that part it is concluded in reverse that d is completed. Only
by this incoherent argument it is possible for d to differ from every line.
Not necessary to mention that in analysis this limit is not created by digits. We have to use finite
definitions for what we never get by digits. The above list does never reach, create, or complete
a string of digits without a tail of infinitely many zeros. And in analysis "never" means never.
§ 425 Why Russell's Paradox is irrelevant in mathematics
We are, like Poincaré and Weyl, puzzled by how mathematicians can accept and publish such
results {{like the Hausdorff Sphere Paradox}; why do they not see in this a blatant contradiction
which invalidates the reasoning they are using? [...] Presumably, the Hausdorff sphere paradox
and the Russell Barber paradox have similar explanations; one is trying to define weird sets with
self-contradictory properties, so of course, from that mess it will be possible to deduce any
absurd proposition we please.
[E. T. Jaynes: "Probability Theory: The Logic of Science", (Fragmentary Edition of March 1996)]
http://www-biba.inrialpes.fr/Jaynes/cappb8.pdf
There are other paradoxes like Socrates' I know that I know nothing or Cantor's diagonal
argument which proves by producing a finitely defined real number that the countable set of
finitely defined real numbers is uncountable. It belongs to the set of all logical tricks but is not in
any way related to determine sizes of sets. All this stuff is simply an amusement based on selfreference.
Some of these tricks even comprehend eternal truths: Nobody has ever checked the ¡0 digits
of Cantor's diagonal that follow behind the last checked digit.
Or: A sausage is better than nothing and nothing is better than the eternal rapture.
Others are so false that even their negation is false: "This sentence contains seven words" is
wrong. "This sentence does not contain seven words" is wrong too.
But such musings should not be confused with mathematics.
§ 426
- Our dissatisfaction with Zermelo's axiomatic in the context of the reality of the Continuum is
rooted in the fundamental sequentiality of its constructions, the sequentiality which implies the
sequential causality of all what could be said about transfinites. The advanced principles of
transfinite set theory are designed to overcome this sequentiality obstruction, but they cannot
eliminate it.
- Ultimately, all modern transfinite set theory represents only a well designed fantasy founded on
Zermelo's axiomatic, the fantasy which pushes to their limits the rich constructionist faculties of
this system. All adaptations of these fantasies to even very modest aspects of the Continuum
realities remain absolutely unsatisfactory.
- This is because, as we claim, the origins of the Continuum are outside all set-theoretical
"explanations".
[E. Belaga "From Traditional Set Theory - that of Cantor, Hilbert, Goedel, Cohen - to Its
Necessary Quantum Extension", IHES/M/11/18 (2011) p. 24]
http://preprints.ihes.fr/2011/M/M-11-18.pdf
§ 427 Is successful diagonalization of a potentially infinite list possible?
William Hughes has claimed that a potentially infinite list of finite definitions of potentially infinite
binary sequences can be subject to diagonalization:
Let L' be a list of finite definitions of potentially infinite 0, 1 sequences.
Let L be the list of potentially infinite 0, 1 sequences.
Define a function of two integers f by
f(n,m) = the mth digit defined by the nth element of L'.
Let dL(n) = f(n,n) +1 (mod 2).
Then dL is a finite definition.
The potentially infinite sequence defined by the definition does not end.
Do not confuse definitions with the potentially infinite 0, 1 sequences they define.
Use induction to show that for every n in Ù dL is not the nth element of L'.
Use an indirect proof to show there is no n in Ù such that dL is the nth element of L'.
However, by definition, dL is in L'. Contradiction.
No. Why does this not contradict the mathematics of potential infinity?
The list L' of every finite definition of numbers can (and must) contain in some line k: "Define a
function of two integers f by f(n,m) = the mth digit defined by the nth element of L'. Let dL(n) =
f(n,n) +1 (mod 2). Then dL is a finite definition."
Further every initial segment of the first j bits of the sequence referenced in line k will be
defined in some line of L'. Insofar the list L' is as complete as a potentially infinite list can be.
The anti-diagonal sequence of L, however, as a potentially infinite sequence of digits does not
define anything since there is always a necessary piece of information missing (in fact even
infinitely many).
This shows that the potentially infinite set of finite definitions of sequences can not be
successfully diagonalized.
§ 428
[...] we cannot have an infinite amount of rooms, this is the same as having an infinite amount of
numbers - we cannot, we may keep counting all we like yet never get any closer to an infinite
amount.
[Amorphos: "Disproving infinity paradoxes; Hilbert's Hotel?" 17 June 2007]
http://www.twcenter.net/forums/showthread.php?105283-disproving-infinity-paradoxesHilbert%92s-Hotel
It isn't just impossible "for us men" to run through the natural numbers one by one; it's
impossible, it means nothing. [...] you can't talk about all numbers, because there's no such thing
as all numbers.
[L. Wittgenstein: "Philosophical Remarks" (1975)]
http://www.press.uchicago.edu/ucp/books/book/chicago/P/bo3615160.html
§ 429 A point of view which the author feels may eventually come to be accepted is that CH is
obviously false. The main reason one accepts the axiom of infinity is probably that we feel it
absurd to think that the process of adding only one set at a time can exhaust the entire universe.
Similarly with the higher axioms of infinity. Now ¡1 is the cardinality of the set of countable
ordinals, and this is merely a special and the simplest way of generating a higher cardinal. The
set C [the continuum] is, in contrast, generated by a totally new and more powerful principle,
namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal
which attempts to build up that cardinal from ideas deriving from the replacement axiom can
ever reach C.
Thus C is greater than ¡n, ¡ω, ¡a, where a = ¡ω, etc. This point of view regards C as an
incredibly rich set given to us by one bold new axiom, which can never be approached by any
piecemeal process of construction. Perhaps later generations will see the problem more clearly
and express themselves more eloquently.
[P. Cohen: "Set Theory and the continuum hypothesis" Dover Publications (2008) p. 151]
http://en.wikipedia.org/wiki/Paul_Cohen_(mathematician)
The continuum is not a set of co-ordinates. This has already become obvious by Cantor's proof
of the same cardinality of the points in a femtometer interval and the whole universe. If someone
does not see the inequality, it is deplorable but does not change the facts.
§ 430
Jede wohldefinierte Antidiagonale einer Cantor-Liste gehört zur abzählbaren Menge der
wohldefinierten reellen Zahlen und, als endlicher Ausdruck, zur abzählbaren Menge der
endlichen Ausdrücke.
Eine undefinierte Folge von Dezimalziffern einer Antidiagonale stellt keine reelle Zahl dar,
sondern höchstens ein Intervall, das sich erst im Unendlichen, also niemals, zu einem Punkt
zusammenzieht.
Merke: Jede versuchte mathematische Definition ohne Endsignal ist ungültig.
Every well-defined antidiagonal of a Cantor list belongs to the countable set of well-defined real
numbers and, as a finite expression, to the countable set of finite expressions.
An undefined sequence of digits of an antidiagonal does not represent a real number but at
most an interval that only in the infinite, i.e. never, contracts to a point.
Note: An attempted mathematical definition without end signal is invalid!
§ 431 Dear Sir, before continuing our discussion, I beg you to give a (formal or reasonable)
explanation of the assertion that set theory allows or requires the use of all natural numbers in
mathematical discourse or in any mathematical application. In my opinion, the axiom of infinity
claims just the contrary, namely: Every natural number belongs to a finite initial segment of the
sequence of natural numbers, and beyond every natural number there are infinitely many
following. It is impossible to derive the use of all natural numbers from the definition of the notion
"set" because such a definition does not exist.
Without any axiomatic or other foundation it is assumed that all ¡0 natural numbers can appear
in mathematical discourse. But the contrary is obvious: Who would ever have used a natural
number that has more predecessors than successors?
§ 432 The complete list is not a square
Cantor's diagonalization argument.
Ok, I've seen this proof countless times. And like I say it's logically flawed because it requires
the a completed list of numerals must be square, which they can' t be.
First off you need to understand the numerals are not numbers. They are symbols that
represent numbers. Numbers are actually ideas of quantity that represent how many individual
things are in a collection.
So we aren't working with numbers here at all. We are working with numeral representations of
numbers.
So look at the properties of our numeral representations of number:
Well, to begin with we have the numeral system based on ten.
This includes the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
How many different numbers can we list using a column that is a single digit wide?
Well, we can only list ten different numbers.
0
1
2
3
4
5
6
7
8
9
Notice that this is a completed list of all possible numbers. Notice also that this list is not
square. This list is extremely rectangular. It is far taller than it is wide.
Let, apply Cantor's diagonal method to our complete list of numbers that are represented by
only one numeral wide.
Let cross off the first number on our list which is zero and replace it with any arbitrary number
from 1-9 (i.e. any number that is not zero) [...]
Ok we struck out zero and we'll arbitrary choose the numeral 7 to replace it.
Was the numeral 7 already on our previous list? Sure it was. We weren't able to get to it using
a diagonal line because the list is far taller than it is wide.
Now you might say, "But who cares? We're going to take this out to infinity!"
But that doesn't help at all.
[...]
We can already see that in a finite situation we are far behind where we need to be, and with
every digit we cross off we get exponentially further behind the list.
Taking this process out to infinity would be a total disaster.
[Divine Insight, Why Cantor's Diagonalization Proof is Flawed (28 June 2012)]
http://debatingchristianity.com/forum/viewtopic.php?t=23975
Thanks to Albrecht Storz for the hint to this source
Usually matheologians confuse infinities. A potentially infinits list is always a square up to every
n. But the presence of the antidiagonal cannot be excluded. Then they switch infinities. Nothing
new. But necessary to mention it over and over again in order to protect newbies from falling into
that trap.
§ 433 The reason for calling matheology matheology
Zeitgeist on the odds of my students to understand transfinite set theory:" if they can be
convinced that what their senses tell them may be not be the whole picture, then they may have
a chance."
Virgil, appearing as Wisely Non-Theist: "There are 'more' real numbers than there are finite
definitions to define them with, so most reals can only be defined collectively, not individually."
Ben Bacarrisse emphasized: "They are not 'entire undefinable'. The set of them can be defined."
He added: "You can know things about the set. For example, that it can't be bijected with N."
William P. Hughes: "A subcollection of a listable collection may not be listable."
Alan Smaill: "After all, since matheology accepts undefinable real numbers, then why are you
trying to suggest that it does not accept undefinable enumerations?"
That is true. I never got a grasp of this idea: Why should undefinable definable enumerations
(also known as lists) be exemptet from the list of unlisted exemptions?
§ 434 Cantor und die Axiome / Cantor and the axioms
Cantor hat niemals versucht, seine Mengenlehre axiomatisch zu formalisieren. Erst am Ende
seiner aktiven Laufbahn, unter Hilberts Einfluss, hat er Axiome dafür überhaupt in Betracht
gezogen. "Er sieht, schon gegen Ende des 19. Jahrhunderts, das Aufkommen eines
formalistischen Denkens, das ihm zutiefst zuwider war" (Meschkowski). Die folgenden
Paragraphen werden jede mir bekannte Erwähnung von Axiomen in Cantors Werk und
Korrespondenz behandeln.
Das sogenannte Cantorsche Axiom (1872) betrifft lediglich die Geometrie:
... ein Axiom hinzuzufügen, welches einfach darin besteht, daß auch umgekehrt zu jeder
Zahlengröße ein bestimmter Punkt der Geraden gehört, dessen Koordinate gleich ist jener
Zahlengröße ... Ich nenne diesen Satz ein Axiom, weil es in seiner Natur liegt, nicht allgemein
beweisbar zu sein. [G. Cantor: "Über die Ausdehnung eines Satzes aus der Theorie der
trigonometrischen Reihen", Math. Annalen 5 (1872) 123 - 132]
Das Archimedische Axiom hielt Cantor für beweisbar und den Euklidschen Satz über den Teil
und das Ganze für fragwürdig :
Also ist das sogenannte "Archimedische Axiom" gar kein Axiom, sondern ein, aus dem linearen
Größenbegriff mit logischem Zwang folgender Satz. [E. Zermelo: "Georg Cantor, Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 409]
{{Ich halte}} an der Ueberzeugung fest, dass das sogenannte "Archimedische Axiom" von mir
bewiesen und daß eine Abweichung von diesem "Axiom" eine Verirrung ist. [Cantor an
Veronese, 7. Sep. 1890]
Sind nicht eine Menge und die zu ihr gehörige Kardinalzahl ganz verschiedene Dinge? Steht
uns nicht erstere als Objekt gegenüber, wogegen letztere ein abstraktes Bild davon in unserm
Geiste ist? {{Zurück zu den Wurzeln: Eine Menge ist ein Objekt (der Realität).}} Der alte, so oft
wiederholte Satz: "Totum est majus sua parte" darf ohne Beweis nur in bezug auf die, dem
Ganzen und dem Teile zugrunde liegenden Entitäten zugestanden werden; dann und nur dann
ist er eine unmittelbare Folge aus den Begriffen "totum" und "pars". Leider ist jedoch dieses
"Axiom" unzählig oft, ohne jede Begründung und unter Vernachlässigung der notwendigen
Distinktion zwischen "Realität" und "Größe" resp. "Zahl" einer Menge, gerade in derjenigen
Bedeutung gebraucht worden, in welcher es im allgemeinen falsch wird, sobald es sich um
aktual-unendliche Mengen handelt und in welcher es für endliche Mengen nur aus dem Grunde
richtig ist, weil man hier imstande ist, es als richtig zu beweisen. [E. Zermelo: "Georg Cantor,
Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p.
416f]
Von Hypothesen ist in meinen arithmetischen Untersuchungen über das Endliche und
Transfinite überall gar keine Rede, sondern nur von der Begründung des Realen in der Natur
Vorhandenen.
Sie hingegen glauben nach Art der Metageometer Riemann, Helmholtz und Genossen auch
in der Arithmetik Hypothesen aufstellen zu können; was ganz- unmöglich ist; darin liegt Ihre
ebenso unglückliche wie verhängnisvolle Täuschung, von welcher ich Sie nicht abbringen kann
und mag. So wenig sich in der Arithmetik der endlichen Anzahlen andere Grundgesetze
aufstellen lassen, als die seit Alters her an den Zahlen 1, 2, 3, ... erkannten, ebensowenig ist
eine Abweichung von den arithmetischen Grundwahrheiten im Gebiete des Transfiniten möglich.
"Hypothesen" welche gegen diese Grundwahrheiten verstoßen, sind ebenso falsch und
widersprechend, wie etwa der Satz 2 + 2 = 5 oder ein viereckiger Kreis. Es genügt für mich,
derartige Hypothesen an die Spitze irgend einer Untersuchung gestellt zu sehen, um von vorn
herein zu wissen, daß diese Untersuchng falsch sein muss.
Und der Erfolg hat es ja bei Ihnen gezeigt, da Sie durch Ihre beklagenswerthen "Hypothesen"
zu dem widersprechenden Begriffe actual unendlich kleiner linearer Größen geführt worden
sind! [Cantor an Veronese, 17. Nov. 1890]
Was Herr Veronese darüber in seiner Schrift giebt, halte ich für Phantastereien und was er
gegen mich darin vorbringt, ist unbegründet.
Ueber seine unendlich großen Zahlen sagt er, daß sie auf anderen Hypothesen aufgebaut
seien, als die meinigen. Die meinigen beruhen aber auf gar keinen Hypothesen, sondern sind
unmittelbar aus dem natürlichen Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei
von WiIlkür, wie die endlichen ganzen Zahlen. [Cantor an Killing, 5. April 1895]
...der Unterschied besteht nur in den "Hypothesen" (Axiomen), die er für das jeweilige System
fordert. Mit einer solchen Sichtweise steht Veronese einer modernen Axiomatik deutlich näher
als Cantor, der aufgrund seiner philosophischen Ansichten Axiome als "Grundwahrheiten"
ansieht. [H. Meschkowski, W. Nilson (Herausgeber): Georg Cantor Briefe , Springer, Berlin
(1991) p. 329]
{{Tatsächlich hat Cantor recht, sofern Mathematik als ernsthafte Wissenschaft aufgefasst wird
und nicht als frivloes Spiel mit Sinnlosigkeiten.}}
Erst die Erkenntnis inkonsistenter Mengen leitete Cantor zum Axiom der transfiniten Arithmetik:
Die Thatsache der "Consistenz" endlicher Vielheiten ist eine einfache, unbeweisbare Wahrheit,
es ist "das Axiom der Arithmetik (im alten Sinne des Wortes)". Und ebenso ist die "Consistenz"
der Vielheiten, denen ich die Alephs als Cardinalzahlen zuspreche, "das Axiom der erweiterten,
der transfiniten Arithmetik". [Cantor an Dedekind, 28. Aug. 1899]
Daß die "abzählbaren" Vielheiten {αν} fertige Mengen sind, scheint mir ein axiomatisch sicherer
Satz zu sein, auf welchem die ganze Functionentheorie beruht. [Cantor an Hilbert, 10. Okt.
1898]
Daß das "arithmetische Continuum" in diesem Sinne eine "Menge" [ist], ist unsere gemeinsame
Ueberzeugung; die Frage ist, ob diese Wahrheit eine beweisbare, oder ob sie ein Axiom ist. Ich
neige jetzt mehr zu der letzteren Alternative, würde mich aber gerne von Ihnen für die andere
überzeugen lassen. [Cantor an Hilbert, 9. Mai 1899]
Unter dem Einfluss Hilberts entsteht schließlich die Cantorsche Axiomatik:
Ich unterscheide in der reinen Mathematik dreierlei Axiome:
1) Die logischen Axiome, die sie mit allen anderen Wissenschaften gemein hat, und die in der
formalen Logik, neuerdings im Logikcalcül systematisch behandelt werden.
2) Die physischen Axiome der Mathematik, z. B. die geometrischen Axiome und die Axiome
der Mechanik. Sie sind dadurch kenntlich, daß ihnen der Charakter der Nothwendigkeit fehlt, sie
können durch andere ersetzt werden (man denke an das Parallelenaxiom Euclid's und die
nichteuclidische Geometrie). Ich nenne sie "physische Axiome", weil sie sich auf besondere
Naturen beziehen, wie etwa auf Raumdinge, Zeitdinge, Kraftdinge, Massendinge etc.
Ausser diesen beiden Arten von Axiomen, existirt noch eine dritte, die bisher unbeachtet
geblieben zu sein scheinen, weil sie mehr versteckt sind, als jene. Ich nenne sie die:
3) Metaphysischen Axiome der Mathematik (ich nenne diese Axiome "metaphysisch", weil sie
sich auf Dinge überhaupt, gleichviel welche Natur sie haben, beziehen); zu diesen gehören vor
Allem die Axiome der Arithmetik, sowohl der endlichen, wie auch der transfiniten Zahlentheorie.
Das Axiom der endlichen Zahlentheorie lässt sich kurz so aussprechen: "Jede endliche
Vielheit ist consistent."
Zur Erläuterung dieses:
a) Der Begriff "endliche Vielheit" ist, wie Sie zugeben werden, ohne Heranziehung des
Zahlbegriffs bestimmbar.
b) Die endlichen Zahlbegriffe resp. Zahlen sind nur unter Voraussetzung der Wahrheit des
soeben formulirten Axioms denkbar resp. möglich.
Das Axiom der transfiniten Zahlentheorie ist dieses: "Jede Vielheit, zu welcher ein signirtes
Alef, ¡γ (wo γ irgend eine Ordnungszahl) gehört, ist consistent". [Cantor an Hilbert, 27. Jan.
1900]
Durch sein untergliedertes Axiomensystem unterscheidet sich Cantor nach eigenen Worten von
Dedekind:
Dedekind geht offenbar von der Meinung aus, daß die Zahlentheorie keine anderen Axiome
voraussetze als die logischen; dasselbe scheinen die Vertreter des Logikcalcüls zu glauben.
In der Vorrede der Dedekindschen Schrift heißt es: die Zahlentheorie "ein Theil der Logik"; die
Zahlen sind ihm "freie Schöpfungen des menschlichen Geistes". [Cantor an Hilbert, 27. Jan.
1900]
Doch schon kurz darauf reduziert Cantor sein System wieder rigoros:
Wie ich die Sache ansehe, so sind folgende zwei Axiome als Grundlage unserer endlichen
Zahlentheorie nothwendig und hinreichend.
I. Es giebt Dinge (d. h. Gegenstände unseres Denkens).
II. Ist V eine consistente Vielheit von Dingen und d ein nicht in V als Theil enthaltenes Ding, so
ist die Vielheit V + d auch consistent.
Diese beiden Axiome liefern mir die unbegrenzte Zahlenreihe 1, 2, 3, 4, ... der endlichen
ganzen Cardinalzahlen und alle Gesetze unter ihnen lassen sich beweisen, ohne Zuhülfenahme
weiterer Axiome. [Cantor an Hilbert, 20. Feb. 1900]
Eine Fortsetzung der axiomatischen Überlegungen in der anschließenden Korrespondenz mit
Hilbert oder anderen Mathematikern ist mir nicht bekannt. Am deutlichsten und zutreffendsten
dürfte Cantors persönliche Position aber in einem frühen Brief an Wundt formuliert sein:
Sie haben vollkommen Recht, wenn Sie den Gauß-Riemann-Lobatschewskischen Räumen den
realen Untergrund absprechen, dagegen ihre volle Berechtigung als "logische Postulate"
zugeben.
Dagegen nehme ich für meine unendlichen Zahlenbegriffe in Anspruch, daß sie frei von
jeglicher Willkür sich durch Abstraktion aus der Wirklichkeit mit derselben Notwendigkeit
ergeben wie die gewöhnlichen ganzen Zahlen, welche bisher allein als Ursprung aller anderen
mathematischen Begriffsbildungen gedient haben. Die transfiniten ganzen Zahlen sind
keineswegs, wie Sie sagen, bloße "Fiktionen" resp. "logische Postulate", wie es die
geometrischen Räume mit n Dimensionen sind, sondern sie haben denselben Charakter der
Realität wie die älteren Zahlen: 1, 2, 3 etc. Um dies zu verstehen, setze ich nichts anderes
voraus, als eine Weltbetrachtung, für welche die Leibnizschen Worte maßgebend sind:
"Je suis lettement pour l'infini actuel, qu'au lieu d'admettre que la nature l'abhorre, comme l'on
dit vulgairement, je tiens quelle l'affecte partout, pour mieux marquer les perfections de son
Auteur. Ainsi je crois qu'il n'y a aucune partie de la matière qui ne soit, je ne dis pas divisible,
mais actuellement divisée" etc. [Cantor an Wundt, 5. Okt. 1883]
"Notwendig und frei von Willkür." An solchen Sachen ist nichts zu deuteln.
Cantor has never attempted to formalize set theory axiomatically. Only toward the end of his
active career, under the influence of Hilbert, he has considered axioms for set theory at all. "He
sees, already toward the end of the 19th century, the appearance of formalistic thinking that he
abhorred" (Meschkowski). The following paragraphs will cover every mentioning, that I am
aware of, of axioms in Cantor's work and correspondence.
The so-called Cantor's axiom (1872) concerns geometry only:
... to add an axiom, requiring that, vice versa, to every numerical value there belongs a certain
point of the straight line the co-ordinate of which is equal to that numerical value. ... I call this
sentence an axiom because it is immanent to its nature that it cannot be proven in general. [G.
Cantor: "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen",
Math. Annalen 5 (1872) 123 - 132]
Cantor considered the Archimedian axiom as a provable theorem and the Euclidean proposition
about the whole and the part as questionable.
Thus the "Archimedean axiom" is not an axiom but a theorem that follows with logical necessity
from the notion of linear magnitude. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer (1932) p. 409]
I maintain the position that the so-called "Archimedean axiom" has been proven by myself and
that a deviation from this "axiom" means going astray. [Cantor to Veronese, 7 Sep. 1890]
Aren't a set and its corresponding cardinal number quite different things? Does not the first face
us as an object whereas the latter is an abstract picture of it in our mind? {{Remember the roots:
A set is an object (of reality).}} The old, so often repeated sentence: "The whole is more than its
part" may be accepted, without proof, only with respect to the entities which the whole and the
part are based upon. Then and only then the sentence is an immediate consequence of the
notions "whole" and "part". Unfortunately, however, this "axiom" has been used innumerably
often without any grounds and neglecting the necessary distinction between "reality" and
"magnitude" just in that meaning which makes it false in general, as soon a actually infinite sets
are involved and which makes it right for finite sets only because we are able to prove it as right
in this domain. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer (1932) p. 416f]
Hypotheses are completely out of the question in my arithmetical investigations about the finite
and the transfinite. Only the reasons for the real that is existing in nature are established there.
You, on the other hand, believe in the manner of the meta geometers Riemann, Helmholtz,
and comrades to be able to establish hypotheses in arithmetic too - which is completely
impossible. This is the cause of your as unfortunate as disastrous delusion which I am neither
able nor willing to dissuade you from. As little as in the domain of finite arithmetics other
fundamental laws can be established than those known from time immemorial for the numbers
1, 2, 3, ..., as little a deviation from the fundamental truths is possible in the domain of the
transfinite. "Hypotheses" violating these fundamental truths are as wrong and contradictory as,
e.g., the expression 2 + 2 = 5 or a square circle. It is sufficient for me to see such hypotheses
being put on top of some investigation in order to know from the outset that this investigation
must be wrong.
And your success has shown it, since your deplorable "hypotheses" have lead you to the
contradictory notion of actually infinitely small linear magnitudes. [Cantor to Veronese, 17 Nov.
1890]
I think the arguing that Mr. Veronese gives in his writings is fantasy, and what I says against me
is unreasonable. He says about his infinitely large numbers that they have been based upon
other hypotheses than mine. But mine are not based upon hypotheses but have been derived
immediately from the notion of set: They are just as necessary and free of arbitrariness as the
finite integers. [Cantor to Killing, 5 April 1895]
... The difference consists only in the "hypotheses" (axioms) which he requires for the respective
system. With that view Veronese is significantly closer to a modern axiomatics than Cantor who,
based upon his philosophical opinions, considers axioms as "fundamental truths". [H.
Meschkowski, W. Nilson (Herausgeber): Georg Cantor Briefe , Springer, Berlin (1991) p. 329]
{{In fact Cantor is right if mathematics is perceived as a serious science and not a frivolous play
about nonsense.}}
Only the recognition of inconsistent sets led Cantor to the consideration of axioms of transfinite
arithmetic.
The fact of the "consistency" of finite multitudes is a simple unprovable truth. It is "the axiom of
arithmetic (in the old meaning of the word)". And similarly is the "consistency" of the multitudes
to which I attach the alephs "the axiom of the extended, the transfinite arithmetic". [Cantor to
Dedekind, 28 Aug. 1899]
That the "countable" multitudes {αν} are completed sets seems to me to be an axiomatically
certain theorem which the whole theory of functions rests upon. [Cantor to Hilbert, 10 Oct. 1898]
It is our common conviction that the "arithmetic continuum" in this sense is a "set". The question
is whether this truth is a provable one or whether it is an axiom. Currently I tend more towards
the latter alternative but I am open to be convinced by you of the other. [Cantor to Hilbert, 9 May
1899]
I distinguish in pure mathematics three kinds of axioms:
1) The logical axioms, that it has in common with all other sciences which in formal logic and
recently in the logical calculus have been treated systematically.
2) The physical axioms of mathematics, for instance the geometrical axioms and the axioms of
mechanics. They can be recognized by their feature of lacking necessity. They can be replaced
by others (think of the parallel axiom of Euclid and and the non-euclidean geometry). I call them
physical axioms because they are related to special natures like space-things, time-things, forcethings, mass-things, etc.
Besides these two kinds of axioms there exists a third kind which hitherto seems to have gone
unnoticed because they are more hidden than those. I call them the
3) Metaphysical axioms of mathematics (I call them "metaphysical" because they are related to
things in general, no matter what nature they may have). T to this category belong in particular
the axioms of arithmetic, both the finite and the infinite numner theory.
The axiom of the finite number theory can briefly be noted as: "Every finite multitude is
consistent".
This as explanation:
a) The notion "finite multitude" is, as you will concede, determinable without reference to the
notion of number.
b) The finite notions of number or numbers themselves are only conceivable or possible when
the truth of the above axiom is assumed.
The axiom of the transfinite number theory is this: "Every multitude, which a signed alef, ¡γ
(with γ some ordinal number) belongs to, is consistent." [Cantor to Hilbert, 27 Jan. 1900]
By his subdivided axiom system Cantor differs, according to his own words, from Dedekind:
Dedkind obviously holds the opinion that number theory assumes no other axioms than the
logical ones; the supporters of the logic calculus seem to believe the same.
In the preface of Dedekind's paper we read: number theory "a part of logic"; the numbers are
for him "free creations of the human spirit". [Cantor to Hilbert, 27 Jan. 1900]
But after a short while already Cantor cuts his axiom system rigorously:
In the way I consider the matter, there are the following two axioms as the foundation of our
finite number theory necessary and sufficient.
I. There are things (i.e., objects of our thinking).
II: If V is a consistent multitude of things and d a thing that is not contained in V, then the
multitude V + d is also consistent.
These two axioms supply me the unlimited series of numbers 1, 2, 3, 4, ... of the finite integer
cardinal numbers, and all laws among them can be proven without using further axioms. [Cantor
to Hilbert, 20 Feb. 1900]
I am not aware of any continuation of the axiomatic reflections in the subsequent
correspondence with Hilbert or other mathematicians. The most apparent and accurate
description of Cantor's personal position, however, might be obtained from an early letter to
Wundt:
You are completely right if you deny the real basis of the Gauß-Riemann-Lobachewsky spaces,
but accept their full legitimacy as "logical postulates".
For my notions of infinite numbers, however, I claim that they result, free of any arbitrariness,
from abstraction from the reality with the same necessity as the usual integers which hitherto
solely have served as the origin of all other mathematical notions. The transfinite integers are by
no means, as you call it, bare "fictions" or "logical postulates" like the geometrical spaces with n
dimensions, but they have the same character of reality as the older numbers 1, 2, 3, etc. In
order to understand this, I do not assume anything else but a world view for which Leibniz' words
set the standard:
"I am so much in favour of the actual infinite. I believe that nature, instead of abhorring it, as is
usually said, uses it frequently everywhere in order to show better the perfectness of its author.
Therefore I believe that there is no piece of matter that not - I don't say is divisible - but actually
divided" etc. [Cantor to Wundt, 5 Okt. 1883]
"Necessary and free of arbitrariness." There are no ifs and buts about such stuff.
§ 435 The rising of the empty set
Bernard Bolzano, the inventor of the notion set (Menge) in mathematics would not have named
a nothing an empty set. In German this word has the meaning of many. Often we find in
German texts the expression große (great or large) Menge, rarely the expression kleine (small)
Menge. Therefore Bolzano apologizes for using this word in case of sets having only two
elements: Auch einen Inbegriff, der nur zwey Theile enthält, erlaube man mir hier eine Menge zu
nennen. (Allow me to call also a collection containing only two parts a set.)
[J. Berg (Hrsg.): Bernard Bolzano, Einleitung zur Grössenlehre, Friedrich Frommann Verlag,
Stuttgart (1975), Bolzano-Gesamtausgabe, Reihe II Band 7. p. 152].
Also Richard Dedekind discarded the empty set. But he accepted the singleton, i.e., the nonempty set of less than two elements: Für die Gleichförmigkeit der Ausdrucksweise ist es
vorteilhaft, auch den besonderen Fall zuzulassen, daß ein System S aus einem einzigen (aus
einem und nur einem) Element a besteht, d. h. daß das Ding a Element von S, aber jedes von a
verschiedene Ding kein Element von S ist. Dagegen wollen wir das leere System, welches gar
kein Element enthält, aus gewissen Gründen hier ganz ausschließen, obwohl es für andere
Untersuchungen bequem sein kann, ein solches zu erdichten. (For the uniformity of the wording
it is useful to permit also the special case that a system S consists of a single (of one and only
one) element a, i.e., that the thing a is elememt of S but every thing different from a is not an
element of S. The empty system, however, which does not contain any element shall be
excluded completely for certain reasons, although it might be convenient for other investigations
to fabricate such. [R. Dedekind: "Was sind und was sollen die Zahlen?" Vieweg, Braunschweig
1887, 8th ed. (1960) p. 2]
Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Es
ist ferner zweckmäßig, ein Zeichen zu haben, welches die Abwesenheit von Punkten ausdrückt,
wir wählen dazu den Buchstaben O; P ª O bedeutet also, daß die Menge P keinen einzigen
Punkt enthält, also streng genommen als solche gar nicht vorhanden ist." (Further it is useful to
have a symbol expressing the absence of points. We choose for that sake the letter O. P ª O
means that the set P does not contain any single point. So it is, strictly speaking, not existing as
such.) [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer (1932) p. 146]
And even Zermelo who made the Axiom II: Es gibt eine (uneigentliche) Menge, die "Nullmenge"
O, welche gar keine Elemente enthält. (Axiom II: There is an (improper) set, the "null-set" O
which does not contain any element.) [E. Zermelo: "Untersuchungen über die Grundlagen der
Mengenlehre I" Mathematische Annalen 65 (1908) p. 263]
Zermelo himself said in private correspondence:
It is not a genuine set and was introduced by me only for formal reasons [Zermelo to Fraenkel,
31 March 1921]
I increasingly doubt the justifiability of the "null set". Perhaps one can dispense with it by
restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal
simplification. [Zermelo to Fraenkel, 9 May 1921]
So it is all the more courageous that Zermelo based his number system completely on the empty
set: { } = 0, {{ }} = 1, {{{ }}} = 2, and so on. He knew at least that there is only one empty set. But
many ways to create the empty set could be devised, like the empty set of numbers, the empty
set of bananas, the empty set of unicorns, the uncountably many empty sets of all real
singletons, and the empty set of empty sets. Is it the emptiest set? Anyhow, zero means nothing.
So we can safely say (pun intended): Nothing is named the empty set.
§ 436 I show with absolute rigour that the cardinality of the second number class (II) is not only
different from the cardinality of the first number class but that it is indeed the next higher
cardinality; (Ich zeige aufs bestimmteste, daß die Mächtigkeit der zweiten Zahlenklasse (II) nicht
nur verschieden ist, von der Mächtigkeit der ersten Zahlenklasse, sondern daß sie auch
tatsächlich die nächst höhere Mächtigkeit ist;) [G. Cantor: "Grundlagen einer allgemeinen
Mannigfaltigkeitslehre", published by the author himself, Leipzig (1883)]
Two questions come to mind:
If one of two statements is false, should the other one be believed?
If an author, who was not a matheologian, published such a grave mistake, how would he be
called by the matheologians of today?
§ 437 I am convinced that the domain of definable numbers is not finished with the finite
magnitudes, (Mit den endlichen Größen ist daher meiner Überzeugung nach der Bereich der
definierbaren Größen nicht abgeschlossen,) [G. Cantor: "Grundlagen einer allgemeinen
Mannigfaltigkeitslehre", Leipzig (1883)]
Obviously Cantor has not been interested in undefinable numbers. Cantors Theorie umfasst
keine (in deutscher oder einer anderen sprechbaren Sprache) undefinierbaren Größen.
§ 438 If I talk about a number in the wider sense, then this happens first in the case that an
infinite sequence of rational numbers a1, a2, ..., an, ... is given by a law such that the difference
an+m - an with growing n becomes infinitely small, (Wenn ich von einer Zahlengröße im weiteren
Sinne rede, so geschieht es zunächst in dem Falle, daß eine durch ein Gesetz gegeben
unendliche Reihe von rationalen Zahlen a1, a2, ..., an, ... vorliegt, welche die Beschaffenheit hat,
daß die Differenz an+m - an mit wachsendem n unendlich klein wird,) [G. Cantor: "Über die
Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen", Math. Annalen 5
(1872) p. 123 - 132]
"Given by a finite law"! Cantor knew that infinite sequences cannot be given in another mode.
Cantor never, in his complete oeuvre and correspondence, accepts undefinable numbers, i.e.,
numbers that cannot be definied in German. He does not even talk about them until 1906
because there was no reason.
The set of those fools of matheology who consider or even accept undefinable real numbers
and who by this perversion of mind have caused mathematics and mathematicians to become
an object of ridicule, honoured not better than the insane inhabitants of a mad house, has been
empty at Cantor's times.
§ 439 The process of correctly constructing notions is in my opinion always the same: One
takes a thing without properties, that beforehand is nothing but a name or symbol A, and
properly endows it with several, even infinitely many understandable predicates, the menaing of
which is well known by means of other already existing ideas and which may not contradict each
other; (Der Vorgang bei der korrekten Bildung von Begriffen ist m. E. überall derselbe; man setzt
ein eigenschaftsloses Ding, das zuerst nichts anderes ist als ein Name oder ein Zeichen A, und
gibt demselben ordnungsmäßig verschiedene, selbst unendlich viele verständliche Prädikate,
deren Bedeutung an bereits vorhandenen Ideen bekannt ist, und die einander nicht
widersprechen dürfen;) [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre",
Leipzig (1883)]
§ 440 The definition of an irrational number always requires a well-defined first-order-infinite set
of rational numbers; (Zur Definition einer irrationalen reellen Zahl gehört stets eine wohldefinierte
unendliche Menge erster Mächtigkeit von rationalen Zahlen;) [G. Cantor: "Grundlagen einer
allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
§ 441 Hessenberg's argument
Hessenberg derives the uncountability of the powerset of Ù from the fact that the set S of all
natural numbers which are not in their image-sets cannot be enumerated by a natural number n.
If this set S is enumerated by n, and if n is not in S, then n belongs to S and must be in S, but
then n does not belong to S and so on. [Gerhard Hessenberg; "Grundbegriffe der Mengenlehre",
Sonderdruck aus den "Abhandlungen der Fries'schen Schule", I. Band, 4. Heft, Vandenhoeck &
Ruprecht, Göttingen (1906) ]
This is a logical paradox of self reference like many others, for instance this: "This set has seven
words" is as wrong as its negation: "This set does not have seven words". Such paradoxa have
nothing to do with cardinal numbers of sets.
Proof: Enumerate Hessenberg's set S by -1. Since -1 is not a natural number, there is no
problem. After having enumerated all subsets of Ù by natural numbers and -1, we can show that
the powerset of Ù is countable.
Well this concerns only one paradoxical set. But there may be others. No problem. How many
paradoxical sets can be defined in a speakable language like English that is suitable to express
all of mathematics (or in any formal language that can be defined in English)? The answer is: At
most countably many. Therefore all negative integers are sufficient to enumerate all possible
paradoxical sets. Finally it is easy to show that all integers belong to a countable set. No
mathematical reason to believe in uncountable sets.
§ 442 If {{the pointset}} P(1) has the cardinality of the second number class (II) [i.e., if P(1) is not
countable], then P(1)) can always, but in only in one way, be separated into two sets R and S,
(Hat aber P(1) die Mächtigkeit der zweiten Zahlenklasse (II) [d. h. ist P(1)) nicht abzählbar], so
läßt sich P(1) stets, und zwar nur auf einzige Weise in zwei Mengen R und S zerlegen,) [G.
Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
Only in one way. That means every point has its own definition. No ambiguity caused by
uncountably many undefinable elements. Proven by Cantor with the same strictness as its other
proofs.
§ 443 By a "manifold" or "set" I understand in general every multitude which can be understood
as a unit, i.e., every embodiment of defined elements which by a law can be connected to
become an entity. (Unter einer "Mannigfaltigkeit" oder "Menge" verstehe ich nämlich allgemein
jedes Viele, welches sich als Eines denken läßt, d. h. jeden Inbegriff bestimmter Elemente,
welcher durch ein Gesetz zu einem Ganzen verbunden werden kann.) [G.Cantor: "Grundlagen
einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
§ 444 A significant difference {{with respect to the related philosophy of Platon, Cusanus,
Bruno}} is that I fix, according to the concept, once and for all the different gradings of the proper
infinite by number classes (I), (II), (III), and so on, and now consider it a task not only to
investigate the relations of the supra-finite numbers mathematically but also, whereever they
appear in nature, to substantiate and to follow them. (Ein wesentlicher Unterschied {{zur
verwandten Philosophie von Platon, Cusanus, Bruno}} besteht aber darin, daß ich die
verschiedenen Abstufungen des Eigentlich-unendlichen durch die Zahlenklassen (I), (II), (III)
usw. ein für allemal dem Begriffe nach fixiere und es nun als Aufgabe betrachte, die
Beziehungen der überendlichen Zahlen nicht nur mathematisch zu untersuchen, sondern auch
allüberall, wo sie in der Natur vorkommen, nachzuweisen und zu verfolgen. [G.Cantor:
"Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
§ 445 Cantor's most famous sentence
By a "set" we mean every gathering together M of certain well-distinguished objects m of our
understanding or our thinking (which are called the "elements" of M) into a whole. (Unter einer
"Menge" verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen
Objekten m unsrer Anschauung oder unseres Denkens (welche die "Elemente" von M genannt
werden) zu einem Ganzen.) [G. Cantor: "Beiträge zur Begründung der transfiniten
Mengenlehre", Math. Annalen 46 (1895) 481-512]
§ 446 Equivalence of sets
The equivalence of sets is the necessary and unmistakable criterion for the equality of their
cardinal numbers. [...] If now M ~ N, this is based on a law of assiging, by which M and N are
mutually uniquely related to each other; here let the element m of M be related to the element n
of N. (Die Äquivalenz von Mengen bildet also das notwendige und untrügliche Kriterium für die
Gleichheit ihrer Kardinalzahlen. [...] Ist nun M ~ N, so liegt ein Zuordnungsgesetz zugrunde,
durch welches M und N gegenseitig eindeutig aufeinander bezogen sind; dabei entspreche dem
Elemente m von M das Element n von N.) [G. Cantor: "Beiträge zur Begründung der transfiniten
Mengenlehre", Math. Annalen 46 (1895) 481-512]
Note: Only definable elements can be uniquely related to each other.
§ 447 As "cardinality" or "cardinal number" of M we denote the general notion which, aided by
our active intellectual capacity, comes out of the set M by abstracting from the constitution of its
different elements m and of the order in which they are given. ("Mächtigkeit" oder "Kardinalzahl"
von M nennen wir den Allgemeinbegriff, welcher mit Hilfe unseres aktiven Denkvermögens
dadurch aus der Menge M hervorgeht, daß von der Beschaffenheit ihrer verschiedenen
Elemente m und von der Ordnung ihres Gegebenseins abstrahiert wird.) [G. Cantor: "Beiträge
zur Begründung der transfiniten Mengenlehre", Math. Annalen 46 (1895) 481-512]
Note: Cardinality requires different elements.
§ 448 Cantor obtained not only a scientific delight from his infinite numbers but also an
aesthetic pleasure
If I understand the infinite as I happened to do here and in my earlier approaches, then this
entails for me a real pleasure, which I greatfully abandon myself to: to observe how the whole
notion of number, which in the finite has only the background of (counting) number, when we
climb to the infinite, so to speak splits into two notions, in that of cardinality, which is
independent of the order, that the set has been given, and the (counting) number, which
necessarily depends on a pattern that the set has been given and by which the set becomes a
well-ordered set. And when I descend back again from the infinite to the finite, then I see as
clearly and beautifully how those two notions re-unite and flow together into the notion of the
finite integer. (Fasse ich das Unendliche so auf, wie dies von mir hier und bei meinen früheren
Versuchen geschehen ist, so folgt daraus für mich ein wahrer Genuß, dem ich mich dankerfüllt
hingebe, zu sehen, wie der ganze Zahlbegriff, der im Endlichen nur den Hintergrund der Anzahl
hat, wenn wir aufsteigen zum Unendlichen, sich gewissermaßen spaltet in zwei Begriffe, in
denjenigen der Mächtigkeit, welche unabhängig ist von der Ordnung, die einer Menge gegeben
wird, und in den der Anzahl, welche notwendig an eine gesetzmäßige Ordnung der Menge
gebunden ist, vermöge welcher letztere zu einer wohlgeordneten Menge wird. Und steige ich
wieder herab vom Unendlichen zum Endlichen, so sehe ich ebenso klar und schön, wie die
beiden Begriffe wieder Eins werden und zusammenfließen zum Begriffe der endlichen ganzen
Zahl.) [G.Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
§ 449 Since every single element m, when its features are left out of account, becomes a
"one", the cardinal number M is itself a certain set composed of nothing but ones, which has
existence in our mind as intellectual image or projection of the given set M. (Da aus jedem
einzelnen Elemente m, wenn man von seiner Beschaffenheit absieht, eine "Eins" wird, so ist die
Kardinalzahl M selbst eine bestimmte aus lauter Einsen zusammengesetzte Menge, die als
intellektuelles Abbild oder Projektion der gegebenen Menge M in unserm Geiste Existenz hat.)
[G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre", Math. Annalen 46 (1895)
481-512]
Note: "every single" element can be identified and its "one" can be indexed by an ordinal
number. Otherwise the "ones" could not be counted.
§ 450 Now we will show that the transfinite cardinal numbers can be ordered according to their
magnitude and that they form in this order a "well-ordered set", like the finite numbers but in an
extended sense. (Es soll nun gezeigt werden, daß die transfiniten Kardinalzahlen sich nach ihrer
Größe ordnen lassen und in dieser Ordnung wie die endlichen, jedoch in einem erweiterten
Sinne eine "wohlgeordnete Menge" bilden.) [G. Cantor: "Beiträge zur Begründung der
transfiniten Mengenlehre", Math. Annalen 46 (1895) 481-512]
§ 451 Therefore all sets are "countable" in an extended sense, in particular all "continua". (Alle
Mengen sind daher in einem erweiterten Sinne "abzählbar", im besonderen alle "Kontinua".) [
Cantor to Dedekind, 28 July 1899]
Note: When a set is countable in normal or extended sense, then its elements must be in one-toone correspondence with the elements of another set that defines countability in normal or
extended sense. One-to-one requires to distinguish each "one" element.
§ 452 Infinite definitions (that do not happen in finite time) are non-things. If Königs theorem
was true, according to which all "finitely definable" numbers form a set of cardinality ¡0, this
would imply that the whole continuum was countable, and that is certainly false. („Unendliche
Definitionen" (die nicht in endlicher Zeit verlaufen) sind Undinge. Wäre Königs Satz, daß alle
"endlich definirbaren" reellen Zahlen einen Inbegriff von der Mächtigkeit ¡0 ausmachen, richtig,
so hieße dies, das ganze Zahlencontinuum sei abzählbar, was doch sicherlich falsch ist.)
[Cantor to Hilbert, 6 August 1906]
Surely Cantor was wrong only in the sense that he didn't point out that the notion of definability
cannot be absolute, but depends upon the language. [B. Tait, FOM, Who was the first to accept
undefinable individuals in mathematics? (2009)]
http://www.cs.nyu.edu/pipermail/fom/2009-March/013468.html
No, Dr. Tait, definability has an absolute meaning. An object is definable whenever it can appear
as an individual in mathematics. It is defined by the framework of its appearance. And if the
whole twaddle of formal languages does not help to define definition, then we define it in English
or German. Every mathematical item that is definable in English or German is "definable in
mathematics" and vice versa.
§ 453 Grades of definition
Some real numbers are extremely well defined: The small natural numbers like 3 or 5 can be
grasped at first glance, even in unary representation.
A real number is very well defined, if its value (compared with the unit) can be determined
without any error, like all rational numbers the representations of which have a complexity that
can be handled by humans or computers.
A real number is well defined, if its value can be determined with an error as small as desired,
i.e., the number can be put in trichotomy with every very well defined rational number. The
irrational numbers with definitions that can be handled by humans or computers belong to this
class.
A real number is more or less defined, if it can be communicated such that a receiver with
more or less mathematical knowledge understands more or less the same as the sender. To this
class belong results of calculations that only in principle can be finished.
Even Non-numbers can be defined like the greatest prime number or the smallest positive
rational number or the reversal of the digit sequence of π or the lifetime of the universe
measured in seconds.
Mathematical objects without definitions, however, cannot exist since all mathematical objects by
definition have no other form of existence than existence by definition.
§ 454 Equality and the axioms of natural numbers
In a discussion about equality it was claimed that 1 + 1 can be same as 1. This is true of course,
as long it remains undefined what equivalence relation is expressed by "being equal".
Consider the expressions 0 + 0 and 0.
With respect to the script they are different. Even the two zeros in 0 + 0 are different, one of
them being that one on the left-hand side and the other one being just the "other". We can
distinguish the zeros. We could not, if they were identical in all respects.
If we know that both expressions are meant to represent numbers, we know that they are equal
with respect to the property of "being numbers" (and not being cars or stars).
With respect to numerical value we cannot know the result unless we know what "+" and "=" are
meaning. As soon as we know the foundations of arithmetic, we see that 0 + 0 = 0. (This
situation is comparable to having apples cut to pieces in closed boxes. Before opening the
boxes, we cannot know in how many pieces the contained apple has been cut.)
With respect to angular diameter sun is as large as moon. With respect to physical diameter sun
is much larger than moon. With respect to volume sun is much, much larger than moon.
Conclusion: Before knowing what kind of comparison is meant, we cannot obtain a result.
With respect to the Peano axioms in their truncated version, we see for instance that S(x) = S(y)
implies x = y. Here equality is not defined, so the expression is meaningless. If the script is
meant, the sequence defined by the axioms could be 0, 0 + 0, 0 + 0 + 0, etc. or 1, 1^1, 1^1^1,
etc. Of course we guess somehow that arithmetical equality is meant as soon as numbers get
involved. That means, the reader is not only expected to be able to read and to understand
written text, but also to decide when two "successors" are equal or different. A reader who is
able to recognize the numerical equality or inequality of numbers would know +1 as well and
obtain the sequence Ù from the three axioms:
1œM
nœMfln+1œM
ÙŒM
If unable to read this text, the prospective reader should learn to read.
If unable to understand the used logic, the prospective reader should learn its basics.
If unable to understand the meaning of "+1" and "=", the prospective reader should learn the
basics of arithmetic.
Then the reader would be far better off than with the five Peano axioms in their truncated version
which do not define the natural numbers unless this definition is taken from elsewhere.
§ 455 Realness of integers according to Cantor
For one we are allowed to consider the integers inasmuch as being really existing as they,
based on definitions, occupy a distinct place in our mind, being extremely well distinguished from
all other pieces of our thinking, having well defined relations to them and in a certain way are
modifying the substance of our mind; allow me to call this kind of reality intrasubjective or
immanent reality. Further realness can be attributed to the the numbers, because they must be
considered as an expression or an image of events or relations in the external world opposite to
the intellect, and since further the different number classes (I), (II), (III), etc. are representing
cardinalities which indeed occur in the physical and intellectual nature. This second kind of
reality I call the transsubjective or also transient reality of the integers.
Einmal dürfen wir die ganzen Zahlen insofern für wirklich ansehen, als sie auf Grund von
Definitionen in unserm Verstande einen ganz bestimmten Platz einnehmen, von allen übrigen
Bestandteilen unseres Denkens aufs beste unterschieden werden, zu ihnen in bestimmten
Beziehungen stehen und somit die Substanz unseres Geistes in bestimmter Weise modifizieren;
es sei mir gestattet, diese Art der Realität unsrer Zahlen ihre intrasubjektive oder immanente
Realität zu nennen. Dann kann aber auch den Zahlen insofern Wirklichkeit zugeschrieben
werden, als sie für einen Ausdruck oder ein Abbild von Vorgängen und Beziehungen in der dem
Intellekt gegenüberstehenden Außenwelt gehalten werden müssen, als ferner die verschiedenen
Zahlenklassen (I), (II), (III) u. s. w. Repräsentanten von Mächtigkeiten sind, die in der
körperlichen und geistigen Natur tatsächlich vorkommen. Diese zweite Art der Realität nenne ich
die transsubjektive oder auch transiente Realität der ganzen Zahlen.
[G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
According to this definition, without at least one atom in the brain devoted to the integer or one
atom outside representing it, the integer has neither intrinsic nor transient reality. Numbers
exceeding the number of 1080 atoms in the universe can have only matheological reality, i.e.,
irreality.
§ 456 David Hilbert on the infinite
Finally we will remember our original topic and draw the conclusion. On balance the complete
result of all our investigations about the infinite is this: The infinite is nowhere realized; it is
neither present in nature nor admissible as the foundation of our rational thinking. This is a
remarkable harmony between being and thinking.
Zuletzt wollen wir wieder unseres eigentlichen Themas gedenken und über das Unendliche das
Fazit aus allen unseren Überlegungen ziehen. Das Gesamtergebnis ist dann: das Unendliche
findet sich nirgends realisiert; es ist weder in der Natur vorhanden, noch als Grundlage in
unserem verstandesmäßigen Denken zulässig - eine bemerkenswerte Harmonie zwischen Sein
und Denken.
[David Hilbert, Über das Unendliche, 24. Juni 1925]
http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=26816
If not in the foundations (i.e., as an axiom) how else could the infinite enter mathematics?
§ 457 The infinite human mind
If it turns out that the mind is able to define and to distinguish, in a certain sense infinite, i.e.,
transfinite numbers, then either the words "finite mind" have to be given an extended meaning
[...] or the precicate "infinite" has to be granted to the human mind in a certain respect, the latter
of which is in my opinions the only right position.
Zeigt es sich aber, daß der Verstand auch in bestimmtem Sinne unendliche, d. i. überendliche
Zahlen definieren und voneinander unterscheiden kann, so muß entweder den Worten
"endlicher Verstand", eine erweiterte Bedeutung gegeben werden [...]; oder es muß auch dem
menschlichen Verstand das Prädikat "unendlich" in gewissen Rücksichten zugestanden werden,
was meines Erachtens das einzig Richtige ist.
[G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
Alas it has turned out that the mind is unable to define or to distinguish infinitely many numbers.
All numbers that ever have been defined as individuals, including real numbers, complex
numbers, and more-complex numbers like tensors, functions, and sets, belong to a finite set and that will remain so forever.
§ 458 Another resolution of Berry's paradox
Quite a lot of resolutions of the Berry paradox have been proposed
http://en.wikipedia.org/wiki/Berry_paradox
Here is another one, based upon the different grades of definition (cp. § 453) resulting in
distinctions like this:
The set { x | x2 - 3x + 2 = 0 } is very well defined.
The set {1, 2} is extremely well defined.
Applied to the Berry paradox we find that "the least natural not nameable in fewer than nineteen
syllables" has been very well defined only, using 18 syllables, but by this definition it is not
immediately clear what number is meant. Some work is required to find a definition that makes
this number extremely well defined: "one-hundred and eleven thousand, seven-hundred and
seventy seven" or 111,777.
The paradox vanishes as soon as definitions are distinguished by their grade.
§ 459 Every set can be well-ordered
That it is always possible to give every well-defined set the form of a well-ordered set, on this,
as it appears to me, fundamental and momentous and by its universality particularly remarkable
law of thinking I will come back in a later treatise.
Daß es immer möglich ist, jede wohldefinierte Menge in die Form einer wohlgeordneten
Menge zu bringen, auf dieses, wie mir scheint, grundlegende und folgenreiche, durch seine
Allgemeingültigkeit besonders merkwürdige Denkgesetz werde ich in einer späteren Abhandlung
zurückkommen. [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig
(1883)]
Proof that every set can be well-ordered.
Beweis, daß jede Menge wohlgeordnet werden kann [E. Zermelo: "Beweis, daß jede Menge
wohlgeordnet werden kann", Mathematische Annalen 59 (1904) 514-516]
New proof of the possibility of a well-ordering {{with the page header}} New proof of the wellordering.
Neuer Beweis für die Möglichkeit einer Wohlordnung {{mit der Seitenüberschrift}} Neuer
Beweis für die Wohlordnung. [Ernst Zermelo: "Neuer Beweis für die Möglichkeit einer
Wohlordnung", Mathematische Annalen 65 (1908) 107-128]
Every set can be well-ordered. This has been proved, to his own satisfaction, by G. Cantor, and
to the satisfaction of matheologians by E. Zermelo - even twice. These proofs claim definitely,
explicitly, and unanimously that it can be done. Meanwhile matheologians keep being satified
with these proofs but explain that it only could be done if it could be done or that there exists a
well-ordering of every set, i.e., it has been done by their Gods and Goddesses, since probably
even a matheolologian would hardly assume that a set can well-order itself.
§ 460 Cantor's criterion for existence and reality of numbers
Mathematics is completely free in its development and only obliged to obey the self-evident
condition that its notions are free of internal contradictions and are related by means of fixed
definitions to the already existig and well-established notions. In particular when introducing new
numbers, mathematics is merely obliged to give definitions of them, by which process they gain
such a definitness and possibly such a relation to the older numbers that they definitely can be
distinguished from each other. As soon as a number satifies all these critera it may be and has
to be considered as existing and having reality in mathematics.
Die Mathematik ist in ihrer Entwicklung völlig frei und nur an die selbstredende Rücksicht
gebunden, daß ihre Begriffe sowohl in sich widerspruchslos sind, als auch in festen durch
Definitionen geordneten Beziehungen zu den vorher gebildeten, bereits vorhandenen und
bewährten Begriffen stehen. Im besonderen ist sie bei der Einführung neuer Zahlen nur
verpflichtet, Definitionen von ihnen zu geben, durch welche ihnen eine solche Bestimmtheit und
unter Umständen eine solche Beziehung zu den älteren Zahlen verliehen wird, daß sie sich in
gegebenen Fällen untereinander bestimmt unterscheiden lassen. Sobald eine Zahl allen diesen
Bedingungen genügt, kann und muß sie als existent und real in der Mathematik betrachtet
werden. [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
However numbers not satisfying these criteria are accepted in matheology.
§ 461 What should hinder us?
[...] forsooth I would not know what should hinder us in our activity of forming new numbers, as
soon as it becomes clear that, for the progress of science, it has become desirable or even
indispensable to include one of these infinitely many number-classes into examination.
[...] ich wüßte aber fürwahr nicht, was uns von dieser Tätigkeit des Bildens neuer Zahlen
zurückhalten sollte, sobald es sich zeigt, daß für den Fortschritt der Wissenschaften die
Einführung einer neuen von diesen unzähligen Zahlenklassen in die Betrachtung
wünschenswert oder sogar unentbehrlich geworden ist. [G. Cantor: "Grundlagen einer
allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]
I know what hinders us in our activity of forming uncountably many numbers: The lack of
definitions, i.e., the lack of names. But if uncountably many are there without having been
created, why should we bother to create anything which is necessarily belonging to a vanishing
minority?
§ 462 Transformations preserving well-ordering
The question by which transformations of a well-ordered set its number of elements is
changed, by which it is not, simply can be answered in this way: Those and only those
transformations do not change the number of elements which can be put down to a finite or
infinite set of transpositions, i.e., of exchanges of two elements.
Die Frage, durch welche Umformungen einer wohlgeordneten Menge ihre Anzahl geändert
wird, durch welche nicht, läßt sich einfach so beantworten, daß diejenigen und nur diejenigen
Umformungen die Anzahl ungeändert lassen, welche sich zurückführen lassen auf eine endliche
oder unendliche Menge von Transpositionen, d. h. von Vertauschungen je zweier Elemente. [E.
Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen
Inhalts", Springer (1932) p. 214]
Can an infinite number of transpositions be finished? Only if infinity can be finished at all. But
that would allow us to obtain the well-ordered set Ù with the largest natural number as the first
element, exchanging, in the natural well-ordering, successively, for every n in Ù, the first number
n0 and the number n, whenever n > n0.
§ 463 David Hilbert on Potential and Actual Infinity
Should we briefly characterize the new view of the infinite introduced by Cantor, we could
certainly say: In analysis we have to deal only with the infinitely small and the infinitely large as a
limit-notion, as something becoming, arising, being under construction, i.e., as we put it, with the
potential infinite. But this is not the proper infinite. This we have for instance when we consider
the entirety of the numbers 1, 2, 3, 4, ... itself as a completed unit, or the points of a length as an
entirety of things which is completely available. This sort of infinity is named actual infinite.
Will man in Kürze die neue Auffassung des Unendlichen, der Cantor Eingang verschafft hat,
charakterisieren, so könnte man wohl sagen: in der Analysis haben wir es nur mit dem
Unendlichkleinen und dem Unendlichengroßen aIs Limesbegriff, als etwas Werdendem,
Entstehendem, Erzeugtem, d. h., wie man sagt, mit dem potentiellen Unendlichen zu tun. Aber
das eigentlich Unendliche selbst ist dies nicht. Dieses haben wir z. B., wenn wir die Gesamtheit
der Zahlen 1, 2, 3, 4, ... selbst als eine fertige Einheit betrachten oder die Punkte einer Strecke
als eine Gesamtheit von Dingen ansehen, die fertig vorliegt. Diese Art des Unendlichen wird als
aktual unendlich bezeichnet. [David Hilbert: "Über das Unendliche", Mathematische Annalen 95
(1926) p. 167]
http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=26816
§ 464 Georg Cantor on Potential and Actual Infinity
In spite of significant differences between the notions of the potential and actual infinite, where
the first is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed
in itself but beyond all finite magnitudes, it happens deplorably frequently that the one is
confused with the other.
[...] improper infinite = variable finite = syncategorematice infinitum on the one side and proper
infinite = transfinitum = completed infinite = being infinite = categorematice infinitum on the other
[...]
Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und aktualen Unendlichen,
indem ersteres eine veränderliche endliche, über alle Grenzen hinaus wachsende Größe,
letzteres ein in sich festes, konstantes, jedoch jenseits aller endlichen Größen liegendes
Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daß das eine mit dem andern
verwechselt wird. (p. 374)
[...] uneigentlichunendlichem = veränderlichem Endlichem = synkategorematice infinitum
einerseits und Eigentlichunendlichem = Transfinitum = Vollendetunendlichem =
Unendlichseiendem = kategorematice infinitum andrerseits (p. 391)
[E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen
Inhalts", Springer (1932)]
§ 465 Bernard Bolzano on Potential and Actual Infinity
[...] a manifold which is larger than every finite one, i.e., a manifold which has the property that
every finite set is only part of it, I shall call an infinite manifold. [...] If they, like Hegel, Erdmann,
and others, imagine the mathematical infinite only as a magnitude which is variable and only has
no limit in its growth (like some mathematicians, as we will see soon, have assumed to explain
their notion) so I agree in their reproach of this notion of a magnitude only growing into the
infinite but never reaching it. A really infinite magnitude, for instance the length of the line not
ending on both sides (i.e. the magnitude of that spatial object containing all points which can be
determined by the purely intellectually imagined relation with respect to two points) need not be
variable, as indeed it is not in this example. And a magnitude that only can be considered to be
larger than considered before and being able of becoming larger than every given (finite)
magnitude, may as well permanently remain a finite magnitude only, as in case of each of the
numbers 1, 2, 3, 4 ....
[...] werde ich eine Vielheit, die grösser als jede endliche ist, d. h. eine Vielheit, die so
beschaffen ist, dass jede endliche Menge nur einen Theil von ihr darstellt, eine unendliche
Vielheit nennen. [...] Wenn sie, wie Hegel, Erdmann u. A. sich das mathematische Unendliche
nur als eine Grösse denken, welche veränderlich ist und in ihrem Wachsthume keine Gränze hat
(was freilich manche Mathematiker, wie wir bald sehen werden, als die Erklärung ihres Begriffes
aufgestellt haben): so pflichte ich ihnen in ihrem Tadel dieses Begriffes einer in das Unendliche
nur wachsenden, nie es erreichenden Grösse selbst bei. Eine wahrhaft unendliche Grösse, z. B.
die Länge der ganzen beiderseits gränzenlosen Geraden (d. h. die Grösse desjenigen
Raumdinges, das alle Puncte enthält, die durch ihr blosses begrifflich vorstellbares Verhältnis zu
zwei gegebenen bestimmt sind), braucht eben nicht veränderlich zu sein, wie sie es denn in dem
hier angeführten Beispiele in der That nicht ist; und eine Grösse, die nur stets grösser
angenommen werden kann, als wir sie schon angenommen haben, und grösser als jede
gegebene (endliche) Grösse zu werden vermag, kann dabei gleichwohl beständig eine bloss
endliche Grösse verbleiben, wie dieses namentlich von jeder Zahlengrösse 1, 2, 3, 4 ...... gilt.
[Bernard Bolzano: "Paradoxien des Unendlichen", Leipzig (1851) 6f]
§ 466 Ernst Zermelo on Potential and Actual Infinity
In contrast to the notion of natural number the field of analysis needs the existence of infinite
sets: "As a consequence, those who are really serious about rejection of the actual infinite in
mathematics should stop at general set theory and the lower number theory and do without the
whole modern analysis." (1909) Infinite domains "can never be given empirically; they are set
ideally and exist only in the sense of a Platonic idea" (1932). In general they can only be defined
axiomatically; any inductive or "genetic" way is inadequate. "The infinite is neither physically nor
psychologically given to us in the real world, it has to be comprehended and 'set' as an idea in
the Platonic sense." (1942) [Heinz.-Dieter Ebbinghaus: "Ernst Zermelo, An Approach to His Life
and Work", Springer (2007)]
§ 467 Georg Cantor on Potential and Actual Infinity
There is no further justification necessary when I in the "Grundlagen", just at the beginning,
distinguish two notions toto genere different from each other, which I call the improper-infinite
and the proper-infinite; they have to be understood as in no way compatible with each other. The
frequently, at all times, admitted union or confusion of these two completely disparate notions
causes, to my firm conviction, innumerable errors; in particular I see herein the reason why the
transfinite numbers have not been discovered before. {{Without this confusion and the chance to
exploit the due errors set theory would have become extinct long ago.}}
Es bedarf also keiner weiteren Rechtfertigung, daß ich in den "Grundlagen" gleich im Anfang
zwei toto genere von einander verschiedene Begriffe unterscheide, welche ich das Uneigentlichunendliche und das Eigentlich-unendliche nenne; sie müssen als in keiner Weise vereinbar oder
verwandt angesehen werden. Die so oft zu allen Zeiten zugelassene Vereinigung oder
Vermengung dieser beiden völlig disparaten Begriffe enthält meiner festen Überzeugung nach
die Ursache unzähliger Irrtümer; im besonderen sehe ich aber hier den Grund, warum man nicht
schon früher die transfiniten Zahlen entdeckt hat. [E. Zermelo: "Georg Cantor, Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 395]
§ 468 Richard Dedekind on Potential and Actual Infinity
Everytime when there is a cut (A1, A2) which is not created by a rational number, we create a
new, an irrational number. {{This is potential infinity.}}
Jedesmal nun, wenn ein Schnitt (A1, A2) vorliegt, welcher nicht durch eine rationale Zahl
hervorgebracht wird, so erschaffen wir eine neue, eine irrationale Zahl. [Richard Dedekind:
"Stetigkeit und Irrationale Zahlen", Vieweg Braunschweig (1872), 6th edn. (1960) p.13]
There are infinite systems. Proof (a similar reflection can be found in § 13 of the Paradoxien
des Unendlichen by Bolzano (Leipzig 1851)) The world of my thoughts, i.e., the collection S of all
things which can be object of my thinking, is infinite. For, if s is an element of S, then the thought
s' that s can be object of my thinking is itself an object of my thinking. {{This is potential infinity.}}
Es gibt unendliche Systeme. Beweis (Eine ähnliche Betrachtung findet sich in § 13 der
Paradoxien des Unendlichen von Bolzano (Leipzig 1851)). Meine Gedankenwelt, d. h. die
Gesamtheit S aller Dinge, welche Gegenstand meines Denkens sein können, ist unendlich.
Denn wenn s ein Element von S bedeutet, so ist der Gedanke s' daß s Gegenstand meines
Denkens sein kann, selbst ein Element von S. [Richard Dedekind: "Was sind und was sollen die
Zahlen?", Vieweg, Braunschweig (1887), 8th edn. (1960) p. 14]
A system S is called infinite, if it is similar to a proper part of itself; otherwise S is called finite
system. [...] S is called infinite if there is a proper part of S into which S can distinctly (similarly)
be mapped. {{A complete infinite system S is actual infinity.}}
Ein System S heißt unendlich, wenn es einem echten Teile seiner selbst ähnlich ist; im
entgegengesetzten Falle heißt S ein endliches System. [...] S heißt unendlich, wenn es einen
echten Teil von S gibt, in welchem S sich deutlich (ähnlich) abbilden lässt. [Richard Dedekind:
"Was sind und was sollen die Zahlen?", Vieweg, Braunschweig (1887), 8th edn. (1960) p. 13]
§ 469 Aristotle on Potential and Actual Infinity
But the phrase 'potential existence' is ambiguous. When we speak of the potential existence of a
statue we mean that there will be an actual statue. It is not so with the infinite. There will not be
an actual infinite. The word 'is' has many senses, and we say that the infinite 'is' in the sense in
which we say 'it is day' or 'it is the games', because one thing after another is always coming into
existence. For of these things too the distinction between potential and actual existence holds.
We say that there are Olympic games, both in the sense that they may occur and that they are
actually occurring. The infinite exhibits itself in different ways - in time, in the generations of man,
and in the division of magnitudes. For generally the infinite has this mode of existence: one thing
is always being taken after another, and each thing that is taken is always finite, but always
different. Again, 'being' has more than one sense, so that we must not regard the infinite as a
'this', such as a man or a horse, but must suppose it to exist in the sense in which we speak of
the day or the games as existing things whose being has not come to them like that of a
substance, but consists in a process of coming to be or passing away; definite if you like at each
stage, yet always different. [...]
It is reasonable that there should not be held to be an infinite in respect of addition such as to
surpass every magnitude, but that there should be thought to be such an infinite in the direction
of division. For the matter and the infinite are contained inside what contains them, while it is the
form which contains. It is natural too to suppose that in number there is a limit in the direction of
the minimum, and that in the other direction every assigned number is surpassed. In magnitude,
on the contrary, every assigned magnitude is surpassed in the direction of smallness, while in
the other direction there is no infinite magnitude. [...] But in the direction of largeness it is always
possible to think of a larger number: for the number of times a magnitude can be bisected is
infinite. Hence this infinite is potential, never actual: the number of parts that can be taken
always surpasses any assigned number. But this number is not separable from the process of
bisection, and its infinity is not a permanent actuality but consists in a process of coming to be,
like time and the number of time. With magnitudes the contrary holds. What is continuous is
divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can
potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is
impossible to exceed every assigned magnitude; for if it were possible there would be something
bigger than the heavens. [...]
Our account does not rob the mathematicians of their science, by disproving the actual
existence of the infinite in the direction of increase, in the sense of the untraversable. In point of
fact they do not need the infinite and do not use it. They postulate only that the finite straight line
may be produced as far as they wish. It is possible to have divided in the same ratio as the
largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will
make no difference to them to have such an infinite instead, while its existence will be in the
sphere of real magnitudes.
[Aristotle: "Physics", part 6 - 7]
http://www.greektexts.com/library/Aristotle/Physics/eng/1327.html
§ 470 Fraenkel et al. on Potential and Actual Infinity
The language in which one deals with the expressions of a given theory (not with the entities
denoted by these expressions!) is called the metalanguage of this theory. In our case the
metalanguage will be ordinary English, supplemented by a few symbols and some rules
governing their use. The language in which the theory itself is formulated is called objectlanguage of this theory. In our case the object-language is a certain extremely restricted sublanguage of ordinary English, again supplemented by a few symbols and their rules. [...] Within
the framework of the first-order predicate calculus we have a (potentially) infinite list of individual
variables x, y, z, w, x', y', z', w', etc. [p 19f]
Dedekind, just like Bolzano four decades before, believed that he had proved the existence of
infinite sets. However, not only are their methods incompatible with the restrictions of our
axiomatic system but they are just those that lead to the logical antinomies.
From the axiomatic viewpoint there is no other way for securing infinite sets {{here actual
infinity is meant}} but postulating them, and we shall express an appropriate axiom in several
froms. While the first corresponds to Zermelo's original axiom of infinity, the second implicitly
refers to von Neumann's method of introducing ordinal numbers. [p. 46]
[A.A. Fraenkel, Y. Bar-Hillel, A. Levy: "Foundations of Set Theory", Elsevier (1973)]
§ 471 Thoralf A. Skolem on Potential and Actual Infinity
In order to obtain something absolutely nondenumerable, we would have to have either an
absolutely nondenumerably infinite number of axioms or an axiom that could yield an absolutely
nondenumerable number of first-order propositions. But this would in all cases lead to a circular
introduction of higher infinities; that is, on an axiomatic basis higher infinities exist only in a
relative sense. [J. van Heijenoort: "From Frege to Gödel: A Source Book in Mathematical Logic,
1879-1931", Harvard University Press, Cambridge, Mass. (1967) p. 296]
§ 472 Gerhard Hessenberg on Potential and Actual Infinity
The present work [...] had originally been scheduled to continue a report which appeared
under the title "The infinite in mathematics" in the first volume of this journal. This report was
concerned with excluding actually infinite magnitudes from the limit methods, in particular from
infinitesimal calculus. The continuation should show that this exclusion does in no way mean to
refrain from considering actual infinity in mathematics. On the contrary, the example of
nondenumerability of the continuum should show the possibility to distinguish different
cardinalities, and Cantor's resulting proof of the existence of transcendental numbers should
show the practical importance of this distinction. [...] it has been shown that neither in the
elementary chapters of mathematics nor in those denoted by "infinitesimal calculus" a really
infinite "magnitude" occurs, that rather the word "infinite" is merely used as an abbreviating
description of important facts of the finite.
Die vorliegende Arbeit [...] war ursprünglich als Fortsetzung eines unter dem Titel "Das
Unendliche in der Mathematik" im ersten Heft dieser Zeitschrift erschienenen Berichtes
gedacht, der sich mit der Ausschaltung der aktual unendlichen Größen aus den
Grenzmethoden, insbesondere aus der Infinitesimalrechnung beschäftigt. Die Fortsetzung sollte
zeigen, daß mit dieser Ausschaltung die Mathematik keineswegs auf die Betrachtung des aktual
Unendlichen überhaupt verzichtet. Vielmehr sollte das Beispiel der Nichtabzählbarkeit des
Kontinuums die Möglichkeit der Unterscheidung verschiedener unendlicher Mächtigkeiten, und
der daraus folgende Cantorsche Beweis der Existenz transzendenter Zahlen die praktische
Bedeutung dieser Unterscheidung dartun. [...] dargetan worden, daß weder in den elementaren
noch in den als "Infinitesimalrechnung" bezeichneten Kapiteln der Mathematik eine wirklich
unendliche "Größe" auftritt; daß vielmehr das Wort "unendlich" lediglich zur abkürzenden
Beschreibung wichtiger Tatsachen des endlichen benutzt wird.
[Gerhard Hessenberg; "Grundbegriffe der Mengenlehre", Sonderdruck aus den "Abhandlungen
der Fries'schen Schule", I. Band, 4. Heft, Vandenhoeck & Ruprecht, Göttingen (1906) Vorwort
und § 1]
§ 473 Thomas Jech on Potential and Actual Infinity
Until then, no one envisioned the possibility that infinities come in different sizes, and moreover,
mathematicians had no use for "actual infinity". The arguments using infinity, including the
Differential Calculus of Newton and Leibniz, do not require the use of infinite sets. [...] Cantor
observed that many infinite sets of numbers are countable: the set of all integers, the set of all
rational numbers, and also the set of all algebraic numbers. {{This proof is due to Dedekind. Had
he observed already that the set of all definable transcendental numbers is countable too (which
is a straight-forward extension of his argument), nobody today would talk about set theory.}}
Then he gave his ingeneous diagonal argument that proves, by contradiction, that the set of all
real numbers is not countable. A consequence of this is that there exists a multitude of
transcendental numbers, even though the proof, by contradiction, does not produce a single
specific example. {{Small wonder! Up to every digit the anti-diagonal is a rational number. It is
impossible to define a transcendental number by its digits.}}
[Thomas Jech: "Set Theory", Stanford Encyclopedia of Philosophy (2002)]
http://stanford.library.usyd.edu.au/entries/set-theory/
§ 474 Solomon Feferman on Potential and Actual Infinity
The notions of forcing and of generic sets were introduced by Paul Cohen to settle the longoutstanding problems of the logical interrelationships of the axiom of constructibility, the axiom of
choice, and the continuum hypothesis, relative to the system of Zermelo-Fraenkel set theory. In
this paper we consider extensions of these notions to other contests, namely that of (1st order)
number theory and of a part of (2nd order) analysis, and obtain some applications there.These
results depend on a general transform lemma concerning forcing [...] By means of this lemma
we are also able to obtain some new applications of Cohen's methods in set theory. The most
interesting of these are the following: (1) No set-theoretically definable well-ordering of the
continuum can be proved to exist from the Zermelo-Fraenkel axioms together with the axiom of
choice and the generalized continuum hypothesis.
[S. Feferman: "Some applications of the notions of forcing and generic Sets", Talk at the
International Symposium on the Theory of Models, Berkeley (1963)]
http://matwbn.icm.edu.pl/ksiazki/fm/fm56/fm56129.pdf
Feferman and Levy showed that one cannot prove that there is any non-denumerable set of real
numbers which can be well ordered. Moreover, they also showed that the statement that the set
of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted.
[Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy: "Foundations of Set Theory", North
Holland, Amsterdam (1973) p. 62]
- I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory
as a philosophy of our subject [...] platonism is the medieval metaphysics of mathematics; surely
we can do better.
- The actual infinite is not required for the mathematics of the physical world.
- The question raised in two of the essays of the volume, Is Cantor Necessary?, is answered
with a resounding no.
[S. Feferman: "In the light of logic", Oxford Univ. Press (1998)]
Feferman zeigt in seinem Aufsatz "Why a little bit goes a long way - Logical foundations of
scientifically applicable mathematics" anhand einiger Fallstudien, dass alle gegenwärtig für
wissenschaftliche Zwecke erforderliche Mathematik in einem Axiomensystem ausgeführt werden
kann, in dem das aktual Unendliche nicht vorkommt.
[W. Mückenheim: "Die Geschichte des Unendlichen", 7th edn., Maro, Augsburg, (2012) p. 108]
§ 475 Henri Poincaré on Potential and Actual Infinity
Why do the pragmatists refuse to admit objects that cannot be defined by a limited number of
words? Because they are of the opinion that an object does not exist unless it has been thought
and that a thought object cannot be comprehended independent of a thinking subject. That is
the core of idealism. And for a thinking subject, be it a man or anything similar, hence a finite
being, the infinite cannot have any other sense than the possibility to create as many objects as
one wishes.
But the Cantorians are realists {{a strange use of this word!}} even with respect to the
mathematical magnitudes. These magnitudes appear to them as having an independent
existence. They do not create geometry, they discover it.
Und warum weigern sich die Pragmatiker, Gegenstände zuzulassen, welche nicht durch eine
beschränkte Anzahl von Worten festgelegt werden können? Deshalb, weil sie der Ansicht sind,
daß ein Objekt nicht existiert, wenn es nicht gedacht ist und daß man ein gedachtes Objekt nicht
unabhängig von einem denkenden Subjekt erfassen kann. Das ist der Kernpunkt des
Idealismus. Und für ein denkendes Subjekt, sei es nun ein Mensch oder irgendein Wesen, das
dem Menschen gleicht, also infolgedessen ein endliches Wesen, kann das Unendliche keinen
anderen Sinn haben als die Möglichkeit, so viele Objekte ins Leben zu rufen, als man will.
Aber die Cantorianer sind Realisten selbst in bezug auf die mathematischen Größen. Diese
Größen scheinen ihnen eine unabhängige Existenz zu besitzen. Sie schaffen die Geometrie
nicht, sie entdecken sie.
[H. Poincaré: "Letzte Gedanken: Die Mathematik und die Logik", übers. von K. Lichtenecker,
Akadademische Verlagsgesellschaft, Leipzig (1913) p. 160f]
§ 476 Detlef Laugwitz on Potential and Actual Infinity
Numbers serve the purposes of counting, mesuring and computing. [...] the logical short circuit
that results from transforming the infinitely increasing number of digits of the potential infinite into
an actual infinite and identifying ◊2 with the never ending decimal representation 1.4142 ...
Zahlen dienen zur Verrichtung des Zählens, des Messens und des Rechnens. [...] den
logischen Kurzschluß, der darin liegt, daß man das potentielle Unendlich der unbegrenzt
wachsenden Stellenzahl zu einem aktualen Unendlich macht und ◊2 der "nicht abbrechenden
Dezimalzahl" 1,4142... gleich setzt.
[Detlef Laugwitz: "Zahlen und Kontinuum", BI, Zürich (1986) p. 16]
(Quoted from my handwritten notes.)
§ 477 Georg Cantor on Potential and Actual Infinity
To exclude this confusion from the outset I denote the smallest transfinite number by a symbol
that differs from the usual symbol ¶ of the improper infinite, namely ω.
In fact ω can be considered somehow as a limit which is approached by the variable, finite
integer ν, but only in the sense that ω is the smallest transfinite ordinal number, i.e., the smallest
fixed and determined number which is is larger than all finite numbers ν; quite like ◊2 is the limit
of certain variable, growing rational numbers. Only this is added: The difference between ◊2 and
its rational approximations becomes arbitrily small whereas ω - ν is always equal to ω; however,
this difference does not change the fact that ω has to be considered as determined and
completed as ◊2, and it does not change that ω has as little traces from of the approaching
numbers ν as ◊2 has not any traces of its rational approximations.
In a certain sense the transfinite numbers themselves are new irrationalities, and indeed the, in
my eyes, best method to define the finite irrational numbers, is quite similar to, I would even like
to say, in principle quite the same as, the above described method to introduce transfinite
numbers. We can strictly say: the transfinite numbers stand or fall with the finite irrational
numbers; they are essentially alike in their basic features because these and those are
determined distinguished shapes and modifications (αϕωρισμενα) of the actual infinite.
Um diese Verwechslung von vornherein auszuschließen, bezeichne ich die kleinste transfinite
Zahl mit einem von dem gewöhnlichen, dem Sinne des Uneigentlich-unendlichen
entsprechenden Zeichen ¶ verschiedenen Zeichen, nämlich mit ω.
Allerdings kann ω gewissermaßen als die Grenze angesehen werden, welcher die
veränderliche endliche ganze Zahl ν zustrebt, doch nur in dem Sinne, daß ω die kleinste
transfinite Ordnungs-Zahl, d. h. die kleinste festbestimmte Zahl ist, welche größer ist als alle
endlichen Zahlen ν; ganz ebenso wie ◊2 die Grenze von gewissen veränderlichen, wachsenden
rationalen Zahlen ist, nur daß hier noch dies hinzukommt, daß die Differenz von ◊2 und diesen
Näherungsbrüchen beliebig klein wird, wogegen ω - ν immer gleich ω ist; dieser Unterschied
ändert aber nichts daran, daß ω als ebenso bestimmt und vollendet anzusehen ist, wie ◊2, und
ändert auch nichts daran, daß ω ebensowenig Spuren der ihm zustrebenden Zahlen ν an sich
hat, wie ◊2 irgend etwas von den rationalen Näherungsbrüchen.
Die transfiniten Zahlen sind in gewissem Sinne selbst neue Irrationalitäten und in der Tat ist
die in meinen Augen beste Methode, die endlichen Irrationalzahlen zu definieren, ganz ähnlich,
ja ich möchte sogar sagen im Prinzip dieselbe wie meine oben beschriebene Methode der
Einführung transfiniter Zahlen. Man kann unbedingt sagen: die transfiniten Zahlen stehen oder
fallen mit den endlichen Irrationalzahlen; sie gleichen einander ihrem innersten Wesen nach;
denn jene wie diese sind bestimmt abgegrenzte Gestaltungen oder Modifikationen
(αϕωρισμενα) des aktualen Unendlichen.
[E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen
Inhalts", Springer (1932) p. 395f]
In fact the decimal representations of irrational numbers require an actually infinite sequence of
digits - more than any finite sequence. But all finite sequences of the rational approximations
also form an infinite sequence leaving out none of the digits belonging to the infinite sequence.
So, a distinction by digits is not possible. Only the "not ending", a finitely defined feature, can
make the difference.
§ 478 Paul Lorenzen on Potential and Actual Infinity
The finite world-models of present natural science clearly show how the power of the idea of
actual infinity has come to an end in classical (modern) physics. In this light the inclusion of the
actual infinite into mathematics which explicitly started by the end of the last century with G.
Cantor appears disconcerting. In the intellectual overall picture of our century - in particular in
view of existentialist philosophy - the actual infinite appears as an anachronism. [...] We
introduce numbers for counting. This does not at all imply the infinity of numbers. For, in what
way should we ever arrive at infinitely-many countable things? [...] (1) Start with I. (2) If x has
been reached, add xI. These rules [...] supply a constructive definition of numbers (namely the
scheme for construction). Now we can immediately say that according to these rules infinitely
many numbers are possible: For every number x there remains to construct xI. We have to
observe that here only the possibility is claimed - and this is secured just by the rules
themselves. [...] To assert however that infinitely many numbers would really be, i.e., really
would have been constructed according to these rules - that would be false of course. [...] In
philosophical terminology we say that the infinite of the number sequence is only potential, i.e.,
existing only as a possibility. [...] In arithmetic - we may be allowed to summarize - there does
not exist a motive to introduce the actual infinite. The surprising appearance of actual-infinity in
modern mathematics therefore can only be understood by including geometry into consideration.
Die endlichen Weltmodelle der gegenwärtigen Naturwissenschaft zeigen deutlich, wie diese
Herrschaft eines Gedankens einer aktualen Unendlichkeit mit der klassischen (neuzeitlichen)
Physik zu Ende gegangen ist. Befremdlich wirkt dem gegenüber die Einbeziehung des Aktual-
Unendlichen in die Mathematik, die explizit erst gegen Ende des vorigen Jahrhunderts mit G.
Cantor begann. Im geistigen Gesamtbilde unseres Jahrhunderts - insbesondere bei
Berücksichtigung des existentialistischen Philosophierens - wirkt das Aktual-Unendliche
geradezu anachronistisch. [...] Wir führen die Zahlen zum Zählen ein. Hieraus folgt keineswegs
die Unendlichkeit der Zahlen. Denn auf welche Weise sollten wir jemals zu unendlich-vielen
zählbaren Dingen gelangen? [...] (1) Man fange mit I an. (2) Ist man zu x gelangt, so füge man
noch xI an. Diese Regeln [...] liefern eine konstruktive Definition der Zahlen (nämlich ihr
Konstruktionsschema). Jetzt können wir sofort sagen, daß nach diesen Regeln unendlich viele
Zahlen möglich sind: zu jeder Zahl x ist ja noch xI zu konstruieren. Man muß darauf achten, daß
hier nur die Möglichkeit behauptet wird - und diese ist gerade durch die Regel selbst gesichert.
[...] Dagegen zu behaupten, daß unendlich viele solche Zahlen wirklich seien, also wirklich nach
dieser Regel konstruiert seien - das wäre natürlich falsch. [...] In philosophischer Terminologie
sagt man, daß das Unendliche der Zahlenfolge nur potentiell, d. h. nur als Möglichkeit existiere.
[...] In der Arithmetik - so wird man zusammenfassend sagen können - liegt kein Motiv zur
Einführung von Aktual-Unendlichem vor. Das überraschende Auftreten von Aktual-Unendlichem
in der modernen Mathematik ist daher nur zu verstehen, wenn man die Geometrie mit in die
Betrachtung einbezieht.
[P. Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) 3-11]
http://books.google.de/books?id=K0duwwznAzQC&pg=PA195&hl=de&source=gbs_toc_r&cad=
4#v=onepage&q&f=false
§ 479 Herbert B. Enderton on Potential and Actual Infinity
There was no objection to a "potential infinity" in the form of an unending process, but an "actual
infinity" in the form of a completed infinite set was harder to accept. [H.B. Enderton: "Elements of
Set Theory", Academic Press, New York (1977) p. 14f]
http://www.amazon.de/Elements-Set-Theory-HerbertEnderton/dp/0122384407#reader_0122384407
§ 480 Edward Nelson on Potential and Actual Infinity
Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral by
prefixing it with S.
Now imagine this potential infinity to be completed. Imagine the inexhaustible process of
constructing numerals somehow to have been finished, and call the result the set of all numbers,
denoted by Ù.
Thus Ù is thought to be an actual infinity or a completed infinity. This is curious terminology,
since the etymology of "infinite" is "not finished".
[Edward Nelson: "Hilbert's Mistake" (2007) p. 3]
https://web.math.princeton.edu/~nelson/papers/hm.pdf
§ 481 Carl Friedrich Gauß on Potential and Actual Infinity
Concerning [a proof Schumacher's for the angular sum of 180° in triangles with two infinitely
long sides] I protest firstly against the use of an infinite magnitude as a completed one, which
never has been allowed in mathematics. The infinite is only a mode of speaking, when we in
principle talk about limits which are approached by certain ratios as closely as desired whereas
others are allowed to grow without reservation.
Was nun aber [einen Beweis Schumachers für die Winkelsumme von 180° in Dreiecken mit
zwei unendlich langen Seiten] betrifft, so protestiere ich zuvörderst gegen den Gebrauch einer
unendlichen Größe als einer Vollendeten, welcher in der Mathematik niemals erlaubt ist. Das
Unendliche ist nur eine Facon de parler, indem man eigentlich von Grenzen spricht, denen
gewisse Verhältnisse so nahe kommen als man will, während anderen ohne Einschränkung zu
wachsen verstattet ist. [Gauß an Schumacher, 12. 7. 1831]
http://gdz.sub.unigoettingen.de/dms/load/img/?PPN=PPN236010751&DMDID=DMDLOG_0068&LOGID=LOG_00
68&PHYSID=PHYS_0222
§ 482 Georg Cantor on Potential and Actual Infinity and Gauß
The erroneous in that piece by Gauss is that he says the completed infinity could not become a
subject of mathematical consideration; [...] I am to my great regret unable to refer, with respect
to the transfinite numbers and what is connected with them, to an important authority like Gauss,
I even find him in this respect among my opponents.
Das Irrthümliche in jener Gauss'schen Stelle besteht darin, dass er sagt, das
Vollendetunendliche könne nicht Gegenstand mathematischer Betrachtungen werden; [...] ich
bin zu meinem grossen Bedauern ausser Stande, mich in Beziehung auf die transfiniten Zahlen
und was mit diesen zusammenhängt auf eine so grosse Autorität wie Gauss berufen zu können,
finde ihn sogar in dieser Beziehung unter meinen Gegnern. [Cantor an Lipschitz, 19. Nov. 1883]
Quite two years ago Mr. Rudolf Lipschitz of Bonn has lead my attention to a certain part of the
correspondence between Gauß and Schumacher, where the former objects to every use of the
actual infinite in mathematics (letter of July 12, 1831); I have answered in great detail and have
rejected in this point the authority of Gauß, which I hold in high esteem in all other relations.
Es sind jetzt gerade zwei Jahre her, daß mich Herr Rudolf Lipschitz in Bonn auf eine gewisse
Stelle im Briefwechsel zwischen Gauß und Schumacher aufmerksam machte, wo ersterer gegen
jede Heranziehung des Aktual-Unendlichen in die Mathematik sich ausspricht (Brief v. 12. Juli
1831); ich habe ausführlich geantwortet und die Autorität von Gauß, welche ich in allen anderen
Beziehungen so hoch halte, in diesem Punkte abgelehnt. [E. Zermelo: "Georg Cantor,
Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p.
371]
If however, based upon a justified aversion against such illegitimate actual infinity (A.I.), in wide
domains of science under the influence of modern epicurean-materialistic mainstream a certain
horror infinity has been developed, which has found its classical expression and foundation in
the mentioned letter by Gauß, it appears to me that the herewith connected uncritical rejection of
the legitimate A.I. is not less a misdemeanour against the nature of things, which have to be
taken as they are. And we cannot but conceive this behaviour as somewhat short-sighted
because it robs us of the possibility to see the A.I. although, in its highest manifestation, it has
created us and maintains us, and in its secondary transfinite forms it surrounds us all-over and
even inhabits our mind.
Wenn aber aus einer berechtigten Abneigung gegen solche illegitime A. U. sich in breiten
Schichten der Wissenschaft, unter dem Einflusse der modernen epikureisch-materialistischen
Zeitrichtung, ein gewisser Horror Infiniti ausgebildet hat, der in dem erwähnten Schreiben von
Gauß seinen klassischen Ausdruck und Rückhalt gefunden, so scheint mir die damit
verbundene unkritische Ablehnung des legitimen A. U. kein geringeres Vergehen wider die
Natur der Dinge zu sein, die man zu nehmen hat, wie sie sind, und es läßt sich dieses Verhalten
auch als eine Art Kurzsichtigkeit auffassen, welche die Möglichkeit raubt, das A. U. zu sehen,
obwohl es in seinem höchsten, absoluten Träger uns geschaffen hat und erhält und in seinen
sekundären, transfiniten Formen uns allüberall umgibt und sogar unserm Geiste selbst
innewohnt. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer (1932) p. 374f]
[...] it seems that the ancients haven't had any clue of the transfinite, the possibility of which is
even strongly rejected by Aristotle and his school as in newer times by d'Alembert Lagrange,
Gauß, Cauchy and their adherents.
[...] daß die Alten keine Ahnung vom Transfiniten gehabt zu haben scheinen, deßen
Möglichkeit sogar von Aristoteles und seiner Schule heftig bestritten wird, wie auch in der
neueren Zeit von d'Alembert Lagrange, Gauß, Cauchy und deren Anhängern.[Cantor an Peano,
21. Sep. 1895]
My opposition to Gauss consists in the fact that Gauss rejects as inconsistent (I mean he does
so unconsciously, i.e., without knowing this notion) all multitudes with exception of the finite and
therefore categorically and basically discards the actual infinite which I call transfinitum, and
together with it he declares the transfinite numbers as impossible, the existence of which I have
founded.
Mein Gegensatz zu Gauss besteht hingegen darin, daß Gauss alle Vielheiten, mit Ausnahme
der endlichen, für inconsistent hält (ich meine unbewusst, d. h. ohne den Begriff zu haben) und
daher kategorisch und principiell dasjenige Acutalunendliche, welches ich Transfinitum nenne,
verwirft, mithin auch die transfiniten Zahlen, deren Existenz ich begründet habe, für unmöglich
erklärt [Cantor an Hilbert, 27. Jan. 1900]
These are Cantor's written opinions on what Gauß could have meant when talking about
something "which never has been allowed in mathematics". Of course the modern masters of
matheology know better and slightly different interpretations. They believe that Gauß would
welcome Cantor's theory and would give three cheers. From that we can derive three different
points of view.
1) Cantor, having been a contemporary of Gauß for 10 years, knew how people of his times
used to express themselves in case they wanted to announce an opinion. Then the masters of
matheology themselves must be rather stupid.
2) Modern masters of matheology show that Cantor, with regard to one of the most important
facts in his whole life, must have been rather stupid.
3) Modern masters of matheology are not stupid but intelligent enough to recognize that their
followers are stupid enough to be deceived by their masters without recognizing it.
§ 483 Ernst Zermelo on Potential and Actual Infinity
But in order to save the existence of "infinite" sets we need yet the following axiom, the contents
of which is essentially due to Mr. R. Dedekind. Axiom VII. The domain contains at least one set
Z which contains the null-set as an element and has the property that every element a of which
corresponds to another one of the form {a} or which with every of its elements a contains also
the corresponding set {a} as an element.
Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden,
seinem wesentlichen Inhalte von Herrn R. Dedekind herrührenden Axiomes. Axiom VII. Der
Bereich enthält mindestens eine Menge Z, welche die Nullmenge als Element enthält und so
beschaffen ist, daß jedem ihrer Elemente a ein weiteres Element der Fom {a} entspricht, oder
welche mit jedem ihrer Elemente a auch die entsprechende Menge {a} als Element enthält. [E.
Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre. I.", Math. Ann. 65 (1908)
261-281, p. 266f]
Axiom VII originally guarantees the existence of a potentially infinite set. That's how it has been
devised by Bolzano and Dedekind: I can think that I can think that I can think ... Never will I have
completed an infinity of thoughts. Only the necessary interpretation of infinite sets as completed
infinities in set theory forces set theorists to erroneously take the axiom in an actual sense. A
countable set S is in bijection with the set Ù when no element s œ S and no element n œ Ù are
remaining unpaired. In potential infinity always nearly all elements would be remaining - if they
existed. Alas that would mean actual infinity.
§ 484 Adolf Abraham Fraenkel and Azriel Levy on Potential and Actual Infinity
The statement limnض 1/n = 0 asserts nothing about infinity (as the ominous sign ¶ seems to
suggest) but is just an abbreviation for the sentence: 1/n can be made to approach zero as
closely as desired by sufficiently increasing the integer n. In contrast herewith the set of all
integers is infinite (infinitely comprehensive) in a sense which is "actual" (proper) and not
"potential". (It would, however, be a fundamental mistake to deem this set infinite because the
integers 1, 2, 3, ..., n, ... increase infinitely, or better, indefinitely.) [p. 6]
While the preceding explanations make it obvious that the attacks of various philosophers
upon the concept of (transfinite) cardinals are unsubstantiated, the attitude of the (neo-)
intuitionists that there do not exist altogether non-equivalent infinite sets is consistent, though
almost suicidal for mathematics. [p. 62]
[A. Fraenkel, A. Levy: "Abstract Set Theory", North-Holland, Amsterdam (1976)]
(Quoted from my handwritten notes.)
§ 485 Henri Poincaré on Potential and Actual Infinity
We'll have to state that the mathematicians in considering the notion of infinity tend toward two
different directions. For the one the infinite flows out of the finite, for them there exists infinity
only because there is an unlimited number of limited possible things. For the others the infinite
exists prior to the finite, the finite constituing a small sector of the infinite.
Wir werden zunächst feststellen, daß die Mathematiker in der Art, wie sie den
Unendlichkeitsbegriff auffassen, zwei entgegengesetzten Richtungen zuneigen. Für die einen
fließt das Unendliche aus dem Endlichen, für sie gibt es eine Unendlichkeit, weil es eine
unbegrenzte Zahl begrenzter möglicher Dinge gibt. Für die anderen besteht das Unendliche vor
dem Endlichen, indem das Endliche sich als ein kleiner Ausschnitt aus dem Unendlichen
darstellt. [H. Poincaré: "Letzte Gedanken: Die Mathematik und die Logik", übers. von K.
Lichtenecker, Akadademische Verlagsgesellschaft, Leipzig (1913) p. 145]
https://archive.org/stream/letztegedanken00lichgoog#page/n162/mode/2up
§ 486 Georg Cantor on Potential and Actual Infinity
{{It must be hard for English-speaking readers to understand Cantor's lengthy and interlocking
sentences. Instead of shortening them, I have inserted parentheses like in mathematics.}}
To the idea (to consider the infinite (not only in form of the unlimited growing and the closely
connected form of the convergent infinite series (introduced first in the seventeenth century) but
also to fix it by numbers in the definite form of the completed-infinite)) I have been forced
logically (nearly against my own will because in opposition to highly esteemed tradition) by the
development of many years of scientific efforts and attempts, and therefore I do not believe that
reasons could be raised which I would not be able to answer.
Zu dem Gedanken, das Unendlichgroße nicht bloß in der Form des unbegrenzt Wachsenden
und in der hiermit eng zusammenhängenden Form der im siebzehnten Jahrhundert zuerst
eingeführten konvergenten unendlichen Reihen zu betrachten, sondern es auch in der
bestimmten Form des Vollendet-unendlichen mathematisch durch Zahlen zu fixieren, bin ich fast
wider meinen Willen, weil im Gegensatz zu mir wertgewordenen Traditionen, durch den Verlauf
vieljähriger wissenschaftlicher Bemühungen und Versuche logisch gezwungen worden, und ich
glaube daher auch nicht, daß Gründe sich dagegen werden geltend machen lassen, denen ich
nicht zu begegnen wüßte. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 175]
§ 487 I never met a mathematician who to a higher degree than Hermite has been a realist in
the sense of Plato, and yet I can claim that I never met a more decided opponent of the
Cantorian ideas. This is the more a seeming contradiction, as he himself stated frankly: I am an
opponent of Cantor because I am a realist.
Niemals bin ich einem Mathematiker begegnet, der in höherem Maße ein Realist im Sinne
Platos war als Hermite und doch kann ich behaupten, daß ich keinem entschiedeneren Gegner
der Cantorschen Richtung begegnet bin. Es ist das ein scheinbarer Widerspruch, um so mehr,
als er selbst aus freien Stücken erklärt: Ich bin ein Gegner Cantors, weil ich ein Realist bin. [H.
Poincaré: "Letzte Gedanken: Die Mathematik und die Logik", übers. von K. Lichtenecker,
Akadademische Verlagsgesellschaft, Leipzig (1913) p. 162f]
https://archive.org/stream/letztegedanken00lichgoog#page/n180/mode/2up
§ 488 Couldn't just this seemingly so fruitful hypothesis of the infinite have straightly inserted
contradictions into mathematics and have fundamentally distroyed the basic nature of this
science which is so proud on its consistency?
Könnte nicht gerade diese scheinbar so fruchtbare Hypothese des Unendlichen geradezu
Widersprüche in die Mathematik hineingebracht und damit das eigentliche Wesen dieser auf ihre
Folgerichtigkeit so stolzen Wissenschaft von Grund auf zerstört haben? [E. Zermelo: "On the
hypothesis of the infinite" Warsaw notes W4, quoted in Heinz-Dieter Ebbinghaus: "Ernst
Zermelo, An Approach to His Life and Work", Springer (2007) p. 292]
§ 489 Russell would certainly reply that not psychology but logic and epistemology are
concerned, and then I would be tempted to answer that neither logic nor epistemology are
independent of psychology. And this declaration would certainly conclude the argument because
it would picture clearly an unbridgeable difference of opinion.
Russell würde mir sicher entgegenhalten, daß es sich nicht um Psychologie, sondern um
Logik und Erkenntnistheorie handelt und ich würde dann dazu geführt werden, zu antworten,
daß weder Logik noch Erkenntnistheorie von der Psychologie unabhängig sind, und dieses
Bekenntnis würde wohl die Auseinandersetzung beschließen, weil es eine unüberbrückbare
Verschiedenheit der Auffassung zutage fördern würde." [H. Poincaré: "Letzte Gedanken: Die
Logik des Unendlichen", übers. von K. Lichtenecker, Akadademische Verlagsgesellschaft,
Leipzig (1913) p. 142f]
https://archive.org/stream/letztegedanken00lichgoog#page/n160/mode/2up
§ 490 People have asked me, "How can you, a nominalist, do work in set theory and logic,
which are theories about things you do not believe in?" ... I believe there is value in fairy tales
and in the study of fairy tales. [A.B. Feferman, S. Feferman: "Alfred Tarski - Life and Logic",
Cambridge Univ. Press (2004), p. 52]
§ 491 [...] further every mathematical notion carries the necessary corrective in itself; if the
notion is unproductive and ineffective, then this will soon become obvious by its uselessness,
and it will be abolished because of lack of success. {{This prediction has proven itself wrong in
case of transfinite numbers.}}
[...] dann aber trägt auch jeder mathematische Begriff das nötige Korrektiv in sich selbst
einher; ist er unfruchtbar und unzweckmäßig, so zeigt er es sehr bald durch seine
Unbrauchbarkeit und er wird dann wegen mangelnden Erfolges fallen gelassen. [E. Zermelo:
"Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts",
Springer, Berlin (1932) p. 182]
§ 492 I must regard a theory which refers to an infinite totality as meaningless in the sense that
its terms and sentences cannot posses the direct interpretation in an actual structure that we
should expect them to have by analogy with concrete (e.g., empirical) situations. This is not to
say that such a theory is pointless or devoid of significance. {{Let alone to say the contrary.}} [A.
Robinson: "Formalism 64" in W.A.J. Luxemburg, S. Koerner (eds.): "A. Robinson: Selected
Papers", North Holland, Amsterdam (1979)]
§ 493 The limit process has won, since the limit is an unavoidable notion the importance of
which is not touched by the assumption or rejection of the infinitely small. But once it is accepted
we see clearly that it makes the infinitely small superfluous.
Der Grenzprozeß trug den Sieg davon; denn der Limes ist ein unvermeidlicher Begriff, dessen
Wichtigkeit von der Annahme oder Verwerfung des Unendlichkleinen nicht berührt wird. Hat man
ihn aber einmal gefaßt, so sieht man, daß er das Unendlichkleine überflüssig macht. [Hermann
Weyl: "Philosophie der Mathematik und der Naturwissenschaft", 7. Aufl., Oldenbourg, München
(2000) p. 64]
§ 494 [...] as such main issues I mention here the sharp separation of the finite from the infinite,
the notion of number of things, the proof that the proof-method known by the name of complete
induction (or the conclusion from n on n + 1) is really evidential, and that also the definition by
induction (or recursion) is definite and free of contradictions.
[...] als solche Hauptpunkte erwähne ich hier die scharfe Unterscheidung des Endlichen vom
Unendlichen, den begriff der Anzahl von Dingen, den Nachweis, daß die unter dem Namen der
vollständigen Induktion (oder des Schlusses von n auf n + 1) bekannte Beweisart wirklich
beweiskräftig, und daß auch die Definition durch Induktion (oder Rekursion) bestimmt und
widerspruchsfrei ist. [Richard Dedekind: "Was sind und was sollen die Zahlen?", Vieweg,
Braunschweig (1887) preface.]
§ 495 To the reviewer it seems unfortunate that classical set theory is developed in a separate
book so that all scruples - or almost all of them - are reserved for the second volume. This might
have the effect that most readers of this present volume will probably not become acquainted
with the criticisms at all. It is true that some hints to such scruples are given, but most students
might not think that they are important. On the other hand, it must be conceded that the lack of
knowledge of the results of foundational research will not mean much to mathematicians who
are not especially interested in the logical development of mathematics. [T. Skolem: "Review of:
A. A. Fraenkel: Abstract Set Theory. Amsterdam & Groningen, North-Holland Publishing
Company, 1953. XII + 479 pp." Mathematica Skandinavica 1 (1953) 313.]
http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=179577
§ 496 I cannot consider the set of positive integers as given, for the concept of the actual
infinite strikes me as insufficiently natural to consider it by itself. [Luzin to Kuratowski, reported
by N.Y. Vilenkin: "In search of infinity", Birkhäuser, Boston (1995) p. 126]
http://yakovenko.files.wordpress.com/2011/11/vilenkin1.pdf
§ 497 Why are the rationals countable? Because there exists a very simple method to construct
all of them. Take two integers. Construct all combinations and enumerate them. Done. (Infinitely
many occur infinitely often. No problem.)
Why were the real numbers believend to be uncountable? Because there exists no simple
method of construction. The reals were thought to be defined by infinite strings of digits. Cantor
had invented that method (again).
Why are the real numbers countable? Because they are not defineable by infinite strings of
digits but only by finite strings of letters like ◊7. These can be expressed in a very simple way,
namely by bits 0 and 1. Construct all finite strings of bits. Enumerate them. Done.
For every string that does not yet point to a real number like ◊7, we can find a real number
which this string points to like ◊8. (infinitely many occur infinitely often. No problem.)
Why does the diagonal argument fail? The list of finite definitions has no diagonal. The list of
defined real numbers has an infinite diagonal. But infinite diagonals like infinite strings of digits
do not define anything. Every real number defines an infinite string. No infinite string defines a
real number (because there is no last digit, and every other digits is not sufficient). So only a
finitely defined list can yield a diagonal. But all finite definitions are in the list. Contradiction.
§ 498 From my French friends I heard that the tendency towards super-abstract generalizations
is their traditional national trait. I do not entirely disagree that this might be a question of a
hereditary disease [...] [V.I. Arnold: "On teaching mathematics" (1997), Transl. A.V. Goryunov]
http://pauli.uni-muenster.de/~munsteg/arnold.html
§ 499 What is a real number?
A real number is an algorithm that supplies an infinite sequence of digits (or bits etc.). This
satisfies the axiom of trichotomy with respect to every rational number.
A general pointer to a real number is a finite expression which has been related to the real
number in at least one physical system of the universe, usually a dictionary or a text book.
A special pointer to a real number contains the space-time coordinates or other identification
properties of actual constructions of infinite sequences of digits (or bits etc.).
All algorithms and all pointers belong as elements to the countable set of all finite expressions.
Therefore any uncountability of the real numbers can be excluded.
§ 500 More than any other question the infinite always has moved so deeply the human soul.
The infinite rather like no other idea has effected so inspiringly and fruitfully on the human mind.
But more than any other notion the infinite is in need of elucidation.
Das Unendliche hat wie keine andere Frage von jeher so tief das Gemüt der Menschen
bewegt; das Unendliche hat wie kaum eine andere Idee auf den Verstand so anregend und
fruchtbar gewirkt; das Unendliche ist aber auch wie kein anderer Begriff so der Aufklärung
bedürftig. [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 163]
§ 501 No punishment, within legal boundaries, would be too severe for you for your
wrongdoings. [anonymous, "What is a real number", sci.math, 9 May 2014]
https://groups.google.com/forum/#!original/sci.math/-GSsWLUKmyo/sldkzJaw9ekJ
Unfortunately I don't know this gentleman. But I will be proud to quote his sentence whenever
teaching my findings with respect to actual transfinity.
§ 502 Modern axiomatic systems unfortunately entail various consequences which are grossly
deviating from reality. Therefore a mathematics connected to reality should be based upon those
foundations from which it has originally emerged, namely from counting units and drawing lines.
Since mathematics owes its creation to abstraction from observations of reality, a statement like
"I + I = II" need not be derived by a proof extending over many pages. This statement is much
better a natural foundation of arithmetic than any axiom devised for this purpose. It can be
proven more compelling by means of an abacus than by any chain of logical conclusions, and be
they even most well-grounded.
Da moderne Axiomensysteme leider zu mancherlei wirklichkeitsfernen Konsequenzen führen,
sollte eine wirklichkeitsnahe Mathematik aus den Grundlagen entwickelt werden, aus denen sie
tatsächlich entstanden ist, nämlich aus dem Zählen von Einheiten und dem Zeichnen von Linien.
Denn die Mathematik verdankt ihre Entstehung der Abstraktion aus Beobachtungen der
Wirklichkeit. Eine Aussage wie "I + I = II" muss nicht aus einem über viele Seiten sich
hinziehenden Beweise hergeleitet werden. Diese Aussage selbst ist eine viel natürlichere
Grundlage der Arithmetik als irgendein dazu erdachtes Axiom. Sie kann mit einem Abakus
zwingender bewiesen werden als durch jede noch so tiefgründige Kette von logischen
Schlüssen. [W. Mückenheim: "Mathematik für die ersten Semester", 3. Aufl, Oldenbourg,
München (2011) Vorwort]
§ 503 [...] gradually and unwittingly mathematicians began to introduce concepts that had little
or no direct physical meaning. [...] after about 1850, the view that mathematics can introduce
and deal with [...] concepts and theories that do not have immediate physical interpretation [...]
gained acceptance. [M. Kline: "Mathematical Thought from Ancient to Modern Times", Oxford
University Press (1972) p. 129ff]
§ 504 Cantor's Theory, if taken seriously, would lead us to believe that while the collection of all
objects in the world of computation is a countable set, and while the collection of all identifiable
abstractions derived from the world of computation is a countable set, there nevertheless "exist"
uncountable sets, implying (again, according to Cantor's logic) the "existence" of a super-infinite
fantasy world having no connection to the underlying reality of mathematics. [David Petry,
"Objections to Cantor's Theory", sci.math, sci.logic, 20 Juli 2005]
http://groups.google.com/group/sci.logic/msg/02ee220b035488f9?dmode=source
{{By the way, Cantor never accepted undefinable numbers, and in fact his diagonal argument
does not use or provide them. Only matheologians trying at any cost to save transfinity have
devised this unmathematical notion.}}
§ 505 As far as I am concerned, I would propose to adhere to the following rules:
1. Never consider other objects than those which can be defined by a finite number of words.
2. Never forget that every proposition about the infinite is only a substitution, an abbreviated
expression of a proposition about the finite.
3. Avoid classifications and definitions which are not well-defined.
Was mich anlangt, so würde ich vorschlagen, an den folgenden Regeln festzuhalten:
1. Niemals andere Objekte der Betrachtung zu unterziehen, als solche, die sich durch eine
endliche Zahl von Worten definieren lassen.
2. Niemals aus den Augen zu verlieren, daß jede Aussage über das Unendliche nur eine
Übertragung, ein gekürzter Ausdruck für eine Aussage über das Endliche ist.
3. Klassifikationen und Definitionen, die nicht wohlbestimmt sind, zu vermeiden.
[H. Poincaré: "Letzte Gedanken: Die Logik des Unendlichen", übers. von K. Lichtenecker,
Akadademische Verlagsgesellschaft, Leipzig (1913) p. 141f]
https://archive.org/stream/letztegedanken00lichgoog#page/n158/mode/2up
§ 506 Sequences generated by algorithms can be specified by those algorithms, but what
possibly could it mean to discuss a "sequence" which is not generated by such a finite rule?
Such an object would contain an "infinite amount" of information, and there are no concrete
examples of such things in the known universe. This is metaphysics masquerading as
mathematics. [N.J. Wildberger: "Set Theory: Should You Believe?" (2005)]
http://web.maths.unsw.edu.au/~norman/views2.htm
§ 507 I understand free variables, for me they are places for substituting individuals. But I do
not understand quantifiers since they refer often to actually infinite universes of abstract objects
and I do not believe in the existence of such universes. [J. Slupecki, private communication
reported in Jan Mycielski: "On the tension between Tarski’s nominalism and his model theory
(definitions for a mathematical model of knowledge)", Annals of Pure and Applied Logic 126
(2004) 215-224]
§ 508 David Hilbert in 1904 [...] wrote that sets are thought-objects which can be imagined prior
to their elements. [At request of he referee {{praised be the referee!}} who asked what is a
thought-object let me add: I understand it to be a thought about an object which may exist or not.
Thus it is an electrochemical event in the brain or/and its record in the memory. In particular it is
a physical thing in space time {{how many can exist in an infinite eternal universe?}}. Of course it
is difficult to characterise any physical phenomena. But we have the ability to recognize thoughts
as identical or different, just as we have the ability to recognize a silent lightning from a
thunderous one. Hence I understand Hilbert's words as follows: mathematicians {{the modern
word is matheologians (because of the close connection with theologians in the belief of the nonexisting or, at least, the not detectable}} imagine sets which do not exist, but their thoughts about
sets do exist and they can arise prior to the thoughts of most elements in those sets. Moreover,
in 1923 he described to some extent the algorithm creating those thoughts {{and he concluded:
"Finally we will remember our original topic and draw the conclusion. On balance the complete
result of all our investigations about the infinite is this: The infinite is nowhere realized; it is
neither present in nature nor admissible as the foundation of our rational thinking. This is a
remarkable harmony between being and thinking."}}] [J. Mycielski: "Russel'sparadox and
Hilbert's (much forgotten) view of set theory" in "One hundred years of Russell's paradox:
mathematics, logic, philosophy" G. Link (ed.), de Gruyter (2004) p. 534]
http://books.google.de/books?id=Xg6QpedPpcsC&pg=PA533&lpg=PA533&dq=Russell%27s+P
aradox+and+++++Hilbert%27s+(much+forgotten)+View+of+Set+Theory&redir_esc=y#v=onepag
e&q=Russell's%20Paradox%20and%20%20%20%20%20Hilbert's%20(much%20forgotten)%20
View%20of%20Set%20Theory&f=false
I am very indebted to Fred Jeffries for pointing to this source.
§ 509 I am convinced that this theory on day will belong to the common property of objective
science and will be confirmed in particular by that theology which is based upon the holy bible,
tradition and the natural disposition of the human race - these three necessarily being in
harmony with each other.
Darum bin i. auch fest überzeugt, daß diese Lehre dereinst Gemeingut der objectiv-gerichteten
Wissenschaft werden u. im Besonderen von derjenigen Theologie bestätiget werden wird,
welche auf d. heil. Schrift, Tradition und auf d. natürliche Beanlagung d. menschl. Geschlechts
sich gründet, welche drei in nothwendiger Harmonie zu einand. stehen. [Georg Cantor an P.
Ignatius Jeiler, 20. Mai 1888]
§ 510 When in the age of the scientific revolution Aristotelian metaphysics became the target of
modernizers like Galileo, logic was considered part and parcel of metaphysics and was
dismissed together with the philosophy Galileo fought against. For him, formalization of logic
was obsolete; what was needed from logic he considered as natural and no real subject of
study, certainly no precondition for founding the new science of physics. {{Has this situation
changed much? It rather has become even worse. Modern logicians "prove" that things can be
done which provably cannot be done. The nimbus of mathematical precision and proof has been
undermined and besmirched.}} [G. Link: "Introduction" to "One hundred years of Russell's
paradox: mathematics, logic, philosophy" G. Link (ed.), de Gruyter (2004) p. 4]
§ 511 I think I have proved above, and it will become plain enough in the course of this paper,
that just as well determined countings as with finite sets can be perfomed with infinite sets,
provided that the sets are given a determined law according to which they become well-ordered
sets.
Ich glaube aber oben bewiesen zu haben und es wird sich dies im folgenden dieser Arbeit
noch deutlicher zeigen, daß ebenso bestimmte Zählungen wie an endlichen auch an
unendlichen Mengen vorgenommen werden können, vorausgesetzt, daß man den Mengen ein
bestimmtes Gesetz gibt, wonach sie zu wohlgeordneten Mengen werden. [Ernst Zermelo
(Hrsg.): "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen
Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor Dedekind. Nebst einem Lebenslauf Cantors von Adolf Fraenkel." Georg Olms
Verlagsbuchhandlung, Hildesheim (1966) p. 174]
§ 512 Concerning the second tranfinite cardinal number ¡1, I am not completely convinced of
its existence. We reach it by considering the collection of ordinal numbers of cardinality ¡0; it is
clear that this collection must have a higher cardinality. But the question is whether it is closed,
that is whether we may talk about its cardinality without contradiction. In any case an actually
infinite can be excluded.
Was nun die zweite transfinite Kardinalzahl ¡1 betrifft, so bin ich nicht ganz überzeugt, daß sie
existiert. Man gelangt zu ihr durch Betrachtung der Gesamtheit der Ordnungszahlen von der
Mächtigkeit ¡0; es ist klar, daß diese Gesamtheit von höherer Mächtigkeit sein muß. Es fragt
sich aber, ob sie abgeschlossen ist, ob wir also von ihrer Mächtigkeit ohne Widerspruch
sprechen dürfen. Ein aktual Unendliches gibt es jedenfalls nicht. [Henri Poincaré: "Über
transfinite Zahlen", Sechs Vorträge über ausgewählte Gegenstände aus der reinen Mathematik
und Mathematischen Physik, auf Einladung der Wolfskehl-Kommission der Königlichen
Gesellschaft der Wissenschaften gehalten zu Göttingen vom 22. - 28. April 1909, Teubner,
Leipzig (1910) p. 48]
§ 513 What is your opinion?
A) Do you accept uncountable alphabets, i.e., lists defining the shape and pronounciation of
letters and determining their place in the order of all letters?
B) Do you accept undefinable real numbers, i.e., real numbers without an algorithm to
determine the infinite decimal representation?
C) Do you accept (A) and (B)?
D) Do you accept neither (A) nor (B)?
§ 514 A survey
In analysis we have sequences (sn) of real numbers sn with (improper) limit limnض sn = ¶.
Every integer part [ sn ] of a positive real number can be expressed by a sum of units like
[ 50/7 ] = 1 + 1 + 1 + 1 + 1 + 1 + 1
Question A) Do you think that it is possible to have
[ limnض sn ] = 0
whereas
limnض [ sn ] = ¶ = 1 + 1 + 1 + ... ?
In set theory we have sequences (sn) of sets sn with cardinality | sn |. The cardinality is a
measure for the number of elements. Contrary to analysis, infinity, called ¡0, is considered to be
the number of really existing elements in an infinite set like Ù. So every element of sn contributes
one unit to | sn |.
Question B) Do you think that it is possible to have
| limnض sn | = 0
whereas
limnض | sn | = ¡0 ?
§ 515 Das Paradoxon des Tristram Shandy vereinfacht und verständlich erklärt /
The paradox of Tristram Shandy simplified and made intelligible
Donald Duck wird nie so reich wie sein Onkel Dagobert. Wenn er etwas Geld erhält, dann gibt er
es sofort wieder aus, bis auf einen Notdollar, den er immer behält,
Er startet mit zwei Dollars, gibt einen aus, erhält einen anderen, gibt einen aus, usw. Seine
Dollarnoten nummeriert er mit Filzstift, damit er immer den ältesten ausgibt. Die folgende
Tabelle zeigt die schrittweise Entwicklung der Nummern:
1, 2
2
2, 3
3
3, 4
4
...
Da er als Comic-Figur unsterblich ist, kann die Mengenlehre zur Berechnung des Grenzwertes
seines Kapitals herangezogen werden:
Der mengentheoretische Grenzwert zeigt an, welche Zahlen Donald ewig besitzt. Das ist die
leere Menge.
Die mengentheoretische Interpretation dieses Grenzwertes sagt aus, dass Donald alle
natürlichen Zahlen ausgibt, die Menge der nicht ausgegeben also leer ist. (Diese Interpretation
ist z. B. für die Nummerierung aller rationalen Zahlen und die Nummerierung aller Zeilen einer
Cantor-Liste erforderlich. Die Menge der nicht nummerierten Zahlen bzw. Zeilen muss leer sein.)
Der tatsächliche Sachverhalt zeigt aber, dass Donald stets eine Zahl behält, die Menge der nicht
ausgegeben also nicht und niemals leer ist. Auch der Grenzwert kann nicht kleiner als 1 $ sein.
Die Lösung dieses offensichtlichen Widerspruchs ist folgende: Die Mengenlehre beweist für jede
Zahl n, dass die Menge der bis zur Zahl n nicht ausgegeben Zahlen leer ist. Ebenso gilt aber
auch für jede Zahl n, dass sie zu einem endlichen Anfangsabschnitt gehört und dass fast alle
natürlichen Zahlen auf sie folgen. Damit ist der Widerspruch aufgelöst. Donald gibt jede Zahl
zurück und behält doch immer eine, denn nach jeder folgen noch unendlich viele.
Donald Duck will never become as rich as his uncle Scrooge McDuck. If he gets some money,
he soon spends it, except one dollar as an emergency ration.
He starts with two dollars, spends one, gets another one, spends one, and so on. He
enumerates his dollar notes with felt pen in order to spend always the oldest dollar. The
following sequence shows the stepwise development of the numbers:
1, 2
2
2, 3
3
3, 4
4
...
Since he, as a cartoon character, lives forever, set theory can be used to calculate the limit of his
wealth.
The set theoretical limit shows which numbers Donald will possess forever. It is the empty set.
The set-theoretical interpretation of this limit says that Donald will spend all natural numbers. So
the set of numbers which he never spends is empty. (This interpretation is required for
enumerating all rational numbers or all lines of a Cantor-list. The set of not enumerated rationals
or lines must be empty.)
Fact is however, that Donald always keeps a number. The set of not spent numbers is not
empty. Even the limit cannot be less than 1 $.
The solution of this obvious contradiction is this: Set theory proves for every natural number n
that the set of not spent numbers up to n is empty. But it is also true that every number n
belongs to a finite initial segment upon which infinitely many numbers will follow. This resolves
the contradiction: Donald returns every number and nevertheless always keeps one, because
every number is followed by infinitely many.
§ 516 On Ducks and Bathtubs
Ben Bacarisse discusses my well known example
Scrooge McDuck every day gets enumerated $2 and returns enumerated $1. If he
happens to return the right numbers, he will get unmeasurably rich. If he happens to return
the wrong numbers, he will go bancrupt.
and adds: But Prof. Mueckenheim explicitly tells us that we should not use the limit of the
cardinalities as the measure of long-term wealth.
http://bsb.me.uk/dd-wealth.pdf
This is not quite true. Of course we can use instead of the cardinality 0 of the empty limit set the
limit ¡0 of the cardinalities to measure wealth. But we cannot talk about wealth unless McDuck
owns enumerated dollar-notes.
If McDuck always returns the smallest nunmber the set theoretic limit is the empty set. This
limit only says that of every finite initial segment {1, ..., n} no element forever remains in
McDucks hands. But we cannot assume that a property owned by every finite segment {1, ..., n}
is automatically inherited by the complete set of natural numbers. If this were true, then McDuck
would be rich in the limit without having any dollar.
I agree with Ben's conclusion that the wealth of McDuck is measured by the limit ¡0 of the
cardinalities, but I do not agree that a person having ¡0 dollars has no dollar. Neither a bathtub
can be consideres full without any water molecule being inside.
This story illustrates in a very lucid way that there is a difference between a statement "for
every number of any finite initial segment" and "for all numbers of Ù". The limit of the sequence
of all finite initial segments differs considerably from a complete infinite set Ù.
If you agree that wealth without dollars is impossible, then you agree too that the enumeration
of all rational numbers never covers all rational numbers. Neither all natural numbers can be
used to enumerate a set.
If you claim that wealth without dollars and wetness without water is possible, then you push
yourself out of any reasonable position - and certainly cannot expect to successfully pass a
university of applied sciences.
But I do not believe that anybody would claim that.
§ 517 Die Bedeutung des Mengenlimes / The meaning of set limits
Die Folge (sn) der Mengen sn = {n+1, n+2, ..., 2n}
{2}
{3, 4}
{4, 5, 6}
...
besitzt den Grenzwert limnض sn = { } (vgl. § 090).
Der Grenzwert limnض |sn| der Kardinalzahlenfolge ist unendlich.
Dies bedeutet keinen Widerspruch, denn der Mengenlimes enthält lediglich Elemente, die
niemals aus den Mengen der Folge verschwinden. Solche Elemente gibt es nicht. Die
Kardinalzahlen dagegen geben die Anzahl der Elemente in den Mengen der Folge an. Der
Grenzwert zeigt, dass mehr Elemente hinzugefügt werden als verschwinden und insbesondere
niemals eine leere Menge vorkommt.
Sollten die Grenzwerte jedoch "im Limes" oder "für ω" realisiert werden, dann ergäbe sich ein
Widerspruch, wie sich schon an der Notation zeigt, denn sω = { } and |sω| = ¶ erlauben, nicht
zwischen limnض |sn| und |limnض sn| zu unterscheiden. Wie diese Realisierung der
Grenzmenge zustande gekommen ist, wäre nicht aus ihr selbst erschließbar.
Deshalb können wir feststellen, dass die Vollendung des Unendlichen nicht gelingen kann.
Allerdings ist sie für die Mengenlehre unerlässlich. Zum Beispiel beruht Cantors "Beweis" der
Existenz transzendenter Zahlen auf einer vollständigen Liste aller algebraischen Zahlen, ebenso
wie der "Beweis" überabzählbarer Mengen oder die Ordinalzahl ω + 1 die Vollständigkeit
"einfach unendlicher", abzählbarer Mengen voraussetzt.
Consider the sequence (sn) of sets sn = {n+1, n+2, ..., 2n}
{2}
{3, 4}
{4, 5, 6}
...
The set limit limnض sn is empty (cp. § 090).
The limit of cardinalities limnض |sn| is infinite.
This is not a contradiction. The set limit indicates those elements which remain in the
sequence forever. There is none. The cardinalities indicate the number of elements in the sets.
The limit shows that by an infinite supply forever elements are inserted into the sets such that
they never run out of elements.
If however these limits should become realized by a set "in the limit" or "at ω", then sω = { } and
|sω| = ¶ would show a contradiction as can be seen by the notation already, because for
realized limits there would be no distinction between limnض |sn| and |limnض sn|.
Conclusion: Infinity is never finished. But set theory needs this completion, for instance in the
"proof" of existence of transcendental numbers by diagonalization of a complete list of all
algebraic numbers or for "proving" the existence of sets that are larger than "simply infinite",
countable sets or by considering ω + 1.
§ 518 What did Fraenkel wish to express with his story of Tristram Shandy?
The appearance that a set, so to speak, can "contain equally many elements" as a proper
subset is in a certain contrast {{that is, so to speak, an understatement}} with the well-known
theorem: The whole is always larger than a part of it. This apparent contrast, already clearly
recognized by Galilei, has historically been an essential obstacle to the admission of the notion
of actual infinity, because it seemed to discredit the infinite sets possessing such a paradoxical
property. In reality, however, this theorem of the whole and its part had been proven only in the
domain of the finite, and there was no reason to expect, that it would maintain its validity in the
giant step that leads from the finite to the infinite {{let alone any reason to accept the contrary}}.
Footnote: Even more paradoxical appears the equivalence between two infinite sets of
apparently very different perimeter, if it is seemingly transferred into the practical life. The
uncomfortable feeling occuring in this case disappears if one realizes that this reality is only
ostensible and that our perception is not adjusted to it. Well-known is so the story of Tristram
Shandy ... {{cp. §. The uncomfortable feeling does not at all disappear when we realize that the
natural numbers have the same well-order as the days or years of Tristram Shandy and that
when enumerating the rational numbers always one settled task implies an infinity of further
tasks. Always infinitely many natural and rational numbers remain unpaired and there is not the
least proof of equinumerousity.}}
[Adolf A. Fraenkel: "Einleitung in die Mengenlehre" 3. Aufl., Springer, Berlin (1928) p. 24]
§ 519 Real analysis has its merits in all domains of physics like classical and quantum
mechanics, thermodynamics, electrodynamics, special and general theory of relativity, atomic
and nuclear physics, astronomy and cosmology. It is further applied in chemistry, biology,
medicine, engineering sciences and even many branches of economic. There are only two
realms where its application is not beneficial, namely the different branches of theology and
transfinite set theory.
§ 520 Set theory as a perpendicular expression of a horizontal desire
The matrix
0.1
0.11
0.111
...
fails to include its limit 1/9 = 0.111... like every strictly monotonically increasing sequence.
According to set theory, this matrix contains all ¡0 horizontal rows with at least one 1. But
there are not ¡0 such vertical rows. That means set theory gives different answers to this
completely symmetrical question. It is inconsistent. Or does it depend on the direction of
gravitation?
In order to be consistent, there should be as many horizontal as vertical rows. In fact, this is
true, since both are potentially infinite ¶. Set theory would only be consistent with ¡0 in both
cases. Alas, this is only a wish. So set theory is also "a perpendicular expression of a horizontal
desire" (G. B. Shaw).
§ 521 Remarkable sequences of sets and their different limits
(an) with an = {n} has limnض {an} = { }, |limnض {an}| = 1, {limnض n} = ¶
(bn) with bn = {-1/n, 1/n} has limnض {bn} = { }, |limnض {bn}| = 2, {limnض (-1/n), limnض (1/n)} = 0
(cn) with cn = {nn} has limnض {cn} = { }, |limnض {cn}| = 1, {limnض (nn)} = ¶
(dn) with dn = {n/n} has limnض {dn} = {1}, |limnض {dn}| = 1, {limnض (n/n)} = 1
(en) with en = {(1+n)/n} has limnض {en} = { }, |limnض {dn}| = 1, {limnض ((1+n)/n)} = 1
The sequences are constructed by always removing the terms with n and introducing the terms
with n+1. In the first three sequences no term stays forever. Only this is expressed by the empty
"limit". Applying actual infinity, however, we could "get ready". Then all natural numbers could
get "exhausted". The empty sets then are the sets "at ω" Then the limits of the sequences of
cardinalities would also apply to the set "at ω", producing a contradiction. Then also numerator
and denominator of (dn) would get "exhausted" like that of sequences (cn) and (en), leaving the
empty limit set "at ω" instead of {1}. And what "finishes" the sequences {1n} or {n0} or {0n} "at ω"?
§ 522 "Every" is not "all"
It is possible for every n in Ù to enumerate the first n rational numbers q1, q2, q3, ..., qn. Set
theorists claim that this proves the possibility of enumerating all rational numbers.
It is possible for every n in Ù to order the first n rational numbers q1, q2, q3, ..., qn by size. Set
theorists do not claim that this proves the possibility of ordering all rational numbers by size.
What is the difference?
§ 523 Can the manner of marking influence the result?
Let (sn) be the sequence of sets sn = {n} with n œ Ù. This sequence has an empty limit set.
Let (tn) be the sequence of sets tn = {I1, I2, I3, ..., In} where we have indexed strokes in order to
distinguish them. sn+1 comes out of sn by adding stroke number n+1. (A unary system is the
historically first manner of marking natural numbers.) This sequence has not an empty limit set.
The number of strokes diverges towards ω, the sets of indices diverge towards |Ù.
§ 524 Are finite cardinal numbers natural numbers?
Cantor has shown how the natural numbers can be defined as cardinal numbers of sets.
First Zermelo and later v. Neumann have shown how the natural numbers can be defined as
ordinal numbers of sets.
Zero, the most unnatural number though, has been raped and mutilated to become a "natural"
number, only in order to justify the unsound idea that a finite initial segment of the ordered set Ù
has a cardinal number surpassing all its elements and to deceive mathematicians with the lie
that this is a natural state and therefore cannot be different in |{1, 2, 3, ... }| = ¡0.
The Löwenheim-Skolem argument has been perverted by defining what "the system thinks".
The countability of all really real numbers has been dampened by imaginating undefinable
"real numbers".
The impossibility of well-ordering uncountable sets has been overridden by "proving" that the
impossible is possible.
But all these desperate attempts to keep set theory free of contrsdictions have been without
success. Some set theorists have recognized that contradictions nevertheless are unavoidable
and now are claiming that finite cardinal numbers are not natural numbers.
So mathematics is completely decoupled and isolated from its asserted "basis".
That's the best inconsistency proof of set theory, isn't it?
§ 525 Let (sn) be a sequence of sets sn of rational numbers q such that for n = 1, 2, 3, ...
sn+1 = (sn » {q | n < q § n+1}) \ {qn+1}
with s1 = {q | 0 < q § 1} \ {q1}
and q1 = 1/1, q2 = 1/2, q3 = 2/1, q4 = 3/1... the positive rational numbers indexed by Cauchydiagonalization of the matrix of positive rational numbers.
The set sn contains the rationals of the interval (0, n] which have not got an index k § n.
When investigating this case for all natural numbers, we get two limits, one for the sequence of
sets and one for the sequence of cardinal numbers:
limnض sn = { } is indicating that no rational remains without index.
limnض |sn | = ¶ is indicating that the set of rational numbers without natural index has infinitely
many elements, not only for every sn but also in the limit.
My questions: Why is the first limit considered more reliable than the second one? Has the
second limit a mathematical meaning? If so what is it?
My answers: limnض sn is meaningless since it is impossible to exhaust an infinite set. There is
an infinite supply; this is indicated by limnض |sn | = ¶.
§ 526 Smallest possible super task
The limit of the sequence (sn ) with sn = {n} is the empty set. This means, among others, that
there is no natural number n that remains in all terms of the sequence. The ordered character of
the natural numbers allows us to understand this sequence as a super task, transferring the set
Ù from a reservoir A containing Ù via an intermediate reservoir B to the final reservoir C. Every
state of B can be represented by a term of the sequence and vice versa.
However, if we introduce the condition that a number n may leave B not before the number n+1
has been inserted into B then we have the same limit, i.e., the whole set Ù will reach C, although
this can be excluded by the definition that B always contains at least one element of Ù.
This contradiction shows that the set limit is not a reasonable notion. It indicates that all natural
numbers with no exception are in C while this is clearly false.
Note that the additional condition is not an artificial hurdle because the condition that every
natural number n is followed by a natural number n+1 is basic to all natural numbers.
§ 527 Limits
A proper limit is a state that is approached better and better by the terms of a sequence.
Is 2limnض n = limnض 2n ?
Neither sequence has a proper limit. These limits are improper and their meaning is only that the
sequences increase beyond any given real number. Neither exists as a real number. In calculus
we cannot decide what ¶/¶ is. But often the unbounded increase on both sides is accepted as
being equal as the improper limit oo. Many write 2ÿ¶ = ¶, for instance. Above equality in this
sense is obvious, when we refrain from using exponential notation. Then both sides simply read
2ÿ2ÿ2ÿ... = 2ÿ2ÿ2ÿ... so that there cannot be a difference.
Now let sn = {n, n+1, n+2, ...}. Why is 0 = |limnض sn| ∫ limnض |sn| = ¶ ?
Also in this case we have improper limits only, showing a never ending process.
limnض sn = { } expresses the fact that n will not be in sets following upon sn.
limnض |sn| = ¶ expresses the fact that infinitely many naturals follow upon every n.
It is very simple. No contradiction. No exhaustion. And therefore no proof of complete bijection or
countability of infinite sets. But many will refuse to understand this because it is so easy to
confuse infinite sets with finite sets and to think that infinite sets could be finished and
enumerated too.
§ 528 The sequence of singletons {n/(n+a)} has limit { } for a ∫ 0 but limit {1} for a = 0. This is
strange. If we refuse to cancel down, then, similar to the case a ∫ 0, the limit should be empty
too, because then the natural numbers should also be exhausted.
§ 529 Contradiction
Set theory is based upon the assumption that every positive rational number q gets a natural
index n in a finite step of this sequence:
sn+1 = (sn » {q | n < q § n+1}) \ {qn+1}
with s1 = {q | 0 < q § 1} \ {q1}
and q1 = 1/1, q2 = 1/2, q3 = 2/1, q4 = 3/1...
sn is the set of positive rational numbers less than n which have not got an index less than n.
The cardinal numbers |sn| = ¶ show that the set of not enumerated rationals is never empty in a
finite step n œ Ù. Other steps are not available for indexing purposes. Contradiction, if "every" is
interpreted as "all".
§ 530 Why Hessenberg's proof fails in infinite infinity *)
Hessenberg derives the uncountability of the powerset of Ù from the limit-set H of all natural
numbers which are not in their image-sets. H cannot be enumerated by a natural number n. If H
is enumerated by n, and if n is not in H, then n belongs to H and must be in H, but then n does
not belong to H and so on. [Gerhard Hessenberg; "Grundbegriffe der Mengenlehre",
Sonderdruck aus den "Abhandlungen der Fries'schen Schule", I. Band, 4. Heft, Vandenhoeck &
Ruprecht, Göttingen (1906) § 24 ]
If "all" is replaced by "every" and if we keep in mind that every natural number is succeeded by
infinitely many natural numbers (and preceded by only finitely many), we get the following
sequential explanation of the "paradox":
Every set Hk = {n1, n2, ..., nk} containing all natural numbers up to nk, which are not mapped on
image-sets containing them, can be mapped by any number m not yet used in the (always
incomplete) mapping. This number m is not in Hk and therefore has to be included as m = nk+1
into the set Hk. Doing so we get the set Hk+1 = Hk » {m}. There remain infinitely many further
natural numbers available to be mapped on Hk+1. Choose one of them, say m'. Of course, m' is
not in Hk+1 and therefore has to be included as m' = nk+2 into Hk+1, such that Hk+2 = Hk+1 » {m'}.
This goes on and on without an end. The mapping is infinite. As long as there is no limit-set H,
there cannot be a contradiction obtained from not finding a natural number to be mapped on H.
___
(*) This headline sounds rather strange, but it is required to distinguish infinities since Cantor
and his disciples have invented finished infinity.
§ 531 What does § 529 show us?
1) The sets sn of the sequence (sn) tell us that always (for all n œ Ù) infinitely many positive
rationals § n remain without index § n. So much is irrefutable. The proof holds for all natural
numbers. Nothing can index further rationals. But the sets sn are never empty.
2) If all natural numbers n could be used and all sets sn could be constructed, then the finally
remaining rationals without index could be indentified. But that is not possible. This is also
irrefutable.
Both points taken together show that not all natural numbers n can be used and, therefore, not
all sn can be constructed. Therefore it is not a logical problem that always something remains. It
is simply the exclusive property of infinity, namely to be never finished.
What is the advantage of this idea over set theory with its finished infinity, besides that it is the
truth? It gets along without finishing the infinite, without exhausting Ù such that an empty set
limnض Ù\{1, 2, 3,..., n} remains. It gets along without undefinable "real" numbers, without the
necessity to distinguish between finite positiv cardinal numbers and natural numbers, without
inaccessible accessories of matheology and without paradoxes of Löwenheim-Skolem or
Banach-Tarski. It gets along without an inexplicable discontinuity of the cardinality functions from
Tristram Shandy (cp. § 077, § 200) to McDuck (cp. § 515) or |limnض sn| (cp. § 529) that
unavoidably always would strike the not initiated thinker. It gets along without an empty limit of
the sequence (sn) of, for all n, infinite sets. Everybody not toughened up in a long study of set
theory would ask: "How can the infinite sequence have an empty limit? What is the reason?
What causes this vacuum?" I think my answer will be accepted by 99 % of all intelligent thinkers.
In fact, I have enjoyed this releasing and satisfying experience for many times.
§ 532 It is easy to demonstrate defective enumerations of the positive rational numbers. The
mapping Ù Ø –+ is not a bijection, for instance, when all indices n œ Ù are used to enumerate all
rational numbers n/1 in –+.
The proof presented in § 529 however proves that a purported enumeration fails. By adapting
this proof in an obvious way we see that every purported enumeration of –+ is condemned to
fail.
It is hard to understand how a method could be accepted in mathematics that, depending on
the choice of indexing, can (purportedly) enumerate an infinite set but as easily can fail. No
scientific application of rational thinking would allow for such an incredible claim.
Astounding delusions often have lead astray sectarians to perverse beliefs and actions even
such as mass murder http://en.wikipedia.org/wiki/Charles_Manson or mass suicide
http://en.wikipedia.org/wiki/Jonestown But mass self-stultification of thousands of assertedly
intelligent mathematicians is certainly unique in the history of the whole universe.
§ 533
Cantor's enumeration of the positive rationals –+ (mentioned in a letter to Lipschitz on 19 Nov.
1883) is ordered by the ascending sum (a+b) of numerator a and denominator b of q = a/b, and
in case of equal sum, by ascending numerator a. Since all fractions will repeat themselves
infinitely often, repetitions will be dropped. This yields the sequence
1/1,
1/2, 2/1,
1/3, 3/1,
1/4, 2/3, 3/2, 4/1,
1/5, 5/1
1/6, 2/5, 3/4, 4/3, 5/2, 6/1
...
It is easy to see that at least half of all fractions of this sequence belong to the first unit interval
(0, 1].
While every positive rational number q gets a natural index n in a finite step of this sequence
there remains always a set sn of positive rational numbers less than n which have not got an
index less than n (cp. § 529).
sn+1 = (sn » {q | n < q § n+1}) \ {qn+1}
with s1 = {q | 0 < q § 1} \ {q1}
All sn are infinite |sn| = ¶. But also their geometric measure is increasing beyond every bound.
This is shown by the following
Theorem. For every k œ Ù there is n0 œ Ù such that for n ¥ n0: (n-k, n] Õ sn.
Proof: Let a/1 be the largest fraction indexed by n. Up to every such n at least half of the natural
numbers are mapped on fractions of the first unit interval. a is continuously increasing, i.e.,
without gaps. Therefore n must be about twice as a, precisely: n-1 ¥ 2(a-1) or n ¥ 2a - 1.
Examples:
a = 1, n = 1
a = 2, n = 3
a = 3, n = 5
a = 4, n = 10
a = 5, n = 12
a = 6, n = 17
...
Therefore for any given k we can take n0 = 2k. Then the interval (n0-k, n0] Õ sn0. This is satisfied
for every n ¥ n0 too.
This means, there are arbitrarily large sequences of undefiled unit intervals (containing no
rational number with an index n or less) in the sets sn.
Remark: It is easy to find a completely undefiled interval of length 101000100000000000 or every
desired multiple in some set sn. Everybody may impartially examine himself whether he is willing
to believe that nevertheless all rational numbers can be enumerated.
Remark: Cantor does neither assume nor prove that the whole set Ù is used for his enumeration
(in fact it cannot be proved). Cantor's argument is this: Every natural number is used, so no
natural is missing. He and most set theorists interpret this without further ado as using Ù.
Remark: Although more than half of all naturals are mapped on fractions of the first unit interval,
never (for no n) more than 1 % of all fractions of this interval will become enumerated. In fact it
can be proven for ever natural number n, that not the least positive interval (x, y] of rational
numbers is ever completely enumerated.
§ 534 Set theorists claim that all rational numbers can be indexed by all natural numbers. In §
533 I have shown not only that every natural number n fails but even that with increasing n the
number of unit intervals of rationals without any rational indexed by a natural less than n
increases without bound, i.e., infinitely. Since nothing but finite natural numbers are available for
indexing, and provably all fail, this task cannot be accomplished.
I don't know what goes on in the heads of matheologians. But I know that it is deliberately
contradicting the magnificent, powerful, and, for all non-matheological purposes, extremely
useful mathematics of the infinite that has been devised by Euler, Gauss, Cauchy, and
Weierstrass. Rational arguments to straighten these matheological assertions are not available.
§ 535 Solution of Berry's Paradox
Berry's Paradox, first mentioned in the Principia Mathematica as fifth of seven paradoxes, is
credited to Mr. G. G. Berry of the Bodleian Library. It uses the least integer not nameable in
fewer than nineteen syllables; in fact, in English it denotes 111,777. But "the least integer not
nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables;
hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen
syllables, which is a contradiction
There is no paradox, if the correct specifications are added: The least integer not nameable in
fewer than nineteen syllables in the usual one-two-three-language is 111,777. It can be named
in the more abstract language applied by Berry by 18 syllables. Languages must not be
confused. That does not mean that any language should be excluded from mathematics! Only
the reduction to one formal language or the invalid assumption that infinite sequences without
finite definition can be subject of mathematical discourse, i.e., mathematics, can raise logical
mischaps like uncountability. The list
0
1
00
01
10
11
000
...
with all possible meanings (less than ¡0) of every binary word (there are ¡0) contains everything,
i.e. every possible notion and meaning that can exist in mathematics. All these meanings belong
to one and the same countable set.
§ 536 The diagonal argument depends on representation
Consider a culture that has not developed decimal or comparable representations of numbers.
Irrational numbers are obtained from geometrical problems or algebraic equations only. They are
defined by the problems where they appear and abbreviated by finite names - just as in human
mathematics. If all rational numbers in an infinite list are represented only by their fractions and
all irrational numbers by their finite names, it is impossible to apply Cantor's diagonalization with
a resulting "anti-diagonal". Such a culture would not fall into the trap of uncountability. (This is
erroneous in human mathematics too, because the infinite decimal representation does never
allow to identify an irrational number. Note the name decimal-fractions.)
§ 537 The diagonal argument requires that Cauchy sequences of irrational numbers contain
their limit.
Cantor's argument constructs from a list (an) of real numbers another real number, the antidiagonal d, that is not contained in the list. The argument is based on the completion of the anti-
diagonal. But this assumption is wrong. The list contains only all finite initial segments d1; d1, d2;
d1, d2, d3; ... of d. d itself is not constructed (and cannot be constructed).
In a list of all rational numbers, the anti-diagonal should be an irrational number. But it is not. It
is only the infinite sequence of all rational approximations. What differs from a list-number is
always merely a rational approximation. All these, however, are already elements of the list, by
definition.
We arrive at a contradiciton, based on the assumption of a rationals-complete list. As a
consequence this assumption, that has no foundation in mathematics, has to be rejected.
§ 538 Sequences and Limits
As the example (1 + 1/n)n sufficiently shows, a Cauchy-sequence has infinitely many (¡0)
rational elements. Since all terms of all Cauchy-sequence are rational, they belong to the
countable set of rational numbers. The limit, if a non-terminating rational or irrational number,
differs from the terms of the sequence. Even in case of simple sequences like 0.999...
discussions about their meaning have often lead to controversies. 0.999... is simply an infinite
Cauchy-sequence. But by writing 0.999... usually the limit is assumed without saying, so that
0.999... = 1.
The same distinction has to be observed with series:
Σn œÙ 1/2n < 1 but limnض Σ1n 1/2n = Σ1¶ 1/2n = 1
Σn œÙ 1/n! < e but limnض Σ1n 1/n! = Σ1¶ 1/n! = e
Σn œÙ 1/10n! < L but limnض Σ1n 1/10n! = Σ1¶ 1/10n! = L
Ignorance of these differences has lead to the "9-problem" in Cantor-list. Provision has been
made that the anti-diagonal cannot have the form 0.999... However, this provision is not
necessary. Cantor's diagonal-argument requires more precision than unwritten limits. Every digit
appearing in a Cantor-list belongs to a Cauchy-sequence - not to its limit! The Cauchy
sequences 1.000... and 0.999... are quite different. The provision shows, however, that set
theorists have been confusing sequences and their limits for about one hundred years. (Cantor
himself did not make this provision.)
What about writing limnض before every line of a Cantor list? Or what about writing every line of
a Cantor-list twice, the second one always equipped with a limnض? Subject and result of
diagonalization are always digits, i.e., rational terms of Cauchy-sequences - whether or not
these sequences stand for themselves or are used as names of irrational numbers. In a
rationals-complete list, this always raises a contradiction. (Here we have assumed the existence
of all terms of a Cauchy-sequence. Of course these can never be written. Therefore they can
never serve as names or definitions of numbers. For that sake only the finite formulas
constructing the infinite sequences and their limits are available.)
By the way, only the confusion about Cauchy-sequence and their limits has lead to the
acceptance of Hessenberg's proof of uncountability of the power set of Ù. Every subset M of Ù
can be denoted by a sequence like 0.1110010101... having, behind the decimal point, 1 at an
index n œ M and 0 at an index n' – M. Being rational numbers, the set of all these Cauchysequences is countable whereas Hessenberg meant to have shown the contrary.
§ 539 I claim that every Cauchy-sequence of rational terms with irrational limit does not contain
its limit among its ¡0 terms. This is also true if the Cauchy-sequence is the sequence of partial
sums of decimal fractions or digits. So every infinite digit sequence is a rational number.
Irrational numbers have no decimal representation. Analogously the infinite digit sequence
0.999..., i.e., the whole sum 9/10n over all natural numbers n œ Ù (not as is usually assumed its
limit), is a rational number less than 1.
Ben Bacarisse objected: You can't define what the "whole sum" means. Remember, in the
specific example you gave, it is a specific rational less than 1. Which one? You don't know.
You can't say. You can't define the notation.
My reply: All natural numbers are finite, although we cannot define a largest one. I can't say
either what a smallest number 10-n would be. Nevertheless, nobody doubts that " n œ Ù:10-n is
a rational number. This holds for all ¡0 terms if ¡0 is a sensible notion. Otherwise it holds in
potential infinity for all infinitely many n œ Ù. For ΣnœÙ 9/10n to be a rational number less than 1
we need only the fact that all partial sums are rationals less than 1. We need not define it other
than by ΣnœÙ 9/10n.
§ 540 Irrational numbers have no representation as decimal fractions.
Consider the digit seqence (dn)nœÙ of an irrational number like 1/π. This sequence has no limit.
But it can be forced to converge by adding factors 10-n. The sequence of decimal-fractions
(dn/10n)nœÙ is a Cauchy-sequence having ¡0 rational terms. The series of decimal-fractions,
here written in the usual decimal notation 0.d1d2d3... the partial sums of which 0.d1, 0.d1d2,
0.d1d2d3, ... are ¡0 rational terms of a Cauchy-sequence not containing its limit 1/π. They are in
bijection with the sequence (dn/10n)nœÙ. The series can also be noted by
(A)
ΣnœÙ dn/10n.
Does it, by magic spell, contain its limit
Σ1¶ dn/10n
(B)
when notation (A) is chosen? Certainly not. Mathematics does not depend on the choice of
notation. But how does the digit sequence (B) differ from the digit-sequence (A)?
Answer: The limit (B) = 1/π does not differ from (A) by any digit. But we know that it is not a
rational number like (A). Conclusion: An irrational number is not a decimal sequence or series. It
has no decimal expansion, it has no representation by digits, not even by infinitely many.
§ 541 Consider the decimal-fractions an = dn/10n of an irrational number x and the Cauchysequence of rational partial sums (sk)kœÙ with sk = a1 + a2 + ... + ak.
If the partial sums are written as s1 + s2 + s3 + ... the limit is not present.
If the partial sums are written as (((a1) + a2) + a3) + ... the limit is probably not present.
If the partial sums are written as a1 + a2 + a3 + ... the limit is probably present.
If the partial sums are written as ΣnœÙ an, then the limit is present: ΣnœÙ an = x.
§ 542 What is a number?
This is one of many possible philosophical positions: Numbers are expressions.
Then different expressions are different numbers. Applying mathematical laws (usually base 10
is chosen) and looking into dictionaries we find out that some of these expressions (like 2 and II
and two and zwei and 6/3 and 17 - 15 which are different numbers) can be exchanged without
changing the meaning of the text. Some expressions, like 2, are more popular than others, like II
or 17 - 15. But it is our choice to use any of these expressions which are numbers.
Another philosophical position says that the set of all expressions which can be exchanged
without changing the meaning of the text is a number. Then the different numbers belonging to
these different expressions are amalgamated into one single number.
No heaven of matheology is required where numbers could exist independent of any
mathematical discourse.
§ 543 Two wrong definitions
A definition is wrong if it conveys or establishes wrong information.
The world laughed about Bill 246 which in 1897 passed in first reading the assembly of the state
of Indiana. It defined π = 3.2. This wrong definition was certainly welcomed by merchants who
had no pocket calculators and disliked the irksome appendix .14.
The world should laugh about the matheological definition of the irrational limit of a rational
sequence (an):
*)
Σn œÙ an = limnض Σ1n an
This definition is wrong because the series a1 '+ a2 '+ a3 '+ ... with purely natural indices, gives
the sequence of the partial sums only, all of which are rational and thus are not the limit. This
sequence, although written as an infinite sum, cannot simultaneously be the limit. π is not a sum
of fractions - not even an infinite one!
*) This wrong definition is necessary in the interest of set theorists who cannot deal in a
consistent way with the actual infinite. The following sequence, written as a triangle, has limit
111.... But the limit is not a term of a strictly monotonic sequence. However, if infinity is actual,
then the height of the triangle is actually 111... and larger than every line determining its width.
This is an inconsistency, not easy to observe and perhaps impossible to understand for
matheologians, but present with no doubt.
1
11
111
...
_____
111...
In order to make this inconsistency disappear, it must be "defined" that the sequence contains or
is its limit.
§ 544 In § 542 we have learned about the possibility of defining a number system such that two
different expressions cannot denote the same number. I do not know of any obstacle to use this
system for common arithmetic and calculus. It is merely a change of names. Instead of talking
about different expressions denoting the same number we talk about different numbers which
however can be replaced by each other according to the common rules. This may be expressed
by the identity symbol too: If x can be replaced by y and vice versa, without changing the
indicated value, then we write x = y, otherwise x ∫ y. For example: 1 + 1 = 2 and 1 + 1 ∫ 1.
No changes of mathematics are to be expected - with one exception: The set of all numbers
appearing in this system is obviously not uncountable.
§ 545 A matter of notation?
The well-known sequence
0.1
0.11
0.111
...
has been written vertically here, because space on the screen is cheap. It contains all digits and
all partial sums of its limit 1/9 - but not the limit itself. When written, on expensive paper, into one
line as 0.1, 0.11, 0.111, ... or, on luxurious vellum, even shorter as 0.111..., it is including, as
before, all digits that are possible. But in the luxurious version we get the limit in addition.
Simply using another notation includes the limit?
Or is it the substratum??
What kind of mathematics could produce such a result???
§ 546 The sequence (Sn) of singletons Sn = {n} has limit limnض Sn = { } with |limnض Sn| = 0.
On the other hand limnض |Sn| = 1. What is the final state after all? Has it cardinality 0 or
cardinality 1? Why should we trust in one of these results? Is there a contradiction?
A contradiction would appear in fact if the set limit described a final state Sω, as the cardinality
limit does. But the set limit does not. It only shows that no natural number remains in the
sequence forever.
For every number n there exists a set Sm such that n has left Sm.
"n $m: n – Sm
This does not allow to conclude that there is ever an empty set appearing.
"n: n – Sm
Therefore the cardinality limit which shows that there remain natural numbers in the sequence
forever is not contradicted.
Meaning and importance of the cardinality limit is demonstrated in cases where no set limit
exists as for circular sequences like (Tn) where Tn = {n mod 2}. It will never be empty as shown
by limnض |Tn| = 1.
§ 547 Let's do something constructive. Let's try to collect all possible ways to define irrational
numbers (or irrationalities, as others say).
§ 548 It is widely held that irrational numbers can be represented by infinite digit-sequences.
We will show that this is incorrect. A digit sequence is only an abbreviated notation for an infinite
sequence of rational partial sums. Irrational numbers are limits, incommensurable with any grid
of decimal fractions.
It is obvious that strictly monotonic sequences do not assume their limit. Rarely the terms of the
sequence and its limit are confused. But this situation changes dramatically when sequences of
partial sums of series are involved. It is customary to identify the infinite sum over all terms of a
series and the limit of this series.
The sequence of rational approximations 3.1415... is purely rational although we cannot find a
fraction m/n = 3.1415... covering all its terms. This disadvatange is shared by sequences like
(1/10n) too. We cannot find a fraction covering its infinitely many terms all of which are rational
with no doubt.
A periodic decimal fraction has as its limit a rational number. A non-periodic decimal
fraction has as its limit an irrational number.
Conclusion: Irrational numbers have no decimal expansion, no representation by digits, not
even by infinitely many. They are incommensurable with every digit-measure. An irrational
number needs a generating formula F in order to calculate every digit of the infinite digit
sequence S and the limit. The formula F may be interpreted as the number as well as the limit.
The implication F fl S cannot be reversed because without F the sequence S cannot be
obained.
The mathematical facts discussed above also apply to all sequences of digits (or bits) appearing
in the folklore version of Cantor's diagonal argument. Digit sequences are never representing
irrational numbers let alone transcendental numbers. Therefore Cantor's diagonal argument
does not concern the cardinality of the set of irrational numbers.
For details see:
http://www.hs-augsburg.de/~mueckenh/GU/Sequences%20and%20Limits.pdf
§ 549 A name denotes a real number if, when given this name, the receiver can show the real
number to be in trichotomy with every rational number. Of course not every receiver can solve
this task when given the name. In fact every reader will fail in some step by practical reasons.
Further names change over time. Before Jones and Euler had introduced the name π nobody
could know that π is the name of a real number. The meaning of a name depends on the applied
decoding or language. All this makes the notion of name of a real number informal. Based on
this disadvantages the notion "named real number" cannot be used in formal theory.
The set of all finite expressions in a given alphabet however can be used in formal theory
and can be shown to be countable. In order to obtain an upper estimate for the set of all named
numbers, we can assume that there are at most ¡0 languages and that every finite expression is
a name of a real number. So every name denotes at most ¡0 real numbers. This results in an
upper estimate of ¡0 named real numbers.
§ 550 Surjections and bijections
Cantor, using his first diagonal argument, by enumerating all positive fractions m/n, maps Ù to
the set –+ of all rational numbers q such that every rational number q is in the image of infinitely
many natural numbers. This mapping is a surjection Ù to –+ but it is tacitly assumed that a
bijection can be obtained from it because every infinite subset of Ù can be put in bijection with
Ù. So Cantor's mapping is called a bijection Ù to –+.
Dedekind, using the notion of height and the fundamental theorem of algebra, by enumerating
all roots r of algebraic equations, maps Ù to the set A of all algebraic numbers x such that every
algebraic number x is in the image of infinitely many natural numbers. This mapping is a
surjection Ù to A but it is tacitly assumed that a bijection can be obtained from it because every
infinite subset of Ù can be put in bijection with Ù. So Dedekind's mapping is called a bijection Ù
to A.
My list of everything, by enumerating all finite expressions u, maps Ù to the set O of all objects
of discourse o such that every object o is in the image of infinitely many natural numbers. This
mapping is a surjection Ù to O including a surjection from a subset of Ù to the subset — of O, but
it is tacitly assumed that a bijection can be obtained from it because every infinite subset of Ù
can be put in bijection with Ù. So this mapping can be called a bijection Ù to —.
§ 551 A small-inaccessible-cardinals-axiom
SIC-Axiom: There exist 10 prime numbers the sum of which is less than 9.
Note that this axiom does not prove constructibility but only existence. Using it we can prove that
we can find 10 prime numbers the sum of which is less than 9. Of course that does not mean
that we can find such numbers - it is only provable that we can find them. Perhaps it will even
turn out provably impossible to find these prime numbers and their correct sum. But the axiom
should be accepted nevertheless, if not for another reason, then at least as an easily
understandable analogon to the axiom of choice.
§ 552 The transfinite hierarchy
The theorem of well-ordering has been contradicted by the assumption of uncountable
sets. (§ 551)
The assumption of uncountable sets has been contradicted by the assumption of
countable sets. (§ 535)
The assumption of countable sets has been contradicted by mathematics. (§ 533)
§ 553 The elements
What is an element? Is it a finite expression? Or is it something supernatural that can be
conjured up by a finite expression? In both cases only a countable set of elements can be called
up in a given language and, thanks to set theory, in all languages that are suitable to do
mathematics, i.e., thinking, talking, and writing mathematics.
It is not even necessary to know what mathematics is in order to know this.
It is not even necessary to know mathematics in order to understand this.
§ 554 Ein albernes Spiel / A silly game
Die Cantorianer behaupten: Zeige mir eine vollständige Liste aller reellen Zahlen, und ich werde
dir eine fehlende reelle Zahl nennen. Gegenbehauptung: Nenne mir eine reelle Zahl, die
angeblich in der Liste fehlt, und ich werde sie in die Liste schreiben. Dann folgen Behauptung
und Gegenbehauptung mit wachsender Frequenz.
The winner is ... der am längeren Hebel sitzt. Zumindest wenn die Behauptung "in jeder Liste
fehlt eine reelle Zahl" mit der Behauptung "ES gibt mehr reelle als natürliche Zahlen" identifiziert
wird - wie das die Cantorianer gern tun.
Niemals in der Geistesgeschichte der Menschheit und vermutlich auch niemals sonst hat ein
derart simpler Fehler eine so enorme Wirkung hervorgerufen: Die transfinite Mengenlehre,
angeblich "die Grundlage aller Mathematik". Doch das messbare Ergebnis ist mager. In
Wirklichkeit gibt es keine einzige Anwendung dieser Lehre, weder in der Mathematik, noch in
anderen Wissenschaften.
Übrigens, eine Ziffernfolge, also eine Reihe von Dezimalbrüchen bezeichnet niemals eine
Irrationalzahl. Zum einen, weil die unendliche Folge nicht komplett genug sein kann (zu jeder
Definition gehört ein Endsignal), und zweitens weil n keinen Grenzwert in Ù erreicht.
Eine streng monotone Reihe oder Partialsummenfolge enthält ihren Grenzwert nicht.
Beispiel:
ΣnœÙ1/10n = (Σ1§k§n1/10k)nœÙ ∫ limnضΣ1§k§n1/10k = 1/9
denn kein Term der Stammfolge (1/10n)nœÙ ist der Grenzwert.
" n œ Ù : 1/10n ∫ limnض1/10n = 0
Cantorians claim: Show me a list of real numbers that assertedly is complete, I will find another
real number. Counterclaim: Show me a real number that assertedly does not fit into the list. I will
put it into the list. Then claim and ounterclaim follow with increasing frequency.
The winner is ... that one who can do the last move. At least if the claim "there is no complete list
of real numbers" is identified with the claim "there (somewhere - not exactly to localize) exist
more real numbers than natural numbers - as Cantorians like to do.
Never before such a simple mistake has stirred up so much ado. Of course the result is meager.
There is no application of this teaching in mathematics or in sciences.
By the way: A digit sequence, i.e., a series of decimal fractions does never denote an irrational
number. Firstly, because an infinite sequence cannot be complete enough to determine a
number (a definition requires an end-of-file signal), and secondly because n does not reach a
limit in Ù.
A strictly monotonic series or sequence of partial sums does not contain its limit.
Example:
ΣnœÙ1/10n = (Σ1§k§n1/10k)nœÙ ∫ limnضΣ1§k§n1/10k = 1/9
because no term of the original sequence (1/10n)nœÙ is the limit.
" n œ Ù : 1/10n ∫ limnض1/10n = 0
§ 555
Es ist offensichtlich unmöglich, die relativen Lagen von 100 Punkten auf der reellen Achse
mittels Dreibit-Wörtern zu beschreiben - selbst wenn ein Axiom diese Möglichkeit behaupten
würde.
Es ist offensichtlich unmöglich, die verschiedenen Lagen von überabzählbar vielen reellen
Zahlen in einer Wohlordnung durch abzählbar viele endliche Wörter zu beschreiben - selbst
wenn ein Axiom diese Möglichkeit behaupten würde.
Trotzdem behaupten die meisten Matheologen diese Möglichkeit mit dem Argument, dass
weder im Auswahlaxiom, noch im Wohlordnungssatz von definierbaren oder beschreibbaren
Elementen die Rede ist.
Das Fehlen dieses Wortes ist nicht verwunderlich, denn eine unbeschreibbare Wohlordnung
wäre nutzlos. Cantor führte die Wohlordnung ein, um Mengen Element für Element zu
vergleichen, was im Falle unbeschreibbarer und damit unbeschriebener "Wohlordnungen"
unmöglich wäre. Zermelo erfand das Auswahlaxiom wonach jede Untermenge einer Menge
ausgewählt werden kann. Eine Untermenge auszuwählen bedeutet, sie von allen anderen zu
unterscheiden. Dazu muss jede Untermenge beschrieben sein, also eine Beschreibung
besitzen. Das ist immer ein endlicher Ausdruck. Aber wie in jeder Religion, folgen die Jünger
nicht immer den Lehren der Gründer. Oft werden die Lehren nicht nur verändert, sondern sogar
in ihr Gegenteil verkehrt. Im Falle der Matheologie sehen wir nur ein besonders krasses Beispiel
der Umkehrung.
Es führt jedoch zu einem seltsamen Ergebnis. Da die Elemente überabzählbarer Mengen nur
im Denken existieren können, kann auch jede Wohlordnung einer solchen Menge nur dort und
nirgendwoanders existieren. Wenn sie im Denken nicht vorhanden ist, so kann sie auch
nirgendwo anders gefunden werden. Dies ist für überabzählbare Mengen bisher stets der Fall
gewesen. Trotzdem behauptet der Matheologe, in seinem Denken existiere eine Wohlordnung,
die nachweislich in seinem Denken nicht existiert.
Und da ist noch ein anderes seltsames Ergebnis: Wenn undefinierbare reelle Koordinaten
akzeptiert werden, dann enthält jede Menge von 100 Punkten auch undefinierbare Punkte - viel
mehr als 99 im Durchschnitt. Und die Standardabweichung ist so klein, dass es immer möglich
ist alle Lagen der beschreibbaren Punkte auf der reellen Achse mit Dreibit-Wörtern zu
beschreiben.
It is obviously impossible to note the relative positions of 100 points on the real line by three-bit
strings - even if an axiom claims it possible.
It is obviously impossible to note the different positions of uncountably many real numbers in a
well-ordering by a countable set of position descriptions - even if an axiom claims it possible.
But it appears possible to most matheologians. Their argument is this: Nowhere, neither in the
axiom of choice nor in the well-ordering theorem we find the word "definable".
The lack of this word is not astonishing because well-ordering would be useless if it were
undefinable. Cantor invented well-ordering in order to compare sets, element by element. This
would be impossible for undefinable, hence undefined "well"-orderings. Zermelo invented the
axiom of choice, stating that every subset of a set can be chosen. To choose a subset means to
distinguish this subset from all other subsets. This implies that every subset has a definition. A
definition is a finite expression. But like in every religion, the followers of the originator do not
always follow his teachings. They often change it, sometimes even inverting it into the contrary.
In case of matheology we see an extremely blatant example of an inversion.
However it leads to a strange result: Since the elements of uncountable sets can exist only in
the mind, also any well-ordering of such a set can exist only there and nowhere else. If this mind
does not know it, the order cannot be found anywhere. This has been always the case for
uncountable sets. Nevertheless the matheologian claims to know that he has in his mind a
property that he knows he cannot know.
And there is another strange result: If the undefinability of real coordinates is acceptable, then
every set of 100 points contains undefinable points - far more than 99 in the average. And the
standard deviation is so small that it is always possible to note the positions of the definable
points on the real line by three-bit strings.