Axiomatic Set Theory
Alexandru Baltag
(ILLC, University of Amsterdam)
Go to
website:
http://alexandru.tiddlyspot.com
then click on Teaching, then click on “Axiomatic Set Theory”
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Course Administration
Instructor: ALEXANDRU BALTAG
Teaching Assistants (and Markers): ZHENHAO LI and
SHENGYANG ZHONG.
Office Hours and Email Inquiries:
Email your questions to BOTH my assistants:
Zhenhao Li, at [email protected] ,
AND
Shengyang Zhong, at [email protected]
with CC to me. Or come to see them during office hours. (Time and
Place TBA.)
In case they cannot answer a question, please email me at
[email protected].
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Course Materials
The main textbook for this course is
K. Devlin, The Joy of Sets, Springer-Verlag, 1993 (Second Edition).
In addition, I will use
J. Barwise and L. Moss, Vicious Circles, CSLI Lecture Notes no. 60,
CSLI Publications, 1996
for the last few lectures of the course (on non-wellfounded sets).
Other relevant materials (book, links, slides): on the website.
Assessment, Course Plan, Important Announcements
Assessment: Four or Five Homework Exercises (worth in total 50%
of the final grade), plus a Final TAKE-HOME EXAM (also worth
50% of the final grade).
Both are individual (i.e. no collaboration)!
Course Plan:See website. It’s tentative, and it will be periodically
updated.
Important Announcements:
February 10: No class! BUT you are all invited to attend Yurii
Khomskii’s PhD defense (in whose committee I am), as a “set
theory lesson”. Look on ILLC news for time and place.
April 3: NO CLASS!
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1.1. Naive Set Theory
• Sets, “Objects”, Elements, Membership: a ∈ x
• Sets as “Properties” or “Classes”: {a|P (a)}, or {a : P (a)}.
• Sets as well-formed, finished collections: totalities that can be
thought of as a single unit.
• Sets of Sets
• Identity of objects or sets: a = b, x = y
• Properties of Identity:
a=a
a=b⇒b=a
a=b∧b=c⇒a=c
a = b ⇒ ∀x(a ∈ x ⇔ b ∈ x)
5
• Leibniz’s principle:
∀x(a ∈ x ⇔ b ∈ x) ⇒ a = b
• Criterion for identity: the Extensionality “Axiom”
x = y ⇔ ∀a(a ∈ x ⇔ a ∈ y)
• Inclusion: x ⊆ y
• Operations with Sets:
x ∩ y, x ∪ y, x − y, {a}, {a, b}, (a, b), x × y,
[
x,
[
i∈I
yi ,
\
x,
\
yi , P(x)
i∈I
• Bynary Relations R ⊆ x × y: reflexive, transitive, symmetric,
antisymmetric, connected, transitive, equivalence relations,
equivalence classes, preorders, partial orders, posets, minimal
elements, well-foundedness, total orders, well-orders.
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• N-ary Relations. Domain and range of a relation.
• Functions: f : x → y. N-ary Functions. Function composition f ◦ g.
Identity function idx . Constant function. Image f [z]. Pre-image
f −1 [z]. Injections, Surjections, Bijections. Inverse f −1 .
• Arbitrary Cartesian Product
Πi∈I xi := {f |f : I →
[
i∈I
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xi ∧ (∀i ∈ I)f (i) ∈ xi }
The Mathematics of Infinity
Pre-history: the Greeks, Aristotle, the horror of infinity; Indians and
the love for infinity; Christian theologians, scholastics etc; Bolzano.
GEORG CANTOR’s Work (1870’s):
• Cantor’s background. Trigonometric Series.
• Dealing with infinity: mathematical and theological arguments.
• Bolzano-Weierstrass Theorem: every bounded but infinite set of
real numbers has a limit point.
• Cantor-Bendixson derivative A0 . Iterated derivatives: A(n) , A(ω)
etc. Perfect sets. Transfinite ordinals ω, ω + 1, 2ω, ω ω etc:
infinite processes.
• Comparing orders: order types. Ordinals as well-ordered types.
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• Comparing Infinities: equipotence, cardinality. Alephs: infinite
cardinals (“powers”) ℵ0 , ℵ1 etc.
• Countable Infinities: Hilbert’s Hotel. Rationals versus natural
numbers!
|N | = |Z| = |Q| = |2N | = |N × N | = ℵ0
• Lines versus Planes versus Cubes etc. “I see it, but I don’t
believe it”.
|R| = |[0, 1]| = |R2 | = |C| = |[0, 1]2 | = |Rn | = |[0, 1]n |
• Reals versus natural (rational, algebraic) numbers:
|R| > |N |
Proving Existence of Transcendental (=non-Algebraic) Numbers.
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Cantor’s Theorem and the Diagonal Method:
|x| < |P(x)|
The Power of Continuum .
• The Continuum Hypothesis (CH). Cantor’s Obsession. Open
Questions and Paradoxes. The Status of the Axiom of Choice
(AC). Kronecker’s Criticism. Madness.
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Cardinals: Proofs
|N | = |Z|:
Bijection given by
n
f (n) = − , if n is even,
2
n+1
, if n is odd.
f (n) =
2
|N | = |2N |:
Bijection given by f (n) = 2n.
N | = |N × N |:
Look at N × N as an infinite bidimensional array. Enumerate its
elements linearly, by going on successively longer diagonals.
Illustrations: “Hilbert’s hotel”.
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|N | = |Q|:
It is enough to show that there is a bijection f : N → Q+ . (Why?)
Arrange positive rational, expressed as integer quotients, in an infinite
bidimensional array
i
A(i, j) =
j
This gives us a surjective map A : N × A → Q+ . Compose this with the
previous enumeration, to get a surjective map F : N → Q+ , i.e. a linear
enumeration of Q+ with repetions. Eliminate the repetions ⇒ a
bijective map from N to Q+ .
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|R| = |(0, 1)|:
A bijection f : R → (0, 1) is given by
1
f (x) = 1 −
, for x ≥ 1,
2x
1
, for x ≤ 1,
f (x) =
2(2 − x)
|R| = |(0, 1)]| = |[0, 1)| = |[0, 1]| = |(0, ∞)|: Exercise.
|R| = |R × R| = |Rn |:
It is enough to show |[0, 1]| = |[0, 1] × [0, 1]|. (Why?) Exercise!
|R| = |2N | = |P(N )|, where 2N = {f |f : N → {0, 1}}.
Exercise! (Hint: use the binary representation of reals; for the second
part, use the characteristic function fS of any subset S ⊆ N .)
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Cantor’s Theorem: proof
Proof of |x| =
6 |P(x)|: Cantor’s “Diagonal Method”.
Let f : x → P(x) be a bijection.
Put
A = {y ∈ x|y 6∈ f (y)}.
Since A ∈ P(x), there exists some a ∈ A such that f (a) = A.
But then we have:
a ∈ A ⇔ a 6∈ f (a) ⇔ a 6∈ A.
Contradiction!
Consequences: |x| < |P(x)|; |N | < |R|; there exist transcendental
numbers; there is no largest cardinal.
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Wosets: what are they good for?
Well-orderings are “well”, because we can do induction on them:
Induction Theorem (Th 1.7.1. in Devlin)
Let (X, ≤) be a woset. If E ⊆ X satisfies
• the smallest element of X is in E: M in≤ X ∈ E;
• Every element of X is in E if all its predecessors are in E:
∀x ∈ X( ∀y(y < x ⇒ y ∈ E) ⇒ x ∈ E ) ;
then E = X.
This generalizes induction on natural numbers.
NOTE (exercise): In fact, the first itemized premise of this theorem is
redundant: it follows from the second itemized premise!
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When are two (well-)orders the same?
An order-isomorphism between two (totally) ordered sets
(X, ≤), (X 0 , ≤0 ) is a bijection f : X → X 0 such that
x < y ⇒ f (x) ≤0 f (y)
for all x, y ∈ X.
If there exists such an order isomorphism, then we write
(X, ≤) ∼
= (X 0 , ≤0 ).
Essentially, this means that the two sets are ordered “in the same way”:
they have the same “order type”.
Ordinals are the order types of wosets.
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Von Neumann Ordinals
In order to formally define ordinals, it would be useful to be able to
choose a “canonical representative” for each (well-)order type:
i.e. to find a special family On of wosets (to be called “ordinals”), such
that every woset is order-isomorphic to a unique woset in On.
Definition: A “von Neumann ordinal” (or ordinal, for short) is a
set that is well-ordered by the membership relation ∈.
Intuition behind definition: an ordinal is simply the set of all its
predecessors.
We (or God) form ordinals in stages:
a newly created ordinals is just the set of all previously formed ordinals.
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Equivalent Definition (Devlin)
Devlin gives an equivalent definition:
Given a woset (X, ≤) and an element a ∈ X, the segment of X
determined by a is the set
Xa = {x ∈ X|x < a}
of all the predecessors of a.
Definition (Devlin): An ordinal is a woset (X, ≤) such that Xa = a for
all a ∈ X.
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Ordinals form a Well-Ordered Collection
The collection of all ordinals will be denoted by On.
As we will see, in standard Set Theory, On is NOT a set.
For two ordinals α, β ∈ On, we define the precedence relation by:
α < β iff α ⊂ β iff α ∈ β,
As a consequence, we’ll indeed always have that any ordinal coincides
with the set of its predecessors:
α = {β ∈ On|β < α}.
Moreover, the collection On of all ordinals is itself well-ordered
by <.
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Examples of von Neumann ordinals
The ordinal 0 has no predecessors, so it is just the empty set:
0 := ∅.
The ordinal 1 has only one predecessor, namely 0, so:
1 := {0} = {∅}.
The ordinal 2 has two predecessors, 0 and 1, so:
2 := {0, 1} = {∅, {∅}}.
More generally, the ordinal corresponding to the natural number n is
the set of all smaller (natural numbered) ordinals:
n := {0, 1, 2, . . . , n − 1}.
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More Examples and Notations: transfinite ordinals
The next ordinal after all the finite ones (= the smallest infinite
ordinal) is defined as the set of all finite ordinals:
ω := {0, 1, 2, . . .} = N.
The next ordinal after that will be
ω + 1 := {0, 1, 2, . . . , ω} = ω ∪ {ω} = N ∪ {ω}.
The series of increasing ordinals goes forever, beyond the confines of
any given system of notation:
0, 1, 2, . . . , ω, ω + 1, . . . , 2ω := ω · 2, . . . , 3ω := ω · 3, . . . , ω · n, . . .
ω
ω 2 := ω · ω, . . . , ω 3 , . . . , ω 4 , . . . , ω n , . . . , ω ω , . . . , ω ω , . . . , ω ω
ω ...
ω1 , . . . , ω2 , . . . , ωn , . . . , ωω , . . . , ωω2 , . . . , ωωω , . . . , ωωω... , . . .
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,...
Successor Ordinals versus Limit Ordinals
For every ordinal α, there is a unique next highest ordinal α + 1.
Ordinals of the form α + 1 are called successor ordinals.
For every set A of ordinals, there is a unique “next” ordinal (=the least
ordinal that is higher than all the ordinals in A). If A has NO last
element (in the <-order), then we denote this “next” ordinal by lim A.
Ordinals of this form are called limit ordinals.
Examples:
lim {0, 1, . . . , n, . . .} = ω,
lim {ω, ω + 1, ω + 2, . . . , ω + n, . . .} = ω · 2,
lim {ω, ω · 2, ω · 3, . . . , ω · n, . . .} = ω · ω.
Every ordinal is either a successor or a limit.
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The Successor Function, set theoretically
Recall that every von Neumann ordinal α coincides with the set of its
predecessors:
α = {β ∈ On|β < α}.
Applying this to a successor ordinal α + 1, we obtain that
α + 1 = {β ∈ On|β < α + 1} = {β ∈ On|β < α} ∪ {α} = α ∪ {α}.
So we can represent set-theoretically the successor function as the
following operation on von Neumann ordinals:
α 7→ α + 1 := α ∪ {α}
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Attempts to Save Math: Logicism, Formalism, Intuitionism
Logicism: Gottlob Frege. All Math is Logic. Logic can axiomatize
infinity.
Formalism: David Hilbert. Infinities are useful fictions. A proper
logical analysis should show that they are consistent with, but not
necessary for Math: they can be eliminated, replaced by finitistic
methods. Once this is proved, infinities can still be safely used, to
shorten the proofs.
”No one will drive us from the paradise which Cantor created for us”.
(D. Hilbert)
Intuitionism, Constructivism etc: Brouwer, Heyting etc. Infinities
are dangerous. They break the laws of classical logic. Only
constructive, finitistic methods should be employed.
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Foundations of Math, Logicism and Paradoxes
Gottlob Frege versus Bertrand Russell (1901):
• invented axiomatic predicate logic
• Logicism: “all Math is Logic”. Deriving all math from logical
axioms.
• Naive Comprehension Principle: for any logically definable
predicate P , we can form the set
{a|P (a)}
• Russell’s Paradox:
{x|x 6∈ x}
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More Paradoxes
Well-founded sets: no infinite descending ∈-chains.
Mirimanoff ’s Paradox:
{x|x is well-founded}
Burali-Forti Paradox: the order type of the set of all ordinals (with
theor natural order).
{x|x is an ordinal}
Cantor’s Paradox: the cardinal of the universe
U = {x|x is a set}
is by definition the largest cardinal, which contradicts Cantor’s
Theorem.
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Towards a Solution: The Set/Class Distinction
Paradoxes show that we have to give up the Naive Comprehension
Principle: not every definable collection of objects (=definable by
some first-order predicate P (x)) forms a set.
Definable collections {x|P (x)} are in general called “classes” in Set
Theory.
So the lesson from the paradoxes is that not every “class” is a “set”.
But what does that mean? What distinguishes sets from “classes”?
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Classes versus Objects
The important distinction is that classes are collections of objects,
but they are not necessarily objects themselves!
Moreover, only “objects” can be collected to form classes.
Syntactically, the variable x in the expression
{x|P (x)}
or in the expression
x∈Y
must denote an object; while the variable Y in
x ∈ Y , or in Y = Z
might stand for a “class” (collection), such as {x|P (x)}, which might
happen to NOT be an object.
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Classes versus Sets
So any definable collection of “objects” is a class. But not all
classes are objects, and hence not all classes can be collected as
members of other classes.
The classes that happen to also be “objects” will be called sets.
The classes which are not objects (and hence are not sets) are called
proper classes.
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Example
Russell’s class, defined as the class (collection) of all the sets that do
not belong to themselves
{x|x 6∈ x}
cannot be a set (since this would lead to Russel’s paradox), so it is a
proper class.
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Axiomatic Systems of Set Theory
All these are only intuitions.
The formal (axiomatic) systems we’ll consider will make some of this
precise.
There are a number of axiomatic proof systems for Set Theory.
The system ZF (from Zermelo-Fraenkel), augmented with the Axiom
of Choice (AC) to form the Zermelo-Fraenkel system ZF C: the
most widely accepted.
The von Neumann-Bernays-Godel system N BG is an alternative
formulation, which is nevertheless completely equivalent to ZF C.
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The system ZF A of Anti-founded (=non-wellfounded) Set
Theory, proposed by P. Aczel, describes a universe of sets that is
wider than the universe of ZF C, allowing for sets that disobey one of
the axioms (Foundation) of ZF C.
However, in a sense ZF A is still equivalent to ZF C!
More precisely, ZF A has the same logical strength (=it is
equi-consistent) with ZF C:
if given any model of ZF C, one can construct a model of ZF A, and
vice-versa.
Hence, the consistency of ZF C is equivalent to the consistency of ZF A.
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There are also systems that are logically stronger than ZF C:
one example is the Morse-Kelley system M K;
another examples are the extensions of ZF C with various large
cardinal axioms;
yet another example is the system ST S of Structural Set Theory.
There are also of course many systems that are weaker than ZF C:
obtained by dropping some of the axioms.
Finally, there are systems of set theory that are not known to be
weaker, nor stronger, nor equivalent to ZF C: an example is Quine’s
system N F of “New Foundations” (very important from a
philosophical perspective and also from a historical and foundational
point of view); another example (much more important for current
research in set theory) is the system obtained from ZFC by replacing
the Axiom of Choice by the Axiom of Determinacy (AD).
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Classes, from a formal-axiomatic perspective
In some axiomatic versions of Set Theory (such as ZF C), there are
strictly speaking no “classes”, at least at a formal level: the allowed
variables x, y, . . . range only over objects
Indeed, in some versions of ZFC, no other objects are allowed but sets,
and so the variables range over sets only.
Nevertheless, at a conceptual, informal level, people continue to talk
about “classes” as ways to denote definable collections of objects, which
might or not be objects (sets) themselves.
The system ZF C interprets such class talk only as “a manner of
speaking”, a metaphor : classes are like “imaginary objects”, that may
be useful as an intuitive way of speaking, but are not to be taken
seriously.
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As we’ll see, in ZF C there is a systematic way to “translate”
statements that involve classes into statements that do NOT
refer to classes in any way.
Other systems, such as N BG, formally allow classes: the variables
x, y, . . . now range over classes. However, only sets can be
members of other classes.
Moreover, N BG still doesn’t fully take classes seriously, since it
restricts the quantifiers (and free parameters) to sets.
In a sense, only definable classes are allowed in N BG.
The Morse-Kelley system M K drops this last restriction, allowing
reference to undefined, arbitrary classes (as free parameters): it is
“impredicative”.
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When can a class be regarded as an object?
The main question is: WHEN does a class (collection of objects)
constitute itself an “object” (hence, a set)?
What is special about those classes/collections that are sets/objects?
Various answers have been proposed:
Sets are “well-formed” collections: built in stages, in a well-ordered
manner, from previously built sets.
Sets are “small” collections (or at least “not very big”).
Sets are “closed”, “finished” collections.
Sets are “well-defined” collections (defined by a predicate that doesn’t
involve any hidden vicious circles).
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A Solution: The Iterative Conception of Sets
Originates in Russell’s Type Theory
(B. Russell, A. N. Whitehead, Principia Mathematica, 1910-1913).
It is the most accepted intuition behind today’s standard setting for Set
Theory:
A set can be constructed only AFTER its elements are given.
Sets are “built” inductively, in transfinite stages, starting from
basic, atomic objects (ur-elements), or even better: starting from
nothing!
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Nothing Versus Ur-elements
Some versions of Set Theory (e.g. the one presented in the book
“Vicious Circles”) allow for the existence of some objects that are
NOT sets. These are called primitive objects, or atoms, or ur-elements.
Ur-elements can be elements of sets, but they themselves havo no
elements.
On the other hand, they are not equal to the empty set ∅.
Hence, ur-elements do NOT respect the Extensionality Axiom!
For this reason, in most standard presentations of Set Theory,
ur-elements are excluded : every object is a set.
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The Cummulative Hierarchy of Sets
In Day 0, God takes nothing and forms the empty set 0 := ∅.
In Day 1, God puts the empty set into a set, creating 1 := {∅} = {0}.
In Day 2, God uses the previous creations to create two sets
2 := {0, 1} = {∅, {∅}} and {{∅}}.
In Day ω, God puts together the infinitely many previous creations in
various ways, creating among other things the set of natural numbers
N = {0, 1, 2, . . .} = {∅, {∅}, {∅, {∅}}, . . .}
but also the set
{∅, {∅}, {{∅}}, . . .}
In Day ω + 1, God creates among other things the real numbers R, i.e.
P(N ).
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Formalizing the Cummulative Hierarchy of Sets
The collection of all the sets created by God before Day α forms the
set-theoretic “universe” Vα on (the morning of) day α.
Before Day 0 there were no sets, so
V0 = ∅
In Day α, God forms sets by collecting in all possible ways his previous
creations, i.e. the objects in Vα :
Vα+1 = P(Vα )
If λ is a limit ordinal, then Vλ consists all sets created before day λ, i.e.
created in any of the previous days; i.e.
[
Vλ =
Vβ
β<λ
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These three equations define the cummulative hierarchy of sets.
As in the Big Bang, the “universe” keeps expanding every day.
Each “universe” Vα becomes just a set in the next universe Vα+1 .
There is no last day, since ordinals never end.
Hence, the set-theoretical universe grows forever: we cannot form a
“set of all sets” in any day!
The intuitive assumption behind this hierarchy is that all sets are
formed in this iterative way: every set is “created” in a particular
day. Hence, there cannot exist any “set of all sets”.
However, we can still in a sense refer to the “real” universe of all sets
as being a (proper) class, namely the class
V = {x|x is a set }.
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Foundation Axiom
Our intuitive assumption (“every set is created in some day”) can
now be captured by the so-called Foundation Axiom:
[
V =
Vα .
α
An equivalent formulation:
Every set is well-founded ( when considered as a structure with
membership ∈ as its binary relation):
i.e. there are NO infinite descending ∈-chains of sets x0 3 x1 3 x2 3 . . ..
Another equivalent formulation:
Every non-empty set has an ∈-minimal element:
If x 6= ∅ is a non-empty set then there is some a ∈ x such that a ∩ x = ∅.
This last one is the formulation in Devlin.
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Null Set Axiom versus Pairs Axiom
Devlin’s first ZFC axiom is the
Null Set Axiom: There exists a set ∅ with no members.
This provides the starting point for building the set hierarchy.
Justification: God creates ∅ in day 0.
In more standard formulations of ZF C, this axiom is replaced by the
Pairs Axiom: For every two sets a, b, there exists a set {a, b}.
Justification: if a and b were created in some days, take the latest of
these two days, say α; then {a, b} is created in day α + 1.
NOTE: In the context of the other ZF C axioms, both the Null Set
Axiom and the Pairs Axiom are redundant (=not independent): they
can be deduced from the other axioms!
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Powerset And Union Axioms
Powerset Axiom: If x is a set, then there is a set P(x)
consisting of all subsets of x.
Justification: if x is created on day α, then all its members were
created before day α; hence, all collections that can be formed from x’s
members (=all subsets of x) are also created in day α (or earlier); the
set P(x) consisting of all these subsets is thus created in day α + 1.
Union Axiom: If x is a set, then there is a set
all elements of all elements of x.
S
x consisting of
Justification: if x is created in day α, then its elements were created
before day α; hence, all the elements of its elements were also created
S
before day α. The set x, collecting all these (elements of elements),
will thus be created in day α (unless it was already created earlier).
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Subset Selection Axiom
Subset Selection Axiom: If x is a set and P (v) is a unary
predicate, then there exists a set
{y ∈ x|P (y)}
consisting of all (and only) the members of x which satisfy
property P .
Justification: If x is created in day α, then all its elements were created
before day α; hence, the collection {y ∈ x|P (y)} is also formed in day α
(unless it was already created earlier).
45
Axiom of Replacement
The Subset Selection Axiom can be strengthened to the
Replacement Axiom: If x is a set and F is a functional
operation on sets, then there is a set
F [x] := {F (y)|y ∈ x}
consisting of all (and only) the F -images of all elements of x.
EXERCISE : This axiom is “stronger than” (i.e. it implies) the Subset
Selection Axiom.
JUSTIFICATION?? Replacement was added later to ZF C than the
other axioms. It is hard, or maybe even impossible, to justify
based on the iterative picture.
46
The Axiom of Infinity
Infinity Axiom: There is a set x such that ∅ ∈ x and which is
closed under taking singletons (i.e. for every a ∈ x, we have
{a} ∈ x).
The essential content of this axiom is the claim that there exists an
“infinite” set.
Justification:
∅ is created in day 0, {∅} in day 1, {{∅}} in day 2 etc; so, in day ω, the
set
{∅, {∅}, {{∅}}, . . .}
is created.
47
The Zermelo-Fraenkel Axioms
The axioms of the system ZF C are:
1. Extensionality Axiom: sets with the same elements are the
same.
2. Null Set Axiom: ∅ is a set.
Alternatively, the Pairs Axiom: if a, b are sets, then {a, b} is a set.
3. Axiom of Infinity: {∅, {∅}, {{∅}}, . . .} is a set.
4. Power Set Axiom: if x is a set then P(x) is a set.
S
5. Union Axiom: if x is a set then x is a set.
6. Subset Selection Axiom: if x is a set and P (y) is a unary
predicate, then {y ∈ x|P (y)} is a set.
48
7. Axiom of Replacement: if x is a set and F is a functional
operation on sets, then {F (y)|y ∈ x} is a set.
8. Foundation Axiom: If x 6= ∅ is a non-empty set, then there exists
a ∈ x such that a ∩ x = ∅.
9. Axiom of Choice (AC): The Cartesian product of a set of
non-empty sets is non-empty:
if {xi |i ∈ I} is a set s.t. ∀i ∈ I : xi 6= ∅, then ×i∈I xi 6= ∅.
49
The Language of Set Theory: LAST
The language LAST presented in Devlin (Ch. 2.1) is one of the various
languages that were used to formalize the system ZF C.
Vocabulary of LAST:
• Constants (=names for sets): a countably infinite collection of
symbols
w0 , w1 , . . . , wn , . . .
(Strictly speaking, these are not really necessary: they can always
be simulated by variables!)
• Variables (for sets): another countably infinite collection of
symbols
v0 , v1 , . . . , vn , . . .
• The membership symbol ∈
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• The equality symbol =
(Again, this is not really necessary: in some presentations of Set
Theory, ∈ is the only primitive relational symbol, and = is defined
in terms of ∈.)
• Logical connectives ∧ (AND), ∨ (OR), ¬ (NOT) (The OR
symbol ∨ is not really necessary: it can be defined as an
abbreviation via ϕ ∨ ψ := ¬(¬ϕ ∧ ¬ψ.)
• Quantifier symbols: ∀ (for all), ∃ (there exists)
• Brackets: (, ).
We use x, y, z, . . . as metavariables, denoting arbitrary constants wn or
variables vn .
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Formulas of LAST:
• x ∈ y, x = y are formulas;
• if ϕ, ψ are formulas, then (ϕ ∧ ψ), (ϕ ∨ ψ), (¬ϕ) are formulas;
• if ϕ is a formula, then (∀vn ϕ), (∃vn ϕ) are formulas.
We drop the brackets whenever there is no ambiguity.
We make the usual conventions concerning free and bound
variables.
A sentence is a formula that contains no free variables.
We write ϕ(v0 , v1 , . . . , vn ) for a formula whose free variables are among
the ones in the list v0 , v1 , . . . , vn .
We make the usual abbreviations:
ϕ ⇒ ψ := ¬ϕ ∨ ψ,
ϕ ⇔ ψ := (ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ).
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More Abbreviations
x ⊆ y := ∀vn (vn ∈ x ⇒ vn ∈ y), where vn 6= x, y ;
[
x=
y := ∀vn (vn ∈ x ⇔ ∃vm (vn ∈ vm ∧ vm ∈ y)),
where vn 6= vm , vn , vm 6= x, y.
x = {y} := ∀vn (vn ∈ x ⇔ vn = y),
where vn 6= x, y;
x = {y, z} := ∀vn (vn ∈ x ⇔ vn = y ∨ vn = z),
where vn 6= x, y, z;
x = (y, z) := ∀vn (vn ∈ x ⇔ vn = {y}∨vn = {y, z}),
[
x = y ∪ z := (x = {y, z}).
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where vn 6= x, y, z;
Formalizing the ZFC axioms
Each of the ZF C axioms can be encoded as formula, or an infinite set
of formulas (an axiom scheme), in LAST.
E.g. the Union Axiom becomes, with the above abbreviations:
[
∀vi ∃vj (vj =
vi ),
or more explicitly (unfolding the abbreviation)
∀vi ∃vj ∀vn (vn ∈ vj ⇔ ∃vm (vn ∈ vm ∧ vm ∈ vi )).
Some ZF C axioms are axiom schemata, e.g. the “Axiom of Subset
Selection” actually consists of infinitely many instances:
∀vi ∃vm ∀vn (vn ∈ vm ⇔ vn ∈ vi ∧ ϕ(vn )).
Homework 1
Do the following exercises from
Devlin (The Joy of Sets, 1993 edition):
Exercise 2.1.1 (page 34): parts (ii), (iii), (v), (vi), (vii);
Exercise 2.2.1 (page 39); Exercise 2.2.3 (page 39);
Exercise 2.2.6 (page 39); Exercise 2.3.1 (page 40);
Exercise 2.3.6 (page 44); Exercise 2.3.8 (page 45).
DUE Friday 24 February before 10:00 am.
Please leave it in SHENGYANG ZHONG’s mailbox at ILLC, or else
email him at
[email protected]
Late submissions will NOT be considered!!
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