Peano Systems and the Whole Number System
Ronald C. Freiwald, Copyright © 2008
Washington University in St. Louis
These notes may be copied and freely distributed provided authorship is acknowledged by including this
cover page, and the following pages are not modified.
We have a good informal picture about how the system of whole numbers works. By the
whole number system we mean to the set = œ Ö!ß "ß #ß ÞÞÞ ×, together with its rules for
arithmetic and for handling inequalities (for example, if +ß ,ß - − = and +  , , then
+  -  ,  -ÑÞ Informally, we know a multitude of facts about behavior involving
whole numbers.  ß † , œ ß  ß and Ÿ . We also know how induction works.
Ultimately, we want to show how the whole number system can be described in terms of
our foundation, set theory. We want to construct a system consisting of sets, ways to
combine them Ð  ß † Ñ and ways to compare them Ð  ß Ÿ Ñ so that the system “acts just
like” the whole number system. As we have said several times, mathematicians don't
care about what the whole numbers “really are.” If we can use set theory to build a
system that “acts just like =”, then all mathematicians can agree to treat that system as =.
More carefully, what do we need to do? When have we got a system “that acts just like
=”? There are so many facts we know about the whole number system that we should
build into this system of sets. There may even be about facts about = that we don't know
but that ought to be included. Our job seems like a hopeless task.
To make things more manageable, it would be very helpful if we had a short list of “the
crucial properties” of =  a list from which we can prove that the other important
properties of = must also inevitably be true. Then, if we can build a system of sets which
has all “the crucial properties” of =, then our new system will include the other
important properties of = automatically.
Fortunately, there is just such a short list  axioms developed by the mathematician
Giuseppe Peano in 1889. The latter part of the 19th century, and the beginning of the20th ,
were an “age of rigor” for mathematics  a period when firm foundations formathematics
were being established. By then, for example, calculus had been around for a couple of
centuries and seemed to work well  at least in skilled and sensitive hands  but it was
clear that there was a lot of vagueness about why it all worked. Part of the problem had
to do with not having firm foundations for the number systems (particularly ‘).
We are going to look at the list of “Peano's Axioms” and try to indicate how all the
informal properties of the whole number system = follow from the properties in the list.
There are many, many details to check. We will check some of the details to indicate how
(with several additional lectures) all the details could be ironed out. In not doing
everything, there is no attempt to “hide” something hard. Any material we leave out is
truly just “more of the same.”
Definition A Peano system c is a collection of objects with the following properties:
P1) There is a special object in c named “!.”
ÐAlthough the name “ !” is intended to suggest “the whole number zero,” we
really know nothing about how the object called “ !” in a Peano system acts
except for what is stated in (or deducible from) the remaining axioms.
P2) For each object B − c , there is exactly one object in c called the successor
of B (for short, we write B to represent the successor of BÑ.
P3) ! is not the successor of any object in c À
ÐaB − c Ñ B Á !
P4) Different objects in c have different successors À
(aB − c ÑÐaC − c ) ÐB Á C Ê B Á C  Ñ
P5) Suppose E © c Þ If ! − E and if ÐaB − c Ñ ÐB − E Ê B − EÑ is true,
then E œ c .
Note: In his 1889 book, Peano went so far as to also include a few other axioms about
how “ œ ” behaves: for example,
ÐaB − c Ñ B œ B and
ÐaB − c ÑÐaC − c Ñ ÐÐB œ CÑ Ê C œ BÑÑ
Our point of view is that “ œ ” is a logical term meaning “is the same thing as” and
that such assumptions about “ œ ” do not really need to be spelled out  although doing
so would certainly be harmless.
A Peano system is an “abstract system” : we are given no information whatsoever about
what the “objects” in c “really are,” and we have no information about how B can be
found for a given B − c . The only things we know about the objects in c and their
successors is what the axioms P1-P5 say about their behavior. Of course, we can
logically deduce (prove) new pieces of information about c (theorems) from those
axioms.
Until a reasonable collection of theorems about a Peano system is built up to use, the
proofs of theorems will usually rely on axiom P5  which we will refer to as the
induction axiom.
The challenge (and the amusement) of proving things about a Peano system is that we
have so little, at the beginning, to work with. We have to fight for each little new fact.
But the more things we prove, the more tools we have to work with and the easier it gets.
Notice that the informal whole number system, =, obeys each of the axioms P1-P5
provided that
i) we interpret the objects B in c to be whole numbers, and
ii) we interpret “successor” B to mean the whole number “B  ".”
Under this interpretation, = is an example of a Peano system. Of course, axiom P5 is
what we called the Principle of Mathematical Induction (PMI) in =.
When we have an abstract system like c and we
i) interpret all the objects and operations in c (such as “successor”)
as representing certain concrete objects and operations, and
ii) all the assumptions about the objects/operations in the abstract system
become true statements about the specific objects in the interpretation
then we say we have found a concrete model for the abstract system. Thus, = is a model
for the abstract Peano system c.
Some Theorems About a Peano System c
To illustrate dealing with an abstract system, we will prove some simple theorems about
c that follow from P1-P5. (Since the theorems follow logically from the axioms P1-P5,
and because P1-P5 (as interpreted) are true statements in a model, each theorem must
also be true (as interpreted) in any model of c (for example, in =). For example, see the
italicized interpretation of Theorem 1 in the model =.)
Theorem 1 For all B − c , either B œ ! or ÐbC − c Ñ B œ C  (that is, every nonzero B
in c is a successor). (Interpreted in the model =, Theorem 1 says that for each nonzero
whole number B, there is a whole number C such that B œ C  "Þ)
Proof Let E œ ÖB − c À B œ ! or ÐbC − c Ñ B œ C  × œ ÖB − c À B œ ! or B is a
successor×Þ We need to show (using P5) that E œ c Þ
i) By definition of E, ! − E
ii) Suppose B − E. Then B − E because B is a successor (namely, the
successor
of B).
By the induction axiom P5, we conclude that E œ c Þ ñ
A corollary is a theorem that follows as a relatively quick and easy consequence of a
previous theorem.
Corollary 2 If B − c and B Á !, then Ðb! C − c Ñ B œ C  Þ
Proof Theorem 1 gives that if B Á !, then Ðb C − c Ñ B œ C 
To show uniqueness, notice that if B œ C and B œ D  , then C œ D  , so C œ D
(using the contrapositive of P4). ñ
Definition If B œ C  in c , we call C the predecessor of B.
Notice that the definition makes sense: we can say the predecessor because (from
Corollary 2) there can't be more than one predecessor for B. Corollary 2 therefore says
that each nonzero element B in c has a unique predecessor.
Theorem 3 For all B − c , B Á B (that is, no object B in c is its own successor).
Proof Homework Exercise
Theorem 4 If B − c , then either B œ ! or B can be obtained from ! by applying the
successor operation to ! a finite number of times.
Proof Let E œ ÖB − c À B œ ! or B can be obtained from ! by applying the successor
operation to ! a finite number of times×Þ
! − E (by definition of E)
Suppose B − E. We prove that B − EÞ
If B œ !, then B − E because B œ ! can be obtained by applying the
successor operation just one time.
If B Á !, then (because B − EÑ B can be obtained from ! by a finite
number of successor operations. But then one additional application of
the successor operation produces B . Therefore B − EÞ
By the Induction Axiom P5), A œ c , which proves the theorem. ñ
Corollary 5 If Bß C − c and B Á C , then one of B or C can be obtained from the other by
applying the successor operation a finite number of times.
Proof If one of B or C is ! (say, B œ !Ñ then Theorem 4 says we can obtain C by applying
the successor operation to B a finite number of times.
If neither B nor C is !, then (by Theorem 4 ) we can obtain both B and C by from !
using the successor operation. Applying the successor operation to !, we arrive first at
(say) B; and then continuing to apply the successor operation an additional number of
times produces C. ñ
NOTE: Theorem 4 and Corollary 5 are italicized because we will not use them in any
proofs that come later. In fact a “purist” might object that if we are trying to formally
develop a theory of Peano systems in order to define the system of whole numbers =, then
we should not be allowed to use an argument that involves doing something “a finite
number of times”  objecting that we can't formally say what “a finite number of times”
means until after we have defined the whole number system.
Nevertheless, it seemed like it would be helpful to include the italicized results to
help build up our intuitive picture of what a Peano system c “looks like”  as discussed
in the next section.
See Theorem 13.
All Peano systems are “the same”
What does a Peano system “look like” ? We can get an idea with a schematic diagram in
which an arrow “ Ä ” points to “the successor.” We start with !, which has no
predecessor:
! Ä ! Ä Ð! Ñ Ä ÞÞÞ
Ä B Ä B Ä ÞÞÞ
Theorem 4 tells us that every nonzero object B − c appears in this diagram eventually,
after applying the successor a sufficient number of times.
When we make the diagram, it will always “keep on going forward”  that is, there will
never be any “backward loops” like
Which axiom says that the first loop is impossible? the second? Why is the third loop
impossible ?
Thus we can informally picture a Peano system c as an “infinite linear chain” starting at
its special element, !:
! Ä ! Ä Ð! Ñ Ä ÞÞÞ
Ä B Ä B Ä ÞÞÞ (and so on, forever)
All Peano systems must look the same. The technical phrase for this is that all Peano
Systems are isomorphic. To be a little more precise, this means that if we have two
Peano systems c and c (we use boldface for the second Peano system and its objects),
then it is possible
i) to pair off all the elements of c with all the elements of c in such a way that so
that each object in one system has a unique “partner” in the other system.
ii) to do this not just with some “random” pairing, but to do it in such a way that !
is paired with ! and the pairing respects the successor operation: if B − c is partnered
with B − c , then B (in c ) is partnered with B Ðin c Ñ) À in other words, “the
successor of partner is the partner of the successor.”
Our images of these systems would then look like this  where vertical arrows indicate
the “pairing”:
! Ä ! Ä Ð! Ñ Ä ÞÞÞ Ä
Î
Î
Î

! Ä ! Ä Ð! Ñ Ä ÞÞÞ Ä
B Ä B Ä ÞÞÞ
Î
Î
B Ä B Ä ÞÞÞ
(and so on, forever)
(and so on, forever)
A slightly different way to think of this isomorphic “pairing” is just to imagine that each
object B in c has been “renamed” subject to the following rules:
i) ! is renamed as !
ii) if B is renamed as B, then B is renamed as B
From this point of view, the “second” Peano System c is just the “same old stuff” but
with new names.
This is an example of an important phenomenon. Sometimes different systems really are
complete look-alikes: one is just the other with elements “renamed” in a way that
respects the operations inside the system (e.g., “successor”). The systems are perfect
“mirror images” of each other  they have exactly the same structure. The words
“structure” and “system” are a little vague, so we can't make a precise mathematical
definition here. But here is an informal definition that may be useful to remember.
Informal Definition Suppose there is a “pairing (or renaming) rule” between two
systems which pairs off all the objects in the two systems with each other in a one-forone way. Suppose, moreover, that this pairing is done in a way that respects all the
operations (like “successor”, for example) in the systems. Then we say that the two
structures are isomorphic and the “pairing rule” is called an isomorphism between the
structures.
Note: “isomorphism” comes from two Greek words,
“isos”
“morphe”
meaning
meaning
“equal” or “same”
“shape” or “form” or “structure” )
To make the definition more precise, we would replace “pairing rule” with “a one-toone, onto function” between the systems. But that additional precision needs to wait
until we say more about functions, one-to-one functions, onto functions, etc.
Students who have already taken Math 309 (Matrix Algebra) should have seen the idea of
“isomorphic systems” before  although the word “isomorphic” might not have been
used. If Z is a finite dimensional vector space with basis U œ Ö," ß ÞÞÞß ,8 ×, then Z is
isomorphic to (“looks just like”) the vector space ‘8 . The “coordinate mapping” pairs
off each vector B − Z with a vector in ‘8 ß namely, B Ð Ð-" ß ÞÞÞß -8 Ñ where -" ß ÞÞÞß -8 are
the coordinates of B with respect to the basis U ÞÑ
This is sufficient detail for what we are going to do. We have argued that any two Peano
systems are isomorphic, so that “if you've seen one Peano system, you've seen them all.”
However, those who are interested are encouraged to also read this optional
(indented) material. Unlike the more informal discussion, above, the following
discussion makes no use of the “picture” and makes no use of the italicized results
Theorem 4 and Corollary 5. The “renaming” or “pairing” rule is defined
inductively without any reference to the figures above.
Define a “renaming” rule (function) V that pairs each element in c with a
“unique partner” in the other Peano system c . The definition of V is done
inductively (that is, using axiom P5):
Let V Ð!Ñ œ !
and, aB − c V ÐB Ñ œ ÐVÐBÑÑ
ЇÑ
Ð‡Ñ tells you how to find V ÐB Ñ (an object in c ) if you already know V ÐBÑ
(an object in c )Þ This defines V for every B − c À
For example, the rule gives
V Ð!Ñ œ !ß
V Ð! Ñ œ ÐVÐ!ÑÑ œ ! ß
V ÐÐ! Ñ Ñ œ ÐV Ð! ÑÑ œ Ð! Ñ , etc.
More precisely, if we let E œ ÖB − c À V ÐBÑ is defined×, then ! − E and
if B − E, then B − E  so, by P5), E œ c .
There are two important observations to make:
1) Different elements Bß C − c get assigned to different partners in
c  that is, if B Á C, then V ÐBÑ Á V ÐCÑ. To see this, we use induction.
Let E œ ÖB − c À aC Ð C Á B Ê V ÐCÑ Á V ÐBÑ Ñ×. We want to
see that E œ c Þ
! − E: To see this, we need to check that if C Á !, then
V ÐCÑ Á V Ð!Ñ œ !Þ In other words, we have to check
that a nonzero C in c gets a nonzero partner in c .
Since C Á !, then (by Corollary 2) C œ D  for
some D − c . Therefore V ÐCÑ œ VÐD  Ñ
œ ÐV ÐDÑÑ Þ That means that V ÐCÑ has a
predecessor V ÐDÑ in c . But ! has no predecessor
in c (by P3), so V ÐCÑ Á !.
If B − E, we must show that B − E, that is: we must
show that if C Á B+ , then V ÐCÑ Á V ÐB Ñ. We do this by
showing the contrapositive: if V ÐCÑ œ V ÐB Ñ, then
C œ B Þ
Suppose V ÐCÑ œ V ÐB ÑÞ Since B Á ! (by P3),
V ÐB Ñ Á V Ð!Ñ (since ! − E) so C Á !. Therefore
C has a predecessor, say C œ D  Þ
Then ÐV ÐBÑÑ œ V ÐB Ñ œ VÐD  Ñ œ ÐV ÐDÑÑ .
By P4), we conclude that V ÐBÑ œ V ÐDÑÞ Since
B − E, this means that B œ DÞ But then
C œ D  œ B .
Therefore, by P5), E œ c Þ ñ
2) Every object in c acquires a partner from c . Again, we use induction.
Let E œ ÖB − c À B œ V ÐCÑ for some C − c ×Þ We need to show that
E œ cÞ
! − E because ! œ V Ð!Ñ
Suppose B − E. Then B œ V ÐCÑ for some C − c Þ
Therefore V ÐC Ñ œ ÐV ÐCÑÑ œ B . In other words, B
is partnered with C from c , so B − EÞ
By P5), E œ c .
Putting observations 1) and 2) together, the rule V gives an exact pairing, onefor-one (V is a “one-to-one, onto function”) between the all the objects in c and
all those in c . By the definition of V , the pairing respects the successor
operation work in the two systems:
VÐB Ñ œ V ÐBÑ
“the partner of the successor” œ “the successor of the partner”
More about a Peano System
We want to convince ourselves that a Peano system captures the essence of our informal
system =. Already, we have a “mental picture” and a few theorems which suggest that
the objects in a Peano system are arranged just like the whole numbers. We want to see
that we can define “addition,” “multiplication,” and “  ” between objects in a Peano
system and that, when we're done, the result acts just like =.
All Peano systems look alike, so let's begin by assigning some convenient names to the
objects in a Peano system. After all, needing to write things like ! becomes
tedious.
There are lots of possible ways to name things.
be:
Naming System
Naming System
Naming System
Naming System
!
œ
!
!
!
!
†
I
"
"
!
‡
II
"!
#
!
¿
III
""
$
For example, some possibilities could
!
ð
IV
"!!
%
!
€
V
"!"
&
!
ß
VI
""!
'
... etc.
...
...
...
The point is that there are lots of ways to invent names for !ß ! , ! ß Þ..etc. It's
important, here, to remember that however we decide to invent names for the objects in
the Peano system, the names themselves don't give us any new information. But,
keeping that in mind, we might as well use names are convenient and that remind us of
how we hope the system is going to work. So we'll use the intuitively familiar symbols
!ß "ß #ß $ß ÞÞÞ as in the fourth row of the table.
Caution For now, !ß "ß #... are now just “marks on paper”  the names
we're giving objects in the Peano system. There's no more reason to say “" plus #
œ $” than there is to say “† plus ‡ œ ¿” À both are just ways of saying (in
different naming systems) that “! plus ! œ ! ”  and in fact, the
statement “! plus ! œ ! ” has no meaning yet at all because we haven't
defined what “plus” means in a Peano system.
We cannot say # † # œ %, because (right now) that statement is just a new
way of writing ! † ! œ !  which, at the moment, has no meaning at
all, because we haven't even defined what it means to “multiply” objects in a
Peano System.
We can, however, use these new names now to record things that we do already know.
For example, the axioms for a Peano system now read:
P1) There is one special object named “!” in c
P2) For each object 8 − c , there is exactly one object in c called its successor
(and denoted 8 Ñ
P3) ! is not the successor of any object, that is, Ða8 − c Ñ 8 Á !
P4) Different objects have different successors, that is
a7, a8 − c Ð7 Á 8 Ê 7 Á 8 Ñ
P5) Suppose E © c Þ If ! − E and if
Ða8 − c Ñ Ð8 − E Ê 8 − EÑ
then E œ c .
Just because of how we named things, statements like these are true:
! œ " (i.e., " is the successor of !) ß
! œ " œ #ß
% œ &
If we had decided instead to use the naming system in the first row of the table, the
following would be true:
œ œ †
œ œ † œ ‡
ð œ €
The theorems we already proved, with the new naming system, can now be written:
Theorem 1 For all 8 − c , either 8 œ ! or 8 œ 7 for some 7 − c
Corollary 2 If 8 − c and 8 Á !, then 8 œ 7 for a unique 7 − c .
Theorem 3 For all 8 − c , 8 Á 8
Theorem 4 If 8 − c , then 8 œ ! or 8 can be obtained from ! by applying the
successor operation finitely often.
Corollary 5 If 7ß 8 − c and 7 Á 8, then one of 7 and 8 can be obtained from
the other by applying the successor operation finitely often.
Defining Arithmetic in a Peano System
Addition
Let c be a Peano system Ðin which we have named the elements !ß "ß #ß ÞÞÞÑ.
First, we want to define addition: what does 7  8 mean? For any given 7 in c , the
definition tells (using P5, the induction axiom) what it means to “add 8, on the right, to
7.”
Definition A Suppose 7 − c . Define
i) 7  ! œ 7 and
ii) a8 − c ß Ð7  8 Ñ œ Ð7  8Ñ
For any given 7, we can use P5) to show that 7  8 has been defined for every 8:
Suppose 7 − c Þ Let E œ Ö8 − T À 7  8 is defined×Þ
By i), ! − E.
If 8 − Eß then 7  8 is defined. So then 7  8 is also defined
because ii) defines 7  8 as the successor of 7  8 in c .
Therefore 8 − EÞ
By P5), E œ c . ñ
Example
Suppose 7 − c .
Then
7!œ7
(by definition Ai)
7  " œ Ð7  ! Ñ because “"” is the name we assigned to !
œ Ð7  !Ñ by Definition Aii
œ 7
by Definition Ai
(Note: thus, the result of our definition of addition is that
“find the successor of 7” is the same as “find 7  ".” )
In the particular case where 7 œ !ß the preceding calculations
show that
!!œ!
!  " œ ! œ "
!  # œ Ð!  "Ñ œ " œ #
If 7 œ ", the preceding calculation shows that "  " œ " œ #
( œ the name we assigned to " ). Similarly
.
2  " œ # œ $
$  " œ $ œ % ,
etc.
ÐBy convention, let's agree that we may also write 7 for Ð7 Ñ Ñ
7  # œ Ð7  " Ñ because “#” is the name we assigned to “" ”
œ Ð7  "Ñ by Definition Aii
œ Ð7 Ñ
by the preceding example
Letting 7 œ "ß #ß Þ.. gives the specific facts
"  # œ " œ # œ $
#  " œ # œ $ œ %
etc.
Similarly, for any 7ß 8, we can reduce and work out the sum
7  8 with enough patience. For example, that &  % œ * Ðgive a
justification for each stepÑ:
&  % œ &  $ œ Ð&  $Ñ œ Ð&  # Ñ œ Ð&  #Ñ
œ Ð&  " Ñ œ Ð&  "Ñ œ Ð&  ! Ñ
œ Ð&  !Ñ œ & œ ' œ ( œ ) œ *
From the definition of addition (Ai), we know that 7  ! œ 7 for any 7 − c Þ BUT
that doesn't mean that we can say !  7 œ 7  because we haven't proved that
addition, as we defined it in c , is commutative. The next theorem is a first step in that
direction.
Theorem 6 Ða8 − c Ñ !  8 œ 8 œ 8  !
Proof Let 8 − Þ We know 8  ! œ 8 by the Definition Ai). What we still
need to prove is that Ða8 − c Ñ !  8 œ 8.
Let E œ Ö8 − c À !  8 œ 8×Þ
Definition Ai) says that, for any 7, 7  ! œ 7. In particular if
7 œ !, then !  ! œ !Þ So 8 œ ! − EÞ
Suppose, for some 8, that 8 − E. Then
!  8 œ Ð!  8Ñ
(by Definition Aii, with 7 œ !Ñ
œ 8
Ðsince 8 − EÑ
Therefore 8 − EÞ
By the induction axiom P5), E œ c . ñ
To prove that addition is commutative and associative, we need first to prove a lemma.
Lemma 7 Ða7 − c ÑÐa8 − c Ñ 7  8 œ Ð7  8Ñ œ 7  8
Proof We already know that Ð7  8Ñ œ 7  8 by Definition Aii). What we still
need to show is that Ða7 − c ÑÐa8 − c Ñ 7  8 œ Ð7  8Ñ
Let 7 − c . We need to show that Ða8 − c Ñ 7  8 œ Ð7  8Ñ
Define E œ Ö8 − c À Ð7  8Ñ œ 7  8×. We want to show that E œ c Þ
Ð7  !Ñ œ 7 œ 7  ! (using Definition Ai), so ! − E
Suppose for some 8, that 8 − EÞ We need to show 8 − E  that is, we need to
show that Ð7  8 Ñ œ 7  8 À
Ð7  8 Ñ œ Ð7  8Ñ
œ Ð Ð7  8Ñ Ñ
œ Ð 7  8 Ñ 
ÐDefinition Aii)
(because 8 − E)
(Definition Aii)
By P5), E œ c Þ ñ
Theorem 8 a) Ða7 − c ÑÐa8 − c ÑÐa: − c Ñ 7  Ð8  :Ñ œ Ð7  8Ñ  :
(Eddition is associative.)
b) Ða7 − c ÑÐa8 − c Ñ 7  8 œ 8  7
(Eddition is commutative.)
Proof a) Suppose 7ß 8 − c Þ We need to show that
Ða: − c Ñ 7  Ð8  :Ñ œ Ð7  8Ñ  :
Let E œ Ö: − c À 7  Ð8  :Ñ œ Ð7  8Ñ  :×Þ We want to show that E œ c .
0 − E: 7  Ð8  !Ñ œ 7  8
œ Ð7  8Ñ  !
Ðby Definition Ai)
Ðby Definition AiÑ, again)
Suppose, for some :, that : − EÞ We show that : must be in EÞ
7  Ð8  : Ñ œ 7  Ð8  :Ñ
(by Definition Aii)
œ Ð7  Ð8  :ÑÑ (by Definition Aii, again)
œ ÐÐ7  8Ñ  :Ñ Ðbecause : − EÑ
œ Ð7  8Ñ  :
(by Definition Aii, again)

Therefore : − EÞ
By P5), E œ c Þ ñ
b) Suppose 7 − c Þ We must show that Ða8 − c Ñ 7  8 œ 8  7.
Let E œ Ö8 À 7  8 œ 8  7×.
7  ! œ 7, by Definition Ai, and we proved in Theorem 6 that
7 œ !  7. Therefore ! − EÞ
Suppose, for some 8, that 8 − E. We show that 8 must be in E.
7  8 œ Ð7  8Ñ
œ Ð8  7Ñ
œ 8  7
Therefore 8 − E.
(by Definition Aii)
(because 8 − E)
(by Lemma 7)
By P5), E œ c Þ ñ
Because addition is associative, we often write things like 7  8  : without
parentheses, because it doesn't matter whether we interpret this as meaning
Ð7  8Ñ  : or 7  Ð8  :ÑÞ
Summary: We have defined addition Ð  Ñ in c . We have proved the necessary theorems
to compute 7  8 for any 7ß 8 − c . Addition turned out to be commutative and
associative, and to have a “neutral” element, ! À 7  ! œ !  7 œ 7 for all 7 − c Þ
In other words (so far as we can see, anyway) c , with addition, behaves exactly like =,
with addition.
In =, we can also multiply. So now we hope to define a multiplication
operation in c that behaves just like multiplication in =.
Multiplication
We also want to define multiplication in c . We do that using addition and the successor
operation. Then we need to look at some theorems about multiplication behaves in c and
how multiplication is connected to addition.
We could try making a definition like
“7 † 8 means the result of adding 7 to itself 8 times.”
But this is an inconvenient way to put it because it doesn't give us a precise formula “
7 † 8 œ ... ” to work with: so what do we do?
We stop and look for motivation. Think about how multiplication works in the
informal system =. In =, 7 † ! œ !, and, if you already know how to find 7 † 8,
there is a formula telling you how to find 7 † Ð8  "Ñ, namely
7 † Ð8  "Ñ œ 7 † 8  7
We use this fact about the informal system =, to inspire our definition of
multiplication in the formal system c . Of course, this makes it likely that
multiplication in c will, in fact, act like multiplication in the informal system, =.
And that's what we want. We are trying to show how to create, from very simple
assumptions, a formal system c that acts like =, so we “build in” what we need to
make the finished product be what we want it to be.
Definition M Suppose 7 − c . We define
i) 7 † ! œ ! and
ii) for any 8 − c , 7 † 8 œ 7 † 8  7
(Sometimes we will just write “78” for “7 † 8Þ”)
Exercise: Suppose 7 − c . Verify (just as we did for addition) that 7 † 8 is defined for
all 8 − c .
Example For any 7 − c ,
7 † " œ 7 † ! œ 7 † !  7 œ !  7 œ 7
7 † # œ 7 † " œ 7 † "  7 œ 7  7
7 † $ œ 7 † # œ 7 † "  7 œ Ð7  7Ñ  7
etc.
For example, $ † " œ $
$ † # œ $  $ œ ' (using earlier work on addition)
% † 3 œ % † # œ % † #  % œ % † "   %
œ Ð% † "  %Ñ  %
œ Ð% † !  %Ñ  % œ ÐÐ% † !  %Ñ  %Ñ  %
œ ÐÐ!  %Ñ  %Ñ  % œ Ð%  %Ñ  %
œ (using all the operations for computing sums)...
œ )  % œ ÞÞÞÞ œ "#
The next lemma gives a useful variation on the equations in Definition M. It is an
analogue (for multiplication) of Lemma 7 (about addition).
Lemma 9 For all 7ß 8 − c ,
a) ! † 7 œ ! œ 7 † !
b) 7 † 8 œ 7 † 8  8
Proof Suppose 7 − c
a) ! œ 7 † ! by Definition Mi). What we need to prove is that
!†7œ!
Let E œ Ö7 − c À ! † 7 œ !×
! − E because ! † ! œ !
(by Definition Mi)
Suppose 7 − EÞ We will show that 7 − EÞ
! † 7 œ ! † 7  !
œ!!
œ!
Ðby Definition MiiÑ
since 7 − E
(by Definition Ai)
Therefore 7 − EÞ
By P5), E œ c .
ñ
b) Let E œ Ö8 − c À 7 † 8 œ 7 † 8  8×
! − E, because 7 † ! œ !
(by Definition Mi)
œ7†!
(by Definition Mi), again)
œ 7 † !  ! (by Definition Ai)
Suppose 8 − E.
We show that 8 − EÞ To do this, we need to show
that
7 † 8 œ 7 † 8  8 Þ
7 † 8 œ 7 † 8  7
œ Ð7 † 8  8Ñ  7
œ 7 † 8  Ð8  7 Ñ
œ 7 † 8  Ð8  7Ñ
œ 7 † 8  Ð7  8Ñ
œ 7 † 8  Ð7  8 Ñ
œ Ð7 † 8  7Ñ  8
œ 7 † 8  8
Therefore 8 − EÞ
By P5, E œ c .
(by Definition Mii)
(because 8 − E)
(by Theorem 8: addition is
associative)
(by Definition Aii)
(by Theorem 8; addition is
commutative)
(by Definition Aii)
(by Theorem 8: addition is
associative)
(by Definition Mii)
ñ
We can now prove a connection between addition and multiplication (the distributive
rule) and see that multiplication is associative and commutative. For convenience, we
agree to write 78 for 7 † 8.
Theorem 10 Ða7 − c ÑÐa8 − c ÑÐa: − c Ñ
a) 7Ð8  :Ñ œ 78  7:
b) 7Ð8:Ñ œ Ð78Ñ:
c) 78 œ 87
Ð † and  are connected by the
distributive rule Ñ
(Multiplication is associative.Ñ
(Multiplication is commutative.Ñ
Proof a) The proof of 1) is an assigned problem in Homework 6. We assume 1) in the
arguments below.
b) Suppose 7ß 8 − c . We need to show that Ða: − c Ñ 7Ð8:Ñ œ Ð78Ñ:
Let E œ Ö: − c À 7Ð8:Ñ œ Ð78Ñ: ×Þ
! − E since 7Ð8 † !Ñ œ 7 † !
œ!
œ Ð78Ñ † !,
We want to show E œ c .
(by Definition Mi)
(by Definition Mi, again)
(by Definition Mi, again)
Suppose, for some :, that : − EÞ Then
7Ð8: Ñ œ 7Ð8:  8Ñ
œ 7Ð8:Ñ  78
(by Definition Mii)
(by part 1 of this theorem: the
distributive rule)
(because : − E)
(by Definition Mii)
œ Ð78Ñ:  78
œ Ð78Ñ:
Therefore : − EÞ
By P5, E œ c . ñ
c) Suppose 7 − c . We need to show that Ða8 − c Ñ 78 œ 87
Let E œ Ö8 − c À 78 œ 87 ×Þ We want to show E œ c .
! − E because 7 † ! œ ! œ ! † 7
(by Lemma 9)
Suppose, for some 8, that 8 − EÞ Then
78 œ 78  7
œ 87  7
œ 8 7
Therefore 8 − EÞ
By P5, E œ c
(by Definition Mii)
(because 8 − EÑ
(by Lemma 9)
ñ
Because multiplication is associative, we often write things like 78: without
parentheses, because it doesn't matter whether we mean Ð78Ñ: or 7Ð8:ÑÞ
Example We proved earlier that (for example) 7 † $ œ Ð7  7Ñ  7Þ By the
commutative law for multiplication, we can now say that 7 † $ œ Ð7  7Ñ  7
œ $ † 7Þ We could also get this fact from addition and the distributive law:
Ð7  7Ñ  7 œ Ð7 † #Ñ  7 œ 7Ð#  "Ñ œ 7 † $Þ
In the proofs that follow, we will now use the definitions of  and † more freely
(without always citing an explicit justification for each and every step). We will also
freely use that multiplication are associative and commutative, and that the distributive
law is true in c . In some arguments, such as the proof of part c) of the following
theorem, we use of results previously proven and don't need to use an induction in the
argument.
Theorem 11 Suppose 7ß 8ß - − c Þ
a) If 7 Á !, then 7  8 Á !Þ
b) (Cancellation for  Ñ If 7  - œ 8  - , then 7 œ 8Þ
(If -  7 œ -  8ß then 7  - œ 8  -ß =9 7 œ 8Þ The theorem
tells us that we can “cancel - on the left” ß tooÞ)
c) If 7 Á ! and 8 Á !, then 78 Á !Þ
Note: We already proved in Theorem 8b) that addition is commutative. Therefore it
doesn't matter here, in part a), if we write “7  8” or “8  7”: 11a) says that if an
object is not !, then its sum with any other object (in either order) is not !.
Suppose 7  8 œ !Þ What can we conclude in c ?
Proof a) Suppose 7 Á !Þ Let E œ Ö8 − c À 7  8 Á !×Þ
! − E because 7  ! œ 7 Á !.
Suppose, for some 8, that 8 − EÞ Then 7  8 œ Ð7  8Ñ Á ! (using P3).
Therefore 8 − E. By P5, E œ c . ñ
b) This proof is an assigned exercise in Homework 6.
c) Suppose 7 Á ! and 8 Á !Þ We know that 8 œ 5  for some 5 (by Theorem 1),
so 78 œ 75  œ 75  7Þ Since 7 Á !, we conclude that 78 Á ! (using part
a) of this theorem). ñ
(Note: Part c) is done without using induction (P5). However, the proof uses other
results (such as Theorem 1) that were proved using the induction axiom P5.
Example For short, we can agree to write “8# ” for “8 † 8”, 8$ for “Ð8 † 8Ñ † 8”, etc.
Show that Ð8  "ÑÐ8  #Ñ œ 8#  $8  #Þ (Justify each step! Be sure that each
“arithmetic calculation” is one that we justified.)
Ð8  "ÑÐ8  #Ñ œ ÐÐ8  "Ñ † 8Ñ  Ð8  "Ñ † # œ Ð8#  " † 8Ñ  # † Ð8  "Ñ
œ Ð8#  8Ñ  Ð#8  #Ñ œ Ð8#  Ð8  #8ÑÑ  #
œ Ð8#  8Ð"  #ÑÑ  #
œ Ð8#  8 † $Ñ  #
œ Ð8#  $8Ñ  #
œ 8#  $8  #
Note: On the surface, it appears that we have shown, without induction, that
Ða8 − c ) Ð8  "ÑÐ8  #Ñ œ 8#  $8  #
In fact, nearly every step in the calculations is justified by a theorem whose proof did use
induction.
The truth is that the proof of every statement of the form
Ða8 − c Ñ T Ð8Ñ
must depend on the induction axiom P5 (either in the proof itself, or in the proofs of
earlier theorems that are used in the proof).
Defining an “Order Relation” in a Peano System
Finally, we can also introduce an “order relation”, Ÿ , into c .
Definition Suppose 7ß 8 − c Þ We say 7 Ÿ 8 Ðequivalently, 8 7Ñ iff
Ðb- − c Ñ Ð7  - œ 8ÑÞ
We say 7  8 Ðequivalently, 8  7Ñ iff 7 Ÿ 8 and 7 Á 8Þ
Example For each 7 − c :
!  7 œ 7 so, by the definition, ! Ÿ 7
7  ! œ 7 so, by the definition, 7 Ÿ 7Þ
Theorem 12 For all 7 − c ß 8 − c ß : − c
a) 7 Ÿ 7
b) if 7 Ÿ 8 and 8 Ÿ :, then 7 Ÿ :Þ
c) if 7 Ÿ 8 and 8 Ÿ 7, then 7 œ 8Þ
Proof a) See the example, above.
b) If 7 Ÿ 8, there is a - such that 7  - œ 8Þ
If 8 Ÿ :, there is a . such that 8  . œ :Þ
Therefore : œ 8  . œ Ð7  -Ñ  . œ 7  Ð-  .Ñß
so 7 Ÿ :Þ
c) If 7 Ÿ 8, there is a - such that 7  - œ 8Þ
If 8 Ÿ 7, there is a . such that 8  . œ 7Þ
Since 7  - œ 8ß
Ð7  -Ñ  . œ 8  . œ 7ß so
7  Ð-  .Ñ œ 7 œ 7  !Þ Theorem 11bÑ lets us cancel the 7
and get
-  . œ !Þ But then, by Theorem 11a), - œ . œ !Þ
Therefore 8 œ 7  - œ 7  ! œ 7Þ ñ
Theorem 13 Ða7 − c Ñ Ða8 − c Ñ Ð 7 Ÿ 8 or 8 Ÿ 7Ñ
Proof Suppose 7 − c Þ We need to show that Ða8 − c Ñ Ð 7 Ÿ 8 or 8 Ÿ 7Ñ.
Let E œ Ö8 − c À 7 Ÿ 8 or 8 Ÿ 7 is true×. We will show that E œ c Þ
By the example above, 8 œ ! Ÿ 7, so ! − EÞ
Suppose, for some 8 − c , that 8 − EÞ ÐSince the statement used in defining E contains
“or”, there are two cases that follow.Ñ
i) If 7 Ÿ 8, then b- − c such that 7  - œ 8. In that case,
7  -  œ Ð7  -Ñ œ 8 so 7 Ÿ 8 and therefore 8 − EÞ
ii) If 8 Ÿ 7, b- − c such that 8  - œ 7Þ
If - œ !, then 8 œ 7ß so 8 œ 7 œ 7  ", so
7 Ÿ 8 and therefore 8 − EÞ
If - Á !, then - has a predecessor . in c À - œ .  .
In that case, 8  . œ 8  .  (by Lemma 7)
œ 8  - œ 7, so 8 Ÿ 7
and therefore 8 − EÞ
In all cases, 8 − EÞ
Therefore, by P5, E œ c Þ ñ
Note: Theorem 13 is the correct, rigorous version of the statement in Corollary 5  a
corollary that was stated in italics since its “proof” was questionable.
Corollary 14 a7a8 − c (7  8 or 7 œ 8 or 7  8Ñ
Proof By Theorem 13, we know 7 Ÿ 8 or 7 8Þ
If 7 Á 8, then (by definition of  Ñß 7  8 or 7  8. ñ
Theorem 15 (Cancellation for multiplication) If 7ß 8ß : − c and : Á ! and 7: œ 8:,
then 7 œ 8Þ (Since multiplication is commutative, if :7 œ :8ß then 7: œ 8:
=9 7 œ 8Þ The theorem tells us that we can cancel a nonzero : “on the left” too.)
Proof (Provide reasons where necessary). ) Suppose 7: œ 8:Þ By Theorem 13,
either there is a - − c such that 7  - œ 8, or there is a - − c such that 8  - œ 7.
If 7  - œ 8, then 7:  -: œ 8:. Since 7: œ 8:, we have 7:  -:
œ 7: œ 7:  !Þ By Theorem 11b), we can cancel “7:” and get -: œ !Þ Since
: Á !, we get - œ ! (by Theorem 11c) and therefore 7 œ 8Þ
If 8  - œ 7, the proof that 7 œ 8 is entirely similar. (You can just interchange
the 7's and the 8's in the preceding paragraph.) ñ
Finally, we need to check that the “order relation” Ÿ interacts nicely with addition and
multiplication (just as Ÿ , +ß  ) interact nicely in the informal system =.) For
example,
Theorem 16 Suppose 7ß 8ß : − c and that 7 Ÿ 8Þ Then
a) 7  : Ÿ 8  :
bÑ 7: Ÿ 8:Þ
Proof
a) Since 7 Ÿ 8, there is a - − c such that 7  - œ 8Þ Then
Ð7  -Ñ  : œ 8  :ß so
Ð7  :Ñ  - œ 8  :, so
7  : Ÿ 8  :Þ ñ
b) This is an assigned problem in the homework.
Looking Back and Looking Forward
Loosely speaking, a formal mathematical system refers to some objects (of unspecified
nature),some axioms that describe precisely how the objects behave, and the body of
definitions and theorems that grow out of the axioms. A Peano system is a formal
mathematical system.
An informal mathematical system is a not very precise, everyday term for how we view a
body of mathematical knowledge, often very well-developed, when we are not bothering
to think about its basis at the axiomatic level.
An example would be our informal system of whole numbers, =, together with all its
arithmetic. We have been taking the point of view that the system of whole numbers, =,
has not yet been carefully defined, even though it is a system that (somehow, informally)
we seem to know a lot about. We have, as yet, no answer for the question “Just what
exactly is the whole number system?”
We believe that the Peano Axioms P1-P5 are true in this informal system, =, if we
interpret the objects in c as whole numbers and interpret successor in c to mean “the
next whole number.” Our understanding  although we only have informal knowledge
about = to judge by  is that the system of whole numbers is a model for a formal Peano
system c .
Moreover we can start with a Peano system, define  , † , Ÿ in c , and prove theorems
about how these “abstract” operations behave. These theorems turn out (when
interpreted in the model =) to be in agreement with how we understand arithmetic in the
informal system, =.
For example, we proved that in c À 7  ! œ 7 œ !  7 and 7 † " œ 7 œ " † 7à that
addition and multiplication are commutative and associative; and that the distributive
law connects addition and multiplication. We also proved some other theorems about
Peano arithmetic, such as the cancellation laws for addition and multiplication. We
defined an order Ÿ in c which interacts nicely with addition and multiplicationÞ
Because mathematicians don't care (unless they are becoming philosophers) what the
whole number “really are”, and because a formal, abstract Peano system c seems to
perfectly mirror the informal system =, mathematicians are perfectly happy to accept as a
formal definition that = is some Peano system or another.
Since all Peano systems “look exactly alike,” it doesn't really matter much which
particular Peano system we define = to be. However, set theory is supposed to be the
foundation for mathematics, so it would be nice to define = to be some particular Peano
system whose objects are sets.
To do this, we would need to have:
i) a collection of sets, and
ii) an operation (called “successor” ) that, within this collection,
assigns to each set B a successor set B and does so in such a way that
the axioms P1-P5) are true.
This would be a Peano system whose objects are actually sets, and which (like any Peano
system) would behave just like the whole number system. We could then define  once
and for all, officially and formally  this particular Peano system to be the system = of
whole numbers. We would call the sets (objects) in = whole numbers.
Suppose this can be done.
i) We are not claiming to have proved that = is this particular Peano system  it
makes no sense to prove a definition. But the official, formal definition for =
does give us a system of sets which behaves, as best we can judge, just like the
informal system = that we started with.
ii) There might also be other ways in which someone could officially define = and
get a system that behaves in the right way. For philosophical or aesthetic reasons,
someone might prefer this other approach. But mathematically, the choice really
doesn't matter: how the whole numbers behave is all the that really counts
mathematically  so we can all manage to live with whatever particular formal
definition for = was chosen.
Getting a Peano system of sets
We want to actually construct a Peano system whose objects are sets in order to complete
the official definition of = in terms of set theory.
The notion of “successor” for sets came up earlier in a Homework Set 3. Here is the
definition again.
Definition For any set B, B œ B  ÖB×Þ The set B is called the successor of BÞ
Notice that
i) We can form the successor of any set whatsoever. For example,
Ö+ß ,× œ Ö+ß ,×  Ö Ö+ß ,×× œ Ö+ß ,ß Ö+ß ,××, and
‘ œ ‘  Ö‘×
ii) Since B − B ß it is always true that B Á g.
iii) If C − Bß then C − B  ÖB× œ B . Therefore B © B is always true.
(Note: iii) tells us that “successor sets” B are rather special. If you pick
a set D “at random” and B − D , then it usually doesn't happen that B © D
is also true.
Example
g
g
œ g  Ög×
œ Ög×


g
œ Ög×
œ Ög×  ÖÖg××
g œ Ögß Ög×× œ Ögß Ög××  ÖÖgß Ög×××
ã
and so on.
œ Ögß Ög××
œ Ögß Ög×ß Ögß Ög××
Definition Suppose M is a collection (set) of sets. M is called inductive if
a) g − M , and
b) if B − Mß then B − MÞ
Then we ask: are there any inductive sets? Informally, it certainly looks like there are.
For example, we could form the set
Ögß g ß g ß g ß ÞÞÞ×
However, this “example” of an inductive set (although fine, informally) seems just a little
shaky if we're being very careful. After all, this set is only inductive because it is
described using a rather casual “etc.” Ðthat is, “ÞÞÞ” ÑÞ
At this point, we should take a brief look at the axioms for set theory itself. This is “as
deep as we ever dig” since set theory, we have agreed, is to be the very foundation for all
mathematics. In other words, all mathematics flows from these axioms.
Axiomatic (that is, formal) set theory starts with an abstract system of unspecified objects
(called sets) and a relation “ − ” among them. We know absolutely nothing about these
objects except that they behave in ways described by 10 axioms  almost always called
the ZFC axioms ( œ “Zermelo-Fraenkel-with Choice” axiomsÑÞ (Of course, these axioms
were chosen to create a formal system of objects that would behave, as best we can tell,
“just like” naive (informal) set theory.)
A careful study of the ZFC Axioms, and the theorems that can be proved from them, is a
whole field of study in itself, usually called “Axiomatic Set Theory.” In order to get the
flavor, here is a partial list of the axioms, omitting some of the more technical ones that
we don't need to think about at all for this course. (The quantifiers in these axioms apply
to the “universe” of sets  so ÐaBÑ means “for all sets B”, etc. Convince yourself that
the English translations given below are correct.)
Zermelo-Fraenkel-with-Choice (ZFC): Axioms for Set Theory
ZFC1 aB aC Ð aD (D − B Í D − CÑ Í B œ CÑÑ
“ Two sets are equal iff they have the same members. ”
ZFC2 bB aC ( C Â B)
“ There is a set with no members (an empty set). ”
If “two” sets both have no members, then they certainly have the same
members (none at all !) so, by ZFC1, they are the same set. In other
words, ZFC1 implies that there is only one empty set. For convenience
we can give it a name: g
ZFC3 aB aC ÐbD (a? Ð? − D Í Ð? œ B ” ? œ CÑÑÑ
“ If B and C are sets, then there is a set D œ ÖBß C×Þ”
ZFC4 aB bC ÐaD ÐD − C Í b, ÐD − , • , − B ÑÑÑ
“ For any set B, there is a set D consisting of the members of the
members of B.”
We can agree to give this set z a name: -B
ZFC& aB bC (aD (D − C Í ÐaA ÐA − D Ê A − BÑÑÑÑ
If we agree to define “D © B” to be shorthand for aA ÐA − D Ê A − BÑ
Then ZFC5 could be written aB bC ÐaD ÐD − C Í D © BÑÑ. That is,
axiom ZFC5 says “every set B has a power set.”
ZFC6 bM Ðg − M • (aC (C − M Ê C  − MÑ Ñ
There exists an inductive set.
plus 4 other more technical axioms (omitted ) À ZFC7, ZFC8,
ZFC10 (AC)
ZFC9,
The axiom ZFC10 is called the Axiom of Choice (AC). It arouses some
philosophical controversy among those mathematicians who worry about
foundations of mathematics, so a few mathematicians omit it from the
list. The 10-axiom system ZFC is the axiom system most
mathematicians would use for set theory (and therefore for all mathematics).
If AC is omitted, then ZFC1-ZFC9 are referred to as the “ZF” axioms.
We may say a little about the Axiom of Choice later in the course.
Although we have taken a naive (informal) approach to set theory, everything that we
have done (or will do) with sets can be justified by theorems provable from the ZFC
axioms.
Returning to the question we were asking: we are convinced (informally) that set theory
should contain an inductive set  for example, the one described informally by
Ögß g ß g ß g ß ÞÞÞ ×
In formal set theory, the existence of an inductive set (at least one, maybe more)
is guaranteed by axiom ZF6  it's built right in so that the formal ZFC system will
contain an object that our intuition expects to be there.
Definition I‡ Choose an inductive set M and.
define = œ ÖB − M À B − N for every inductive set N ×Þ
This set, =, together with the successor operation for sets, is going to turn out to
be a Peano system. And it is going to be the particular Peano system that we use
as the official definition for the system of whole numbers. For now, choosing to
call this set “=” is in anticipation of what's going to happen later. But for the
moment, it is nothing but called =, which just happens to have the same name as
the system of whole numbers.
By definition, = © M , but the definition tells us that, in fact, = is a subset of every
inductive set N (including, for example, N œ MÑÞ Therefore you could think of = as
= œ +ÖN À N is an inductive set×Þ
As a matter of fact, axiomatic set theory contains many inductive sets. But
i) The definition of = would make sense even if there were only one
inductive set, M .
ii) Convince yourself that if a different inductive set M w were used instead
of M in the definition, then the resulting set = w would the same would be the
same: = w œ =Þ
Although = is an intersection of inductive sets, we can't assume just assume that makes =
itself an induction set. But the next theorem confirms that, in fact, it is.
Theorem 17 = is an inductive set.
Proof a) g is in every inductive set N ß so g − =Þ
b) Suppose B − =. Then (by definition of =) B is a member of every inductive
set N Þ Therefore B is also a member of every inductive set N (by definition of
“inductive set”). Hence B − =Þ
Therefore = is inductive. ñ
Since = © N for every inductive set N , and because we now know that = itself is an
inductive set, we can now say that = is the smallest inductive set.
Since g − =; therefore g − =à therefore g − =à therefore g − =à and so on.
Caution: We are using the same notation B for “successor of a set B” as we
used for “successor” in a Peano system. But don't let the notation deceive: we
have no right to assume that the successor operation for sets obeys the rules
axioms P1-P5. We need to check whether that is true.
(It does turn out to be true; when we have proved that, then we will know that the
set =, with the set successor operation, is a Peano system.)
Definition ! œ g
(We are simply agreeing that ! will be another name for g.)
Since g − =, we now have
P1: There is a special object in = named !.
( ! is the set gÞ )
Because = is inductive, the successor set B for each set B in = is also in =. Therefore
P2: For every object (set) B − =, there is a successor B in =.
We remarked earlier that B Á g for every set B. Therefore
P3: For all B − =, B Á ! Ð œ gÑ
Suppose E © =, and suppose that
i) ! Ð œ gÑ − E, and
ii) ÐaB − =Ñ ÐB − E Ê B − EÑ
then E, by definition, is an inductive set. But = is the smallest inductive set, so = © EÞ
Therefore E œ =Þ This shows that P5 holds in =.
P5: Suppose E © = À
i) if ! Ð œ gÑ − E, and
ii) if ÐaB − =Ñ ÐB − E Ê B − EÑ
then E œ =Þ
To show that =, with the set successor operation, is a Peano system, we now only need to
prove P4: if Bß C − = and B Á Cß then B Á C  Þ That takes a little more work.
We will be using another definition introduced earlier (in Homework Set 4).
Definition A set 5 is called transitive iff ÐaBÑÐ aCÑ ÐB − C − 5 Ê B − 5Ñ
(Less formally, B is transitive if “every member of a member of 5 is a member of 5 .”
For example Ögß Ög×ß Ögß Ög× ×× is transitive but Ög, ÖÖg×× × is not transitive.
Theorem 18 For any set E, the following are equivalent:
i) E is transitive
ii) -E © E
iii) + − E Ê + © E
iv) E © c ÐEÑ
Proof This theorem was an exercise in homework. See the Homework 4 Solutions
online if you're uncertain about the proof. ñ
Theorem 19 If 5 is a transitive set, then 5 œ -Ð5  ÑÞ
Remember: -Ð5  Ñ just means the “set of all members of the sets that are in the
collection 5  .”
Proof i) Suppose B − -Ð5  Ñ. Then B − D , where D − 5  , that is, where D − 5  Ö5×.
if D − 5 , then B − D − 5 so B − 5 because 5 is assumed to be transitive
if D − Ö5×, then D œ 5 so B − 5
Either way, we have D − 5 . Therefore -Ð5  Ñ © 5 .
ii) Suppose B − 5 . Since 5 − 5  ß B − -Ð5  ÑÞ Therefore 5 © -Ð5  ÑÞ
Hence -Ð5  Ñ œ 5 . ñ
Alternate Proof (a little slicker: think about each step) -Ð5  Ñ œ - Ð5  Ö5×Ñ œ - 5
 - Ö5× œ -5  5 . Since 5 is transitive, -5 © 5 (by Theorem 18, with E œ 5 ).
Therefore -5  5 œ 5Þ ñ
The next theorem gives us lots of examples of transitive sets.
Theorem 20 If 5 − =, then 5 is transitive.
Proof We use the fact the P5 is true in the set =. Let E œ Ö5 − = À 5 is transitive×Þ
i) ! Ð œ gÑ is transitive, so ! − EÞ
ii) Assume 5 − E, We will show that 5  − E  that is, we will show that
5  is also a transitive set.
Since 5 is transitive, Theorem 19 says that -Ð5  Ñ œ 5 . Since 5 © 5 
any set 5 , transitive or not), we have -Ð5  Ñ œ 5 © 5  Þ
(for
transitive
Since -Ð5  Ñ © 5  , Theorem 18 Ðwith E œ 5  Ñ says that 5  is a
set. Therefore 5  − EÞ
By P5, E œ =Þ ñ
Now we can finally show that the remaining Peano axiom, P4, is true in the set =.
Theorem 21 Suppose Bß C − =Þ If B Á C , then B Á C  Þ
Proof (We prove the contrapositive.) Since B and C are in =, they are transitive (by
Theorem 20). Thus, if B œ C , then Theorem 19 gives us that B œ -B œ -C  œ C
ñ
We have now achieved the objective. The set =, as given in Definition I‡ , with the set
successor operation, is a Peano system.
Definition = The Peano system = is called the set of whole numbers. The
members of = are called whole numbers.
Names for the whole numbers
We can name the objects in = using the same system we used for any abstract Peano
system:
Member of =
Name
(set)
g
g œ Ög×
g œ {g} œ Ögß Ög××
g œ Ögß Ög×× œ Ögß Ög×ß Ögß Ög×××
ã
!
"
#
$
Thus, in our official definition of =, the whole numbers !ß "ß #ß $ÞÞÞ as really being sets
(or, more precisely, names of sets).
Some interesting (amusing?) observations show up in the list above:
i)
!œg
" œ Ög× œ Ö!×
# œ Ögß Ög×× œ Ö!ß "×
$ œ Ögß Ög×ß Ögß Ög××× œ Ö!ß "ß #×
The pattern suggests a theorem (which we won't take the time to prove):
every whole number is the set of preceding whole numbers !
ii) g − Ög× − Ögß Ög×× − Ögß Ög×ß Ögß Ög××× − ..., in other words
! − " −
#
−
Also, for example, 0 − 3,
$
− ÞÞ
2−$
If you continued the list of successors and names, you would keep
observing that
7  8 Ðas defined in a Peano system) Í 7 − 8
That is also a theorem that can be proved.
Definition 8 is called a natural number if 8 − = and 8 Á !Þ The set of natural
numbers  is defined to be the set =  Ö!×.
Therefore, natural numbers can also be thought of as sets.
Conclusion: Having done all this, the point is not that in the future you should always
be thinking of the whole numbers as sets. In fact, you ordinarily should think of the
whole numbers the way you always have.
The real points are that
1) There is a very small collection of axioms (P1-P5) from which
all aspects of the whole number system (including
arithmetic and rules for inequalities) can be carefully and
systematically proven, and that
2) the whole numbers, and their arithmetic, can be “built” from set
theory  in accordance with the view the sets should be a
foundation for everything we need in mathematics.
We will see soon that the set of integers can be built from the set of
whole numbers (sets)  so each integer will be a set. The set of
rationals from the set of integers (sets)  so each rational number
will be a set; and so on.