Definition The natural or counting
numbers are the elements of ω.
We use the blackboard bold face letter N to re present the set of natural
numbers. As the definition indicates they are synonymous with the ordinal
number ω.
We note here that when the natural or counting numbers are developed
via the Peano postulates, the set of natural numbers begins with 1 and do not
include the number 0. The Whole numbers, indicated by W, are considered
to be the set of numbers that are the union of the natural numbers and
{0}. However in the set theoretic development of numbers it is much more
convenient to consider the natural numbers as the ordinal number ω, and not
specify any particular set as being whole numbers. (The Peano postulates
can be found in most elementary number theory texts, and also in Naive Set
Theory by Paul R. Halmos.)
Integers
The next set of numbers we develop we shall call Integer Numbers
and we will indicate this set by the bold face letter Z (From the German
word for counting, zahlen). For the purpose of brevity the integer numbers
are referred to as integers. The rationale for the development of this
set is
that we may wish to answer questions such as: What number added to 2 is
0? This can be expressed symbolically by x + 2 = 0. We realize of course
that 
, thus the question has a vacant answer
in the
natural numbers. We extend the natural numbers to a larger set in which
this question and other questions like it have non-vacuous answers.
We define an equivalence relation on the cartesian product of the natural
numbers with themselves, i.e., N × N, by
The rationale for this definition is that each ordered pair represents a
difference . Using our previous (to the study of set theory) concept of sub-
traction we see
a + d = b + c 
a - b = c - d.
We leave it to the reader to verify that this relation is an equivalence
relation. Also the reader should note and verify that this relation is not an
equivalence relation for α × α where α > ω.
Definition The integers are the collection of equivalence classes
of N×N
with respect to the equivalence relation
(a, b) ≡ (c, d) if a + d = b + c.
We will indicate the equivalence class of (a, b) by [a, b], that is
We now wish to define an order and an arithmetic for the integers . First
we need a pair of lemmas.
Lemma 7.1 
Proof Let 
and
let k ∈ N. If
k ≥ 2, then 
. Thus a + 1 = b + 1=> a = b,
and
a+k-1 = b+k-1 => a = b, for all a, b ∈ N. We now as sume a +k = b+k,
which implies a+k-1+1 = b+k-1+1, which implies a+k-1 = b+k-1,
which implies a = b. For k = 1 we have a+1 = b+1=> a = b by Theorem
2.8. For k = 0, we have a+0 = a and b+0 = b, thus a+0 = b+0=> a = b.
All three cases imply k ∈ A. Thus by transfinite induction we have the
desired result.
Lemma 7.2 a + n < b + n => a < b 
n ∈
N.
Proof: The proof is identical to the proof of the previous lemma by re-
placing "=" with "<", except possibly the case k = 1. For the case k = 1,
. If a 
b,
then there exists x ∈ a such
that x 
b. Since
we must have x = b. But either a ∈ b
or
a = b, which contradicts the axiom of regularity. We thus conclude a
b. If
a = b we also have the obvious contradiction to the axiom of regularity, thus
There is a natural order that may be defined on the integers.
Definition [a, b] < [c, d] iff a + d < b + c:
Since [a, b] and [c, d] represent equivalence classes, and the numbers a, b, c,
d
are specific values we must verify that the definition is valid regardless of
what
pair is chosen to represent each equivalence class. That is, we must show that
the ordering is well defined.
Theorem 7.3 The ordering of the integers is well defined.
Proof Let [a, b] < [c, d], [a, b] = [x, y], and [c,
d] = [z ,w]. We have
a + d < b + c, a + y = b + x, and c + w = d + z
=>a + d + x + w < b + c + x + w = a + d + y + z
=>
x + w < y + z
=> [x, y] < [z, w].
We define addition and multiplication of integers as fol lows
[a, b] + [c, d] 
= [a + c, b + d]
[a, b] · [c, d] = [ac + bd, ad + bc].
We now demonstrate that these operations are well defined.
Theorem 7.4 Addition and multiplication of integers are well defined.
Proof Let (a, b) ≡ (x, y), and (c, d) ≡ (z, w). We thus have
a + c + y + w = b + d + x + z
=> (a + c, b + d) ≡ (x + z, y + w)
=> [a, b] + [c, d] = [x, y] + [z, w]:
Hence addition is well defined.
For multiplication we have
a + y = b + x and c + w = d + z
=> ac + cy = bc + cx, bd + dx = ad + dy,
xw + cx = dx + xz and yz + dy = cy + yw
=> ac + bd + xw + yz + cy + dx + cx + dy =
ad + bc + xz + yw + cy + dx + cx + dy
=> ac + bd + xw + yz = ad + bc + xz + yw
=> (ac + bd, ad + bc) ≡ (xz + yw, xw + yz)
=> [a, b] · [c, d] = [x, y] · [z,w].
Hence multiplication is well defined.
We must now develop the usual properties of the integers .
