10.1 AN AXlOMATlZATlON FOR THE SYSTEM OF POSITIVE INTEGERS 331and yields ultimately to the existence of a complete ordered field (ie., the
reals) as a theorem.
As we begin this development, we should be mindful of what must be
expected from any successful axiomatization of N; the 11 field axioms (recall
Definition 2, Article 9.1) are a helpful guide in this direction. Of these
axioms, the system of positive integers, as we know it intuitively, should
satisfy all except three: the additive identity, the additive inverse, and the
multiplicative inverse axioms (i.e., field axioms 4, 5, and 10). Thus our ini-
tial assumptions must be of a character that we are able to derive from them
as theorems such familiar arithmetic properties as commutativity of addi-
tion and distributivity of multiplication over addition and, beyond these,
properties such as multiplicative cancellation. There are also familiar prop-
erties of N involving order that must be derivable as theorems if our
axiomatization is to be satisfactory.
As indicated earlier, our point of departure is a single two-part axiom.
AXIOM 1
There exists a set N and a mapping a: N -+ N satisfying:(a) a is one to one, but not onto. In particular, there exists an element of N,
which we denote by the symbol 1, such that 1 4 im (a).
(b) If S is a subset of N satisfying the properties:
(I) 1 E S, and
(11) for all m E N, if m E S, then o(m) E S,
then S = N.The mapping m + o(m) from (a) of the axiom is called a successor func-
tion and may be thought of (in the interest of developing a sense of familiarity
with the set N) as sending any positive integer to "the next" positive integer;
for this reason, the image o(m) of an element m E N is called the successor
of m. We refrain at this stage from writing a(m) = m + 1, because there is
not as yet any operation of addition available to us. Condition (b) of Axiom
1 is known as the induction postulate and should remind you of the principle
of mathematical induction, the basis of our work in Article 5.4.
Let us begin bx noting, for the record, several immediate consequences
of Axiom 1. The set N is nonempty since 1 E N; in fact, N must be infinite
since a is a one-to-one mapping of N into a proper subset of itself, namely,
N - (1) (recall Definition 2, Article 8.3). The stipulation that a is a one-
to-one mapping means of course that if m, n E N with o(m) = o(n), then
m = n. This was the fourth original postulate of Peano. The requirement
that 1 4 im (a) means that 1 is not the successor of any element of N.
No doubt you have already noticed the absence of algebraic operations
from the definition of N. Clearly a major obstacle to the plan to prove as
theorems the familiar arithmetic properties of N is the need to create from