In this chapter our purpose is to establish all the familiar properties of the natural numbers (= positive integers) and to obtain the ring of integers by starting from the five axioms of Peano. We also demonstrate that the five axioms are equivalent to the axioms of an ordered integral domain whose positive elements are well-ordered. Either of these sets of axioms determines a unique ring (up to isomorphism) called the ring of integers.

Peano's axioms

The traditional method of describing the set N of natural numbers axiomatically is by means of the following axioms of Peano:

1. (i) 1 ∈ N.

2. (ii) For each a ∈ N there exists a unique a′ ∈ N called the successor of a. (In other words there exists a map a ↦ a′ of N into itself, called the successor map.)

3. (iii) a′ ≠ 1 for any a ∈ N.

4. (iv) For all a,b ∈ N, a′ = b′ ⇒ a = b. (In other words, the successor map a ↦ a′ of N into itself is 1-1.)

5. (v) Let S be a subset of N such that (a) 1 ∈ S; (b) if a ∈ S, then a′ ∈ S. Then S = N.

The fifth axiom is called the axiom of induction or the first principle of induction and is the basis of the proofs of many theorems in mathematics.

We write 1′ = 2, 2′ = 3, 3′ = 4, and so on.