Matematica

CONSTRUCTION OF NUMBER SYSTEMS

N. MOHAN KUMAR

1. Peano's Axioms and Natural Numbers

We start with the axioms of Peano.

Peano's Axioms. N is a set with the following properties.

(1) N has a distinguished element which we call `1'.

(2) There exists a distinguished set map  : N ! N.

(3)  is one-to-one (injective).

(4) There does not exist an element n 2 N such that (n) = 1. (So,

in particular  is not surjective).

(5) (Principle of Induction) Let S  N such that a) 1 2 S and b)

if n 2 S, then (n) 2 S. Then S = N.

We call such a set N to be the set of natural numbers and elements

of this set to be natural numbers.

Lemma 1.1. If n 2 N and n 6= 1, then there exists m 2 N such that

(m) = n.

Proof. Consider the subset S of N dened as,

S = fn 2 N j n = 1 or n = (m); for somem 2 Ng:

By denition, 1 2 S. If n 2 S, clearly (n) 2 S, again by denition

of S. Thus by the Principle of Induction, we see that S = N. This

proves the lemma. 

We dene the operation of addition (denoted by +) by the following

two recursive rules.

(1) For all n 2 N, n + 1 = (n).

(2) For any n;m 2 N, n + (m) = (n + m).

Notice that by lemma 1.1, any natural number is either 1 or of the

form (m) for some m 2 N and thus the dention of addition above

does dene it for any two natural numbers n;m.

Similarly we dene multiplication on N (denoted by
, or sometimes

by just writing letters adjacent to each other, as usual) by the following

two recursive rules.

(1) For all n 2 N, n
1 = n.

1

...