Peano's Axioms

Have you ever wondered about how the Natural Numbers (1, 2, 3, ...) are defined mathematically?

Well, no you probably haven't. Most people don't.

After all, the Natural Numbers are...um...natural. They're a fundamental part of how we think about numbers. But believe it or not their official definition (originally stated by Peano as a set of axioms) is fairly complex!

The following is quoted from Edmund Landau, Foundations of Analysis, Chelsea, 1951, pp. 1-18 (via http://www.ms.uky.edu/~lee/ma502/notes2/node7.html). 

We assume the following to be given:

A set (i.e. totality) of objects called natural numbers, possessing the properties--called axioms--to be listed below.

Before formulating the axioms we make some remarks about the symbols = and ≠ which will be used.

Unless otherwise specified, small italic letters will stand for natural numbers...

If x is given and y is given, then:
either x and y are the same number; this may be written x=y (to be read ``equals'');
or x and y are not the same number; this may be written x≠y ( to be read ``is not equal to'').
Accordingly, the following are true on purely logical grounds:

1. x=x for every x.
2. If x=y then y=x.
3. If x=y, y=z then x=z.

Thus a statement such as a=b=c=d, which on the face of it means merely that a=b, b=c, c=d, contains the additional information that, say, a=c, a=d, b=d.... Now, we assume that the set of all natural numbers has the following properties:

Axiom 1:

1 is a natural number. That is, our set is not empty; it contains an object called 1 (read ``one'').

Axiom 2:

For each x there exists exactly one natural number, called the successor of x, which will be denoted by x'.

Axiom 3:

We always have x' ≠ 1 . That is, there exists no number whose successor is 1. That is, there exists no number whose successor is 1.

Axiom 4:

If x'=y' then x=y. That is, for any given number there exists either no number or exactly one number whose successor is the given number.

Axiom 5 (Axiom of Induction):

Let there be given a set M of natural numbers, with the following properties:

I.

1 belongs to M.

II.

If x belongs to M then so does x'.

Then M contains all the natural numbers.