Peano's Axioms
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' not equal 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.
If we define the successor:
x' = x + 1,
Then since 1 exists, we let x = 1,
x' = (1) + 1
x' = 2
The successor of 1 is 2.
Since 2 exists, we let x = 2,
x' = (2) + 1
x' = 3
The successor of 2 is 3.
Since 3 exists, we let x = 3,
and so on ... until we reach 227.
Therefore if 1 exists, 227 also exists by the proof of induction.