Peano axioms

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The arithmetic of Peano defines the natural numbers.

Peano's axioms contain the notion of a set. They define the (non-negative) integers as:

1.	0 is a natural number
2.	Every integer has a unique successor.
3.	There is an integer which is not a successor of any integer.
4.	Two distict integers cannot have the same successor.
5.	If M is a set of integers such that 0 is in M and such that if an integer n is in M then its successor is in M, then every integer is in M.

The Italian mathematician Giuseppe Peano has contributed much to the number theory.