Peano axioms

The Peano axioms are a set of axioms that formally describes the natural numbers (0, 1, 2, 3 ...). They were proposed by the Italian mathematician Giuseppe Peano in 1889. They consist of a few basic — and intuitively obvious — properties that, however, are sufficient to define the natural numbers:

There is a smallest natural number (either 0 or 1), starting from which all natural numbers can be reached by moving finitely often to the "next" number (obtained by adding 1).

The axioms

Today the Peano axioms are usually formulated as follows:

1. Zero is a natural number.
2. Every natural number has a unique successor that also is a natural number.
3. Zero is not the successor of any natural number.
4. Different natural numbers have different successors.
5. If it is true that

(a) Zero has property P, and

(b) if any given natural number has property P then its successor also has property P

then all natural numbers have property P.

The last axiom is called the axiom (or rule) of induction.