Peano axioms

The Peano axioms are a set of axioms that formally describes the natural numbers (0, 1, 2, 3 ...). Together, they describe some of the most important properties of the natural numbers: their infinitude, zero as the smallest natural number and the rule of induction. They were proposed by the Italian mathematician Giuseppe Peano in 1889.

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 n 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.