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).
Today the Peano axioms are usually formulated as follows:
- Zero is a natural number.
- Every natural number has a unique successor that also is a natural number.
- Zero is not the successor of any natural number.
- Different natural numbers have different successors.
- 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.