Peano axioms

Peano axioms, in number theory, five axioms introduced in 1889 by Italian mathematician Giuseppe Peano. Like the axioms for geometry devised by Greek mathematician Euclid (c. 300 bce), the Peano axioms were meant to provide a rigorous foundation for the natural numbers (0, 1, 2, 3,…) used in arithmetic, number theory, and set theory. In particular, the Peano axioms enable an infinite set to be generated by a finite set of symbols and rules.

The five Peano axioms are:

1. Zero is a natural number.
2. Every natural number has a successor in the natural numbers.
3. Zero is not the successor of any natural number.
4. If the successor of two natural numbers is the same, then the two original numbers are the same.
5. If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.

The fifth axiom is known as the principle of induction because it can be used to establish properties for an infinite number of cases without having to give an infinite number of proofs. In particular, given that P is a property and zero has P and that whenever a natural number has P its successor also has P, it follows that all natural numbers have P.