Peano axioms

The Peano axioms are a set of formal axioms describing 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.

The axioms

The axioms can be formulated as follows:

1. Zero is a natural number.
2. Every natural number has a successor, which is also a natural number.
3. Zero is not the successor of any natural number.
4. Different natural numbers have different successors.
5. If Zero has property P, and if it can be shown that:

(a) If a given natural number n has property P,

(b) It follows that its successor Sn also has P,

Then all natural numbers have the property P.

The last axiom is called the rule of induction.