Axiom:Peano's Axioms/Formulation 1

Axioms
Peano's Axioms are a set of properties which can be used to serve as a basis for logical deduction of the properties of the natural numbers.

Let there be given a set $P$, a mapping $s: P \to P$, and a distinguished element $0$.

Historically, these two presuppositions were considered the first two of Peano's Axioms. The other three can be formulated as follows:

Also defined as
There is nothing special about the symbol $0$ for the non-successor element. Another popular choice is $1$.

Also see

 * Equivalence of Formulations of Peano's Axioms

They were formulated by Peano, and were later refined by Dedekind.