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