Axiom:Peano's Axioms/Formulation 2

Axioms
Peano's Axioms are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: \map s n = n + 1$ and $0$ as an element of $\N$.

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

Historically, the existence of $s$ and the existence of $0$ were considered the first two of Peano's Axioms:

The other three are as follows:

Also see

 * Equivalence of Formulations of Peano's Axioms