Peano axioms

Summary

The Peano axioms are axioms for the natural numbers. They use a set $\mathbb{N}$, an element $0$ of $\mathbb{N}$ and a function $S \colon \mathbb{N} \to \mathbb{N}$, the successor function. The axioms state three properties of these data. First, $S(n) \neq 0$ for each $n$ in $\mathbb{N}$. Second, $S$ is injective: if $S(m) = S(n)$, then $m = n$. Third, the induction axiom holds: if a subset $A$ of $\mathbb{N}$ contains $0$ and contains $S(n)$ for each $n$ in $A$, then $A = \mathbb{N}$.