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 $A = \mathbb{N}$.

Remarks

Giuseppe Peano published the axioms in 1889. Richard Dedekind gave a similar system in 1888. The induction axiom speaks about all subsets of $\mathbb{N}$, thus it is a second-order axiom. First-order Peano arithmetic replaces it with an axiom schema. The axioms are the base for the principle of mathematical induction and for definition by recursion.

Explanations

L1 human

The Peano axioms say what is special about counting: you start at $0$ and you always go to the next number. Two different numbers never have the same next number, and you never come back to $0$. The induction axiom says that you reach every natural number in this way.