Let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers}, with successor map $S$ as in that definition. We take as an axiom the following \textbf{principle of induction}.

If $A\subseteq\mathbb{N}$ satisfies

1. $1\in A$, and
2. $S(n)\in A$ for every $n\in A$,

then $A=\mathbb{N}$.