Well-ordering principle

Well-ordering principle: Every nonempty subset $S\subseteq\mathbb{N}$ has a least element (with respect to the usual order on $\mathbb{N}$); that is, there exists $m\in S$ such that $m\le s$ for all $s\in S$.

This is exactly the statement that $(\mathbb{N},\le)$ is a well-ordered set . It is equivalent in strength (over standard basic assumptions) to mathematical induction .