Well-ordering principle for ℕ

Well-ordering principle for $\mathbb{N}$: If $S\subseteq\mathbb{N}$ is a subset and $S\neq\emptyset$ (see empty set ), then there exists $m\in S$ such that

This is exactly the statement that $(\mathbb{N},\le)$ is a well-ordered set . It is equivalent to the principle of mathematical induction .

Proof sketch (idea): If a nonempty $S\subseteq\mathbb{N}$ had no least element, one can define “$n$ is not in $S$” inductively for all $n$, forcing $S$ to be empty; conversely, well-ordering yields induction by applying least-element arguments to the set of counterexamples.