Well-ordered set

A well-ordered set is a pair $(X,\le)$ where $\le$ is a total order on $X$ such that every nonempty subset $S\subseteq X$ has a least element: there exists $m\in S$ with

Equivalently, every nonempty subset has a minimum (a lower bound that belongs to the set).

Well-ordering is the key hypothesis in transfinite recursion and is central in the well-ordering theorem ; for $\mathbb{N}$ it is expressed by the well-ordering principle .

Examples:

• $(\mathbb{N},\le)$ with the usual order is well-ordered.
• $(\mathbb{Z},\le)$ with the usual order is not well-ordered (the subset of negative integers has no least element).
• Any finite totally ordered set is well-ordered.