Well-ordered set

A well-ordered set is a set $X$ equipped with a total order $\le$ such that every nonempty subset $A\subseteq X$ has a least element; that is, there exists $m\in A$ with $m\le a$ for all $a\in A$.

Well-orderings are central in statements like the well-ordering theorem , which asserts that every set can be well-ordered. The well-ordering principle is the special case asserting that $\mathbb{N}$ is well-ordered by its usual order.

Examples:

• With the usual $\le$, the natural numbers form a well-ordered set.
• Any finite set becomes well-ordered after choosing a total order; for instance, $\{a_1,\dots,a_n\}$ is well-ordered by declaring $a_1<a_2<\cdots<a_n$.