View source for Well-ordering theorem

In [[mathematics]], the '''well-ordering theorem''' states that every set can be [[well-ordered]].

== Description ==

A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering.

This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice.

[[Ernst Zermelo]] introduced the Axiom of Choice as an "unobjectionable logical principle" to prove the well-ordering theorem.

This is important because it makes every set susceptible to the powerful technique of [[transfinite induction]].

The well-ordering theorem has consequences that may seem [[Paradox|paradoxical]], such as the [[Banach–Tarski paradox]].

== See also ==

* [[Mathematics]]
* [[Paradox]]
* [[Ernst Zermelo]]
==  External links ==

* [https://en.wikipedia.org/wiki/Well-ordering_theorem Well-ordering theorem] @ Wikipedia