In mathematics, the
well-ordering theorem states that every
set can be
well-ordered. 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 paradoxical, such as the
Banach–Tarski paradox.