Well-ordering Theorem

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.