  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 banachâtarski paradox. ==history== georg cantor considered the well-ordering theorem to be a "fundamental principle of thought." most mathematicians however find it difficult to visualize a well-ordering of, for example, the set r of real numbers. in 1904, gyula kÅnig claimed to have proven that such a well-ordering cannot exist. a few weeks later, felix hausdorff found a mistake in the proof. it turned out, though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the zermeloâfraenkel axioms is sufficient to prove the other, in first order logic (the same applies to zorn's lemma). in second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem. ==statement and sketch of proof== for every set x, there exists a well-ordering with domain x. the well-ordering theorem
Axiom_of_choice Theorems_in_the_foundations_of_mathematics