Choice, Axiom of

An axiom of set theory which states that from any family of sets a new set may be created that contains one element chosen from each set in the family. This axiom has been shown to be independent of the other axioms of set theory. It is equivalent to many other statements, including Zorn’s Lemma, the well-ordering principle, the Hausdorff Maximality Theorem, and the assertion that the Cartesian product of an infinite family of sets is non-empty.

Because the Axiom of Choice permits non-constructive proofs it is rejected by intuitionism, and careful mathematicians refer explicitly to its use in proofs that require it.