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.
- [MLA] “Choice, Axiom of.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 24 Mar 2013. Web. 24 Mar 2013. <http://platonicrealms.com/>
- [APA] Choice, Axiom of (24 Mar 2013). Retrieved 24 Mar 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/Axiom-of-Choice/
See also: Gödel’s Theorems