Weaker versions of this mathematical statement include the Boolean prime ideal theorem, while concepts requiring it include Tychonoff's theorem, trichotomy involving the cardinal numbers, and that every vector space has a basis.
Weaker versions of this mathematical statement include the Boolean prime ideal theorem, while concepts requiring it include Tychonoff's theorem, trichotomy involving the cardinal numbers, and that every vector space has a basis. Proofs requiring it are always nonconstructive. The Banach-Tarski paradox, Baire category theorem, well-ordering theorem and Zorn's lemma are all equivalent to, for 10 points, this fundamental assertion that is logically independent of Zermelo-Fraenkel set theory, that states for a set of non-empty sets X, one can select a member from each set in X.
Weaker versions of this mathematical statement include the Boolean prime ideal theorem, while concepts requiring it include Tychonoff's theorem, trichotomy involving the cardinal numbers, and that every vector space has a basis.
{axiom of choice}