Axiom:Axiom of Choice

Comment
Although it seems intuitively obvious ("surely you can just pick an element?"), when it comes to infinite sets of sets this axiom leads to non-intuitive results, notably the famous Banach-Tarski Paradox.

For this reason, the Axiom of Choice (often abbreviated AoC or AC) is often treated separately from the rest of the Zermelo-Fraenkel Axioms.

Set theory based on the Zermelo-Fraenkel axioms is referred to ZF, while that based on the ZF axioms including the AoC is referred to as ZFC.

Additional forms
The following are equivalent, in ZF, to the Axiom of Choice:


 * Zorn's Lemma
 * Kuratowski's Lemma
 * Hausdorff's Maximal Principle


 * Tukey's Lemma


 * Tychonoff's Theorem
 * Kelley's Theorem


 * Vector Space has Basis

Also see

 * Equivalence of Formulations of Axiom of Choice


 * Banach-Tarski Paradox