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 Z-F 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