Axiom:Axiom of Choice

Axiom
For every set, we can provide a mechanism for choosing one element of any non-empty subset of the set.


 * $\forall x \in a: \exists y: P \left({x, y}\right) \implies \exists y: \forall x \in a: P \left({x, y \left({x}\right)}\right)$

That is, one can always create a choice function for selecting an element of any set.

Alternative Version
Let $\left \langle {X_i} \right \rangle_{i \in I}$ be a family of sets all of which are non-empty, indexed by $I$ which is also non-empty.

Then there exists a family $\left \langle {x_i} \right \rangle_{i \in I}$ such that:
 * $\forall i \in I: x_i \in X_i$

That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty.

Comment
Although it seems intuitively obvious ("surely you can just pick an element?"), when it comes to transfinite 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 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.

Also see

 * Equivalence of Versions of Axiom of Choice