\begin{definition}[Definition:Choice Function/Use of Axiom of Choice]
The Axiom of Choice (abbreviated '''AoC''' or '''AC''') is the following statement:
:''All $\mathbb S$ as above have a choice function.''
It can be shown that the AoC it does not follow from the other usual axioms of set theory, and that it is relative consistent to these axioms (i.e., that AoC does not make the axiom system inconsistent, provided it was consistent without AoC).
Note that for any given set $S \in \mathbb S$, one can select an element from it (without using AoC). AoC guarantees that there is a choice function, i.e., a function that "simultaneously" picks elements of all $S \in \mathbb S$.
 
AoC is needed to prove statements such as "all countable unions of finite sets are countable" (for many specific such unions this 
can be shown without AoC), and AoC is equivalent to many other mathematical statements such as "every vector space has a basis".
{{WIP|The above needs to be rewritten more tersely. Examples of its use should be moved either to the AoC page itself or onto the specific pages where the statements themselves are used.<br/>Note that the recent amendment to this page which added a considerable quantity of material was made by an anonymous editor and therefore we can not enter into a discussion with him/her.}}
Category:Definitions/Set Theory
\end{definition}