Axiom:Axiom of Choice

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

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

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

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 theorem.

For this reason, the Axiom of Choice ("AoC") is treated separately from the rest of the Zermelo-Fraenckel axioms.

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

It can be shown that the Axiom of Choice holds for all finite /countable sets. (TODO: Check this, and provide a proof.)