Axiom of choice
Set theory
The assumption that for any collection of nonempty sets there is a function selecting exactly one element from each.
What is usually left out
For finite collections this needs no axiom — it follows by induction. The entire content is in the infinite case, and specifically the case where no rule for choosing is available. It is independent of the other ZFC axioms, and it implies results many find counterintuitive, including the Banach–Tarski decomposition.