What is the Axiom of Choice?

Mathematics is based on a foundation of axioms, or assumptions. One of the most important and widely-used set of axioms is called Zermelo-Fraenkel set theory with the Axiom of Choice, or ZFC. These axioms define what a set is, which are fundamental objects in mathematics. And the Axiom of Choice is arguably one of the most important and interesting axioms of ZFC. But what does it really say? And how is it used?