The Axiom of Choice (AC) is perhaps the most controversial axiom in mathematics. It states that given any collection of non-empty sets, we can select exactly one element from each set to form a new set.

While this seems intuitive for finite collections, its application to infinite collections leads to some surprising and counterintuitive results, such as the Banach-Tarski paradox, which proves that it’s possible to take a solid ball, cut it into a finite number of pieces, and reassemble those pieces to create two identical copies of the original ball!

The axiom is independent of the other axioms of set theory - meaning that we can choose to accept or reject it, and either way we’ll still have a consistent mathematical system. This has led to fascinating discussions about what we mean by mathematical truth and existence.