Mathematics
The Axiom of Choice and Its Equivalents in Set Theory
Quick fact
The Axiom of Choice implies the Banach-Tarski paradox: a solid ball can be split into five pieces and reassembled into two identical balls, seemingly doubling its volume—a result possible only because the pieces are non-measurable sets.
Why this is interesting
You can pick one sock from each of infinitely many pairs of socks. That seems obvious—but do you need a rule for which sock you pick? This simple question leads to one of mathematics' most controversial and powerful axioms.