Mathematics
The Axiom of Choice and Its Equivalents like Zorn's Lemma
Quick fact
The Axiom of Choice is equivalent to Zorn's Lemma and the Well-Ordering Theorem, all of which are independent of the standard axioms of set theory. One surprising consequence is the Banach-Tarski Paradox, where a solid ball can be split into finitely many pieces and reassembled into two identical copies, seemingly doubling its volume.
Why this is interesting
You can pick a shoe from each of infinitely many pairs of shoes—but can you pick a sock from each pair when the socks are indistinguishable? That simple question leads to one of the most powerful and controversial principles in all of mathematics.