The Banach-Tarski Paradox: Duplicate a Sphere with Math?

This astonishing result, known as the Banach-Tarski Paradox, relies on the Axiom of Choice, a fundamental principle in set theory. The 'pieces' are not physical objects that could be cut with a knife; they are non-measurable sets of points, so infinitely fragmented they don't have a conventional volume. While a theoretical existence proof, it highlights how mathematical infinities can defy our physical intuition, demonstrating that volume can behave counter-intuitively when dealing with certain abstract sets. It does not imply you can physically duplicate gold!