Mathematics
The Banach-Tarski Paradox and the Axiom of Choice
Quick fact
The Banach-Tarski paradox shows that a solid sphere can be split into just five pieces that, when rearranged, form two complete spheres of the same size as the original—seemingly doubling its volume. The catch? The pieces are non-measurable sets, so they don't have a well-defined volume, and the construction relies on the Axiom of Choice.
Why this is interesting
You have a solid ball. Could you cut it into a few pieces and reassemble it to get two identical balls of the same size? Mathematically, yes—if you accept the Axiom of Choice. But how can that be?