The Banach-Tarski Paradox

Stefan Banach and Alfred Tarski proved in 1924 that a solid ball can be decomposed into a finite number of pieces and reassembled, using only rotations and translations, into two solid balls identical to the original. This is not a physical trick — it relies on the Axiom of Choice and non-measurable sets that have no well-defined volume. The paradox does not violate physics, but it reveals a profound tension between geometric intuition and the axioms of set theory. Mathematics can contain truths that cannot be visualised or physically realised.