Banach-Tarski Paradox

The Banach-Tarski Paradox is a theorem in set-theoretic geometry stating that a solid ball in three-dimensional space can be decomposed into a finite number of disjoint pieces that can then be reassembled, using only rotations and translations, into two solid balls each identical to the original. The pieces themselves are not solids in any ordinary sense but infinite, intricately scattered sets of points whose individual volumes cannot be consistently defined. Named for Stefan Banach and Alfred Tarski, the theorem depends critically on the axiom of choice, which permits the construction of the non-measurable sets the decomposition requires, and it is considered a paradox only because it defies geometric intuition rather than because it is logically inconsistent.