Definition
Banach–Tarski Paradox
In Zermelo–Fraenkel set theory with the axiom of choice, a solid ball
of positive radius can be partitioned into finitely many disjoint sets and reassembled by rotations and translations into two disjoint solid balls , each congruent to : where each
is a rigid motion and denotes a union of pairwise disjoint sets. No piece is stretched, and the reassembled pieces do not overlap. The pieces cannot all be Lebesgue measurable: ordinary volume is not defined for every piece. The result therefore does not contradict conservation of volume for measurable sets and does not describe a physical cutting procedure.