The Banach–Tarski paradox is that a ball can be decomposed into a finite number of point sets and reassembled into two balls identical to the original.

In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is a partitioning of the set into two subsets, along with an appropriate group of functions that operate on some universe (of which the set in question is a subset), such that each partition can be mapped back onto the entire set using only finitely many distinct functions (or compositions thereof) to accomplish the mapping. Since a paradoxical set as defined requires a suitable group $G$, it is said to be $G$-paradoxical, or paradoxical with respect to $G$.

Paradoxical sets exist as a consequence of the Axiom of Infinity. Admitting infinite classes as sets is sufficient to allow paradoxical sets.