Paradoxical decomposition

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 ${\displaystyle G}$, it is said to be ${\displaystyle G}$-paradoxical, or paradoxical with respect to ${\displaystyle G}$.

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

The most famous, and indeed motivational, example of paradoxical sets is the Banach–Tarski paradox, which divides the sphere into paradoxical sets for the special orthogonal group. This result depends on the axiom of choice.

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