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 two families of disjoint subsets, along with appropriate group actions that act 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. A set that admits such a paradoxical decomposition where the actions belong to a group ${\displaystyle G}$ is called ${\displaystyle G}$-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.

Definition

Suppose a group ${\displaystyle G}$ acts on a set ${\displaystyle A}$. Then ${\displaystyle A}$ is ${\displaystyle G}$-paradoxical if there exists some disjoint subsets ${\displaystyle A_{1},...,A_{n},B_{1},...,B_{m}\subseteq A}$ and some group elements ${\displaystyle g_{1},...,g_{n},h_{1},...,h_{m}\in G}$ such that:^[1]

${\displaystyle A=\bigcup _{i=1}^{n}g_{i}(A_{i})}$ and ${\displaystyle A=\bigcup _{i=1}^{m}h_{i}(B_{i})}$

Examples

See also

• Pathological (mathematics)

References