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 exactly 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
, it is said to be
-paradoxical, or paradoxical with respect to
Paradoxical sets exist as a consequence of the Axiom of Infinity. Admitting infinite classes as sets is sufficient to allow paradoxical sets.
An example of a paradoxical set is the natural numbers. They are paradoxical with respect to the group of functions generated by the natural function :
Split the natural numbers into the odds and the evens. The function maps boths sets onto the whole of . Since only finitely many functions were needed, the naturals are -paradoxical.
- S. Wagon, The Banach–Tarski Paradox, Cambridge University Press, 1986.