Paradoxical set

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 $G$, 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.

Examples

Natural numbers

An example of a paradoxical set is the natural numbers. They are paradoxical with respect to the group of functions $G$ generated by the natural function $f$:

Split the natural numbers into the odds and the evens. The function $f$ maps boths sets onto the whole of $mathbb\{N\}$. Since only finitely many functions were needed, the naturals are $G$-paradoxical.

References

* S. Wagon, The Banach–Tarski Paradox, Cambridge University Press, 1986.