Paradoxical set

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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.

[edit] Examples

[edit] 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 $f$ :

$f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ (n-1)/2, & \mbox{if }n\mbox{ is odd} \end{cases}$

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.