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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 $G$.## Examples

### Natural numbers

## References

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 $G$ generated by the natural function $f$:

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

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.

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

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday April 25, 2008 at 09:47:16 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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