Definitions

Paradoxical set

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:

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.

References

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