- 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) =egin{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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