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[[Image:Banach-Tarski Paradox.svg|thumb|right|350px|The [[Banach–Tarski paradox]] is that a ball can be decomposed into a finite number of point sets and reassembled into two balls identical to the original.]]

In [[set theory]], a '''paradoxical set''' is a set that has a '''paradoxical decomposition'''. A paradoxical decomposition of a set is atwo [[partitioning]]families of thedisjoint setsubsets, into subsetsalong with an appropriate [[group actions]] of functions that operateact on thesome elements[[Universe (mathematics)|universe]] (of which the set in question is a subset), such that each subsetpartition can be mapped back intoonto the entire set using only finitely many distinct functions (or compositions thereof) to accomplish the mapping. A Sinceset that admits such a paradoxical setdecomposition aswhere definedthe requiresactions abelong suitableto a group <math>G</math>, it is said to becalled <math>G</math>-paradoxical, or paradoxical with respect to <math>G</math>.

Paradoxical sets exist as a consequence of the [[Axiom of Infinity]]. Admitting infinite classes as sets is sufficient to allow paradoxical sets.

== Example Definition==

Suppose a group <math>G</math> acts on a set <math>A</math>. Then <math>A</math> is <math>G</math>-paradoxical if there exists some disjoint subsets <math>A_1,...,A_n,B_1,...,B_m \subseteq A</math> and some group elements <math>g_1,...,g_n,h_1,...,h_m \in G</math> such that:<ref>{{cite book|last1=Wagon|first1=Stan|last2=Tomkowicz|first2=Grzegorz|title=The Banach–Tarski Paradox|title-link= The Banach–Tarski Paradox (book)|date=2016|publisher=Cambridge University Press |isbn=978-1-107-04259-9|edition=Second}}</ref>

<math>A = \bigcup_{i=1}^n g_i(A_i)</math> and <math>A = \bigcup_{i=1}^m h_i(B_i)</math>

An example of a paradoxical set is the [[natural numbers]]. They are paradoxical with respect to the group of functions <math>\{\frac{x}{2},\frac{x-1}{2}\}</math>.

== Examples ==

Split the natural numbers into the odds and the evens. If you apply the function <math>f(x)=\frac{x-1}{2}</math> to the odds and <math>f(x)=\frac{x}{2}</math> to the evens; each of these will give back the entire set of the natural numbers.

=== Free group ===

The [[Free group]] ''F'' on two generators ''a,b'' has the decomposition <math>F = \{e\} \cup X(a) \cup X(a^{-1}) \cup X(b) \cup X(b^{-1})</math> where ''e'' is the identity word and <math>X(i)</math> is the collection of all (reduced) words that start with the letter ''i''. This is a paradoxical decomposition because <math>X(a) \cup aX(a^{-1}) = F = X(b) \cup bX(b^{-1}).</math>

=== Banach–Tarski paradox ===

{{main|Banach–Tarski paradox}}

The most famous example of paradoxical sets is the [[Banach–Tarski paradox]], which divides the sphere into paradoxical sets for the [[special orthogonal group]]. This result depends on the [[axiom of choice]].

== See also ==

* [[Pathological (mathematics)]]

== References ==

<references />

{{DEFAULTSORT:Paradoxical Set}}

[[Category:Set theory]]

[[Category:Geometric dissection]]