Paradoxical set

In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is two families of disjoint subsets, along with appropriate group actions that act 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. A set that admits such a paradoxical decomposition where the actions belong to a group $G$ is called $G$ -paradoxical or paradoxical with respect to $G$ .

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.

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

Definition

Suppose a group $G$ acts on a set $A$ . Then $A$ is $G$ -paradoxical if there exists some disjoint subsets $A_{1},...,A_{n},B_{1},...,B_{m}\subseteq A$ and some group elements $g_{1},...,g_{n},h_{1},...,h_{m}\in G$ such that:[1]

$A=\bigcup _{i=1}^{n}g_{i}(A_{i})$ and $A=\bigcup _{i=1}^{m}h_{i}(B_{i})$

Examples

Free group

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

Banach–Tarski paradox

The most famous, and indeed motivational, 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.

gollark: Oh yes, add warning signs too, obviously.

gollark: So leave it and ignore it in a specific place. Maybe the Moon or something, if space travel gets cheaper.

gollark: Apparently the recycling of solar panels isn't very efficient or cost-effective right now.

gollark: It's still quite small, so you can shove it in a box somewhere and ignore it, or apparently run much of it through a breeder reactor to reuse it.

gollark: The nuclear waste problem isn't even that much of an issue compared to vast amounts of degraded solar panels, it's much lower in volume.

References

Wagon, Stan; Tomkowicz, Grzegorz (2016). The Banach–Tarski Paradox (Second ed.). ISBN 978-1-107-04259-9.

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[1] Wagon, Stan; Tomkowicz, Grzegorz (2016). The Banach–Tarski Paradox (Second ed.). ISBN 978-1-107-04259-9.