Banach–Tarski paradox

E400162

mathematical theorem paradox in mathematics result in set-theoretic geometry

The Banach–Tarski paradox is a theorem in set-theoretic geometry stating that a solid ball in 3‑dimensional space can be decomposed into finitely many non-measurable pieces and reassembled into two identical copies of the original ball, highlighting counterintuitive consequences of the axiom of choice.

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All labels observed (2)

Label	Occurrences
Banach–Tarski paradox canonical	3
Hausdorff paradox	2

Statements (50)

Predicate	Object
instanceOf	mathematical theorem ⓘ paradox in mathematics ⓘ result in set-theoretic geometry ⓘ
appliesTo	3-dimensional Euclidean space ⓘ
assumes	axiom of choice ⓘ
author	Alfred Tarski ⓘ Stefan Banach ⓘ
category	paradoxes of infinity ⓘ paradoxes of set theory ⓘ
consequenceOf	axiom of choice ⓘ
contradicts	intuitive notion of volume ⓘ
contrastWith	Jordan measure and classical geometric measure of volume ⓘ
dependsOn	non-amenability of the free group on two generators ⓘ
dimension	3 ⓘ
doesNotApplyTo	1-dimensional Euclidean space ⓘ 2-dimensional Euclidean space ⓘ
field	axiomatic set theory ⓘ geometric group theory ⓘ measure theory ⓘ set theory ⓘ
hasConsequence	Lebesgue measure cannot be defined on all subsets of R^3 ⓘ no finitely additive, rotation-invariant, total measure on all subsets of the 3-ball exists ⓘ
hasGeneralization	paradoxical decompositions of any bounded subset of R^3 with non-empty interior ⓘ
implies	existence of non-measurable sets in R^3 ⓘ
interpretation	does not allow physical realization because pieces are non-measurable and highly non-constructive ⓘ
involves	axiom of choice ⓘ finite decomposition ⓘ non-measurable sets ⓘ rigid motions ⓘ rotations and translations ⓘ
namedAfter	Alfred Tarski ⓘ Stefan Banach ⓘ
originalTitle	Sur la décomposition des ensembles de points en parties respectivement congruentes ⓘ
proofTechnique	choice of representatives from orbits using the axiom of choice ⓘ equidecomposability ⓘ group actions ⓘ
publishedIn	Fundamenta Mathematicae ⓘ
relatedTo	Hausdorff paradox ⓘ Tarski’s theorem on amenable groups ⓘ Vitali set ⓘ amenable groups ⓘ
requires	non-measurable subsets of Euclidean space ⓘ
shows	a ball can be decomposed into finitely many pieces and reassembled into two balls of the same size ⓘ existence of paradoxical decompositions of the 3-ball ⓘ volume is not preserved for non-measurable sets ⓘ
statementAbout	decomposition of a solid ball in 3-dimensional space ⓘ
uses	free subgroup of SO(3) ⓘ group of rotations SO(3) ⓘ paradoxical decomposition ⓘ
yearProved	1924 ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

axiom of choice → implies → Banach–Tarski paradox ⓘ

Stefan Banach → notableWork → Banach–Tarski paradox ⓘ

Stefan Banach → eponymOf → Banach–Tarski paradox ⓘ

Felix Hausdorff → knownFor → Banach–Tarski paradox ⓘ

this entity surface form: Hausdorff paradox

Felix Hausdorff → notableConcept → Banach–Tarski paradox ⓘ

this entity surface form: Hausdorff paradox