Banach–Tarski paradox
E400162
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Banach–Tarski paradox canonical | 5 |
| Hausdorff paradox | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T3913054 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Banach–Tarski paradox Context triple: [axiom of choice, implies, Banach–Tarski paradox]
-
A.
Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
-
B.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
C.
Smale’s paradox
Smale’s paradox is a result in differential topology showing that a sphere can be turned inside out in three-dimensional space through smooth deformations without tearing or creasing, challenging intuitive notions of geometry.
-
D.
Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
-
E.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Banach–Tarski paradox Target entity description: 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.
-
A.
Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
-
B.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
C.
Smale’s paradox
Smale’s paradox is a result in differential topology showing that a sphere can be turned inside out in three-dimensional space through smooth deformations without tearing or creasing, challenging intuitive notions of geometry.
-
D.
Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
-
E.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Banach–Tarski paradox Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Banach–Tarski paradox
linked to: Banach–Tarski paradox
subject linked to:
Alfred Teitelbaum