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 | 3 |
| 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hausdorff paradox
this entity surface form:
Hausdorff paradox