von Neumann paradox in set theory
E15214
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Banach–Tarski paradox | 2 |
| Banach–Tarski-type decompositions | 1 |
| Hausdorff paradox | 1 |
| von Neumann paradox in set theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T131676 — 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: von Neumann paradox in set theory Context triple: [John von Neumann, notableConcept, von Neumann paradox in set theory]
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A.
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.
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B.
von Neumann universe
The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.
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C.
Burali-Forti paradox
The Burali-Forti paradox is a foundational logical contradiction in set theory that arises from considering the set of all ordinal numbers, showing that such a totality cannot consistently exist as a set.
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D.
Russell’s paradox
Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.
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E.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
- 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: von Neumann paradox in set theory Target entity description: The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
-
A.
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.
-
B.
von Neumann universe
The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.
-
C.
Burali-Forti paradox
The Burali-Forti paradox is a foundational logical contradiction in set theory that arises from considering the set of all ordinal numbers, showing that such a totality cannot consistently exist as a set.
-
D.
Russell’s paradox
Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.
-
E.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical paradox
ⓘ
result in set theory ⓘ set-theoretic paradox ⓘ |
| appliesTo |
non-amenable groups
ⓘ
sets with a free group action ⓘ |
| assumes |
axiom of choice for selecting representatives of orbits
ⓘ
existence of a free subgroup of the acting group ⓘ |
| concerns |
decomposition of a set into finitely many pieces
ⓘ
reassembly of pieces into subsets of the same cardinality as the original set ⓘ |
| demonstrates | existence of paradoxical subsets of equal size to the original set ⓘ |
| field |
foundations of mathematics
ⓘ
group theory ⓘ measure theory ⓘ set theory ⓘ |
| formalizes | paradoxical decompositions in terms of group actions ⓘ |
| hasConsequence |
intuitive notion of volume fails for all subsets under choice
ⓘ
some sets cannot be assigned a finitely additive, group-invariant probability measure ⓘ |
| historicalContext | early 20th century ⓘ |
| implies | non-existence of a countably additive, translation-invariant measure on all subsets of some spaces ⓘ |
| influenced |
development of amenability theory
ⓘ
modern ergodic theory ⓘ study of non-measurable sets ⓘ |
| involvesConcept |
cardinality
ⓘ
equidecomposability ⓘ finitely additive measure ⓘ group action on a set ⓘ invariant measure ⓘ non-measurable set ⓘ paradoxical subset ⓘ |
| isAbstractVersionOf |
Banach–Tarski-type decompositions
ⓘ
linked to:
von Neumann paradox in set theory
|
| isGeneralizationOf | earlier paradoxes about rotations on the sphere ⓘ |
| logicalStatus | consistent with Zermelo–Fraenkel set theory plus the axiom of choice (ZFC) ⓘ |
| motivationFor |
definition of amenable groups
ⓘ
study of invariant means ⓘ |
| namedAfter | John von Neumann ⓘ |
| relatedTo |
Banach–Tarski paradox
ⓘ
linked to:
von Neumann paradox in set theory
Hausdorff paradox ⓘ
linked to:
von Neumann paradox in set theory
Vitali set ⓘ countable group actions ⓘ free group on two generators ⓘ non-amenable group ⓘ paradoxical decomposition ⓘ |
| shows |
counterintuitive consequences of the axiom of choice
ⓘ
existence of paradoxical decompositions under certain group-theoretic conditions ⓘ failure of finitely additive invariant measures on some groups ⓘ |
| usesAxiom | axiom of choice ⓘ |
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: von Neumann paradox in set theory Description of subject: The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: von Neumann paradox in set theory
linked to: von Neumann paradox in set theory
linked to: von Neumann paradox in set theory
linked to: von Neumann paradox in set theory