von Neumann paradox in set theory

E15214

mathematical paradox result in set theory set-theoretic paradox

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.

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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]

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
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	von Neumann paradox in set theory self-linksurface differs ⓘ surface form: Banach–Tarski-type decompositions
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	von Neumann paradox in set theory self-linksurface differs ⓘ surface form: Banach–Tarski paradox von Neumann paradox in set theory self-linksurface differs ⓘ surface form: Hausdorff paradox 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

Referenced by (5)

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

John von Neumann → notableConcept → von Neumann paradox in set theory ⓘ

von Neumann paradox in set theory → relatedTo → von Neumann paradox in set theory self-linksurface differs ⓘ

this entity surface form: Banach–Tarski paradox

von Neumann paradox in set theory → relatedTo → von Neumann paradox in set theory self-linksurface differs ⓘ

this entity surface form: Hausdorff paradox

von Neumann paradox in set theory → isAbstractVersionOf → von Neumann paradox in set theory self-linksurface differs ⓘ

this entity surface form: Banach–Tarski-type decompositions

Alfred Tarski → knownFor → von Neumann paradox in set theory ⓘ