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.
Aliases (3)
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
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|
| 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
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|
| isGeneralizationOf |
earlier paradoxes about rotations on the sphere
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|
| logicalStatus |
consistent with Zermelo–Fraenkel set theory plus the axiom of choice (ZFC)
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|
| motivationFor |
definition of amenable groups
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study of invariant means → |
| namedAfter |
John von Neumann
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|
| relatedTo |
Banach–Tarski paradox
→
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
→
|
Referenced by (5)
| Subject (surface form when different) | Predicate |
|---|---|
|
von Neumann paradox in set theory
("Banach–Tarski paradox")
→
von Neumann paradox in set theory ("Hausdorff paradox") → |
relatedTo |
|
von Neumann paradox in set theory
("Banach–Tarski-type decompositions")
→
|
isAbstractVersionOf |
|
Alfred Tarski
("Banach–Tarski paradox")
→
|
knownFor |
|
John von Neumann
→
|
notableConcept |