Cantor’s paradox
E14593
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.
Aliases (1)
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
logical paradox
→
result in set theory → set-theoretic paradox → |
| assumes |
existence of a set of all sets
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|
| avoidedIn |
Zermelo–Fraenkel set theory
→
von Neumann–Bernays–Gödel set theory → |
| clarifies |
distinction between sets and proper classes
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|
| concerns |
hierarchy of infinities
→
infinite sets → |
| conclusion |
naive set theory with unrestricted comprehension is inconsistent
→
no set can contain all sets as elements → |
| contradicts |
existence of a maximal cardinality
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|
| derives |
contradiction in cardinalities
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|
| field |
mathematical logic
→
set theory → |
| formalizes |
impossibility of a set of all cardinalities
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|
| hasKeyIdea |
self-application of Cantor’s theorem to a supposed universal set
→
|
| hasProofMethod |
cardinality comparison
→
diagonalization → |
| historicalPeriod |
late 19th century
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|
| holdsIn |
naive set theory
→
|
| implies |
collection of all sets must be a proper class in axiomatic set theory
→
universe of sets is not itself a set → |
| influenced |
Zermelo–Fraenkel set theory
→
axiom of separation → axiom schema of replacement → |
| isDiscussedIn |
foundations of mathematics
→
philosophy of mathematics → |
| mainClaim |
the set of all sets cannot exist
→
there is no universal set in standard set theory → |
| motivated |
development of axiomatic set theory
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|
| namedAfter |
Georg Cantor
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|
| relatedTo |
Burali-Forti paradox
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Russell’s paradox → proper class → universal set → |
| shows |
a universal set would have cardinality strictly less than its power set and equal to it
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power set of any set has strictly greater cardinality than the set → |
| usesConcept |
Cantor’s theorem
→
cardinality → diagonal argument → power set → |
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
Burali-Forti paradox
("Cantor paradox")
→
Cantor’s theorem → Russell’s paradox → |
relatedTo |