Cantor’s paradox

E14593

logical paradox result in set theory set-theoretic 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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All labels observed (3)

Label	Occurrences
Cantor’s paradox canonical	2
Cantor paradox	1
Cantor's paradox	1

Statements (42)

Predicate	Object
instanceOf	logical paradox ⓘ result in set theory ⓘ set-theoretic paradox ⓘ
assumes	existence of a set of all sets ⓘ
avoidedIn	Zermelo–Fraenkel set theory ⓘ von Neumann–Bernays–Gödel set theory ⓘ
clarifies	distinction between sets and proper classes ⓘ
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 ⓘ
derives	contradiction in cardinalities ⓘ
field	mathematical logic ⓘ set theory ⓘ
formalizes	impossibility of a set of all cardinalities ⓘ
hasKeyIdea	self-application of Cantor’s theorem to a supposed universal set ⓘ
hasProofMethod	cardinality comparison ⓘ diagonalization ⓘ
historicalPeriod	late 19th century ⓘ
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 ⓘ
namedAfter	Georg Cantor ⓘ
relatedTo	Burali-Forti paradox ⓘ Russell’s paradox ⓘ proper class ⓘ universal set ⓘ
shows	a universal set would have cardinality strictly less than its power set and equal to it ⓘ power set of any set has strictly greater cardinality than the set ⓘ
usesConcept	Cantor’s theorem ⓘ cardinality ⓘ diagonal argument ⓘ power set ⓘ

Referenced by (4)

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

Russell’s paradox → relatedTo → Cantor’s paradox ⓘ

Burali-Forti paradox → relatedTo → Cantor’s paradox ⓘ

this entity surface form: Cantor paradox

Cantor’s theorem → relatedTo → Cantor’s paradox ⓘ

naive set theory → isInconsistentBecauseOf → Cantor’s paradox ⓘ

this entity surface form: Cantor's paradox