Cantor’s theorem

E78328

mathematical theorem result in set theory

Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.

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Observed surface forms (1)

Surface form	Occurrences
Cantor’s diagonal argument	1

Statements (43)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in set theory ⓘ
appliesTo	every set ⓘ finite sets ⓘ infinite sets ⓘ
consequence	for any infinite cardinal κ, 2^κ > κ ⓘ hierarchy of ever-larger infinities ⓘ no set is equinumerous with its power set ⓘ
coreConcept	cardinality ⓘ infinite sets ⓘ power set ⓘ uncountability ⓘ
domain	foundations of mathematics ⓘ
field	set theory ⓘ
formalExpression	∀S (\|S\| < \|P(S)\|) ⓘ ∀S ¬∃f : S → P(S) such that f is surjective ⓘ
historicalPeriod	late 19th century ⓘ
holdsIn	ZF ⓘ surface form: ZFC Zermelo–Fraenkel set theory ⓘ most standard axiomatic set theories ⓘ
implies	the existence of infinitely many distinct infinite cardinalities ⓘ the real numbers are uncountable ⓘ the set of all subsets of the natural numbers is uncountable ⓘ there is no largest cardinal number ⓘ there is no largest infinity ⓘ
importance	fundamental result in set theory ⓘ key to understanding different sizes of infinity ⓘ
influenced	development of modern set theory ⓘ philosophy of mathematics ⓘ
isPartOf	the study of cardinal numbers ⓘ the theory of infinite sets ⓘ
namedAfter	Georg Cantor ⓘ
proofIdea	assume a surjection from S to P(S) and derive a contradiction using a specially constructed subset ⓘ
relatedTo	Cantor’s theorem self-linksurface differs ⓘ surface form: Cantor’s diagonal argument Cantor’s paradox ⓘ cardinal arithmetic ⓘ continuum hypothesis ⓘ
states	for any set S, the power set P(S) has strictly greater cardinality than S ⓘ for every set S, \|S\| < \|P(S)\| ⓘ there is no surjection from a set S onto its power set P(S) ⓘ
status	proven ⓘ
usesMethod	diagonal argument ⓘ proof by contradiction ⓘ

Referenced by (3)

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

Georg Cantor → knownFor → Cantor’s theorem ⓘ

Cantor’s theorem → relatedTo → Cantor’s theorem self-linksurface differs ⓘ

this entity surface form: Cantor’s diagonal argument

Cantor’s paradox → usesConcept → Cantor’s theorem ⓘ