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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All labels observed (3)

Label	Occurrences
Cantor’s diagonal argument	2
Cantor’s theorem canonical	2
Cantor's theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T624849 — 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: Cantor’s theorem
Context triple: [Cantor’s paradox, usesConcept, Cantor’s theorem]

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 paradox in set theory
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.
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. Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
E. 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.
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: Cantor’s theorem
Target entity description: 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.

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 paradox in set theory
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.
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. Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
E. 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.
F. None of above. chosen

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 ⓘ

How these facts were elicited

Referenced by (5)

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

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

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–Bernstein–Schröder theorem → relatedTo → Cantor’s theorem ⓘ

this entity surface form: Cantor's theorem

Everything and More: A Compact History of Infinity → exploresConcept → Cantor’s theorem ⓘ