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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Cantor’s paradox canonical | 2 |
| Cantor paradox | 1 |
| Cantor's paradox | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T124578 — 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 paradox Context triple: [Russell’s paradox, relatedTo, Cantor’s paradox]
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A.
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.
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B.
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.
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C.
Barber paradox
The Barber paradox is a self-referential logical puzzle about a barber who shaves all and only those who do not shave themselves, illustrating a contradiction similar to Russell’s paradox.
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D.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
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E.
On Contradiction
"On Contradiction" is a 1937 philosophical essay by Mao Zedong that systematically applies and develops Marxist dialectical materialism to analyze the nature and role of contradictions in social and historical processes.
- 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 paradox Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Barber paradox
The Barber paradox is a self-referential logical puzzle about a barber who shaves all and only those who do not shave themselves, illustrating a contradiction similar to Russell’s paradox.
-
D.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
-
E.
On Contradiction
"On Contradiction" is a 1937 philosophical essay by Mao Zedong that systematically applies and develops Marxist dialectical materialism to analyze the nature and role of contradictions in social and historical processes.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cantor’s paradox Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cantor paradox
this entity surface form:
Cantor's paradox