Burali-Forti paradox
E14267
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Burali-Forti paradox canonical | 10 |
| Burali-Forti | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T124577 — 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: Burali-Forti paradox Context triple: [Russell’s paradox, relatedTo, Burali-Forti paradox]
-
A.
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.
-
B.
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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C.
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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D.
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.
-
E.
liar paradox
The liar paradox is a classic self-referential logical puzzle arising from sentences that declare their own falsehood, leading to a contradiction about whether they are true or false.
- 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: Burali-Forti paradox Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
liar paradox
The liar paradox is a classic self-referential logical puzzle arising from sentences that declare their own falsehood, leading to a contradiction about whether they are true or false.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
antinomical paradox
ⓘ
logical paradox ⓘ set-theoretic paradox ⓘ |
| appearsIn |
foundations of mathematics literature
ⓘ
standard textbooks on set theory ⓘ |
| classification | paradox of size ⓘ |
| concerns | totalities that are too large to be sets ⓘ |
| consequence |
distinction between sets and proper classes
ⓘ
necessity of restricting set formation axioms ⓘ the class of all ordinals is too large to be a set ⓘ |
| contradicts | definition of the set of all ordinals ⓘ |
| describes | inconsistency of the set of all ordinal numbers ⓘ |
| field |
foundations of mathematics
ⓘ
mathematical logic ⓘ set theory ⓘ |
| formalContent | If Ω is the set of all ordinals, then Ω itself would be an ordinal greater than every ordinal in Ω, leading to a contradiction ⓘ |
| historicalContext | arose in the study of transfinite numbers after Cantor’s work ⓘ |
| historicallyInfluenced |
Zermelo’s formulation of the axiom of separation
ⓘ
development of class theories in set theory ⓘ |
| implication |
hierarchical cumulative universe of sets
ⓘ
no universal set of all sets of standard set theory ⓘ |
| involvesConcept |
Russell-style self-reference
ⓘ
ordinal number ⓘ proper class ⓘ set of all ordinals ⓘ transfinite ordinal ⓘ well-ordering ⓘ |
| leadsTo | ordinal strictly larger than every ordinal in the supposed set of all ordinals ⓘ |
| logicalForm | reductio ad absurdum argument ⓘ |
| motivatedDevelopmentOf |
Zermelo set theory
ⓘ
Zermelo–Fraenkel set theory ⓘ axiomatic set theory ⓘ |
| namedAfter | Cesare Burali-Forti ⓘ |
| originalLanguage | Italian ⓘ |
| relatedTo |
Cantor’s paradox
ⓘ
surface form:
Cantor paradox
Russell’s paradox ⓘ
surface form:
Russell paradox
naive comprehension schema ⓘ |
| resolvedIn |
Morse–Kelley set theory by class–set distinction
ⓘ
von Neumann–Bernays–Gödel set theory ⓘ
surface form:
Zermelo–Fraenkel set theory by treating the collection of all ordinals as a proper class
von Neumann–Bernays–Gödel set theory ⓘ
surface form:
von Neumann–Bernays–Gödel set theory by class–set distinction
|
| shows | naive set theory with unrestricted comprehension is inconsistent ⓘ |
| statement | The collection of all ordinal numbers cannot form a set without contradiction ⓘ |
| typicalFormalizationFramework | first-order axiomatic set theory ⓘ |
| usesAssumption |
every well-ordered set is order-isomorphic to a unique ordinal
ⓘ
the set of all ordinals, if it existed, would itself be well-ordered ⓘ |
| yearProposed | 1897 ⓘ |
How these facts were elicited
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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: Burali-Forti paradox Description of subject: 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.
Referenced by (12)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Burali-Forti
subject surface form:
Cesare Burali-Forti
subject surface form:
Cesare Burali-Forti
subject surface form:
Cesare Burali-Forti
this entity surface form:
Burali-Forti