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]
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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.
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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.
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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 paradox
ⓘ
linked to:
Cantor’s paradox
Russell paradox ⓘ
linked to:
Russell’s paradox
naive comprehension schema ⓘ |
| resolvedIn |
Morse–Kelley set theory by class–set distinction
ⓘ
Zermelo–Fraenkel set theory by treating the collection of all ordinals as a proper class ⓘ
linked to:
von Neumann–Bernays–Gödel set theory
von Neumann–Bernays–Gödel set theory by class–set distinction ⓘ
linked to:
von Neumann–Bernays–Gödel set theory
|
| 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
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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.