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.
Aliases (1)
- Burali-Forti ×1
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
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|
| 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
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|
| describes |
inconsistency of the set of all ordinal numbers
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|
| field |
foundations of mathematics
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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
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|
| historicalContext |
arose in the study of transfinite numbers after Cantor’s work
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|
| historicallyInfluenced |
Zermelo’s formulation of the axiom of separation
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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
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|
| logicalForm |
reductio ad absurdum argument
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|
| motivatedDevelopmentOf |
Zermelo set theory
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Zermelo–Fraenkel set theory → axiomatic set theory → |
| namedAfter |
Cesare Burali-Forti
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|
| originalLanguage |
Italian
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|
| relatedTo |
Cantor paradox
→
Russell paradox → naive comprehension schema → |
| resolvedIn |
Morse–Kelley set theory by class–set distinction
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Zermelo–Fraenkel set theory by treating the collection of all ordinals as a proper class → von Neumann–Bernays–Gödel set theory by class–set distinction → |
| shows |
naive set theory with unrestricted comprehension is inconsistent
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|
| statement |
The collection of all ordinal numbers cannot form a set without contradiction
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|
| typicalFormalizationFramework |
first-order axiomatic set theory
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|
| 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
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|
Referenced by (8)
| Subject (surface form when different) | Predicate |
|---|---|
|
Cantor’s paradox
→
Russell’s paradox → |
relatedTo |
|
Zermelo set theory
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|
addressesProblem |
|
Morse–Kelley set theory by class–set distinction
→
|
avoidsParadox |
|
Cesare Burali-Forti
("Burali-Forti")
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hasFamilyName |
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Cesare Burali-Forti
→
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notableFor |
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Cesare Burali-Forti
→
|
notableWork |
|
Cesare Burali-Forti
→
|
paradoxNamedAfter |