Burali-Forti paradox

E14267

antinomical paradox logical paradox set-theoretic 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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All labels observed (2)

Label	Occurrences
Burali-Forti paradox canonical	10
Burali-Forti	2

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

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: 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.

Russell’s paradox → relatedTo → Burali-Forti paradox ⓘ

Cantor’s paradox → relatedTo → Burali-Forti paradox ⓘ

Morse–Kelley set theory by class–set distinction → avoidsParadox → Burali-Forti paradox ⓘ

Cesare Burali-Forti → notableFor → Burali-Forti paradox ⓘ

Cesare Burali-Forti → notableWork → Burali-Forti paradox ⓘ

Cesare Burali-Forti → paradoxNamedAfter → Burali-Forti paradox ⓘ

Cesare Burali-Forti → hasFamilyName → Burali-Forti paradox ⓘ

this entity surface form: Burali-Forti

Zermelo set theory → addressesProblem → Burali-Forti paradox ⓘ

naive set theory → isInconsistentBecauseOf → Burali-Forti paradox ⓘ

Cesare → notableFor → Burali-Forti paradox ⓘ

subject surface form: Cesare Burali-Forti

Cesare → notableIdea → Burali-Forti paradox ⓘ

subject surface form: Cesare Burali-Forti

Cesare → hasFamilyName → Burali-Forti paradox ⓘ

subject surface form: Cesare Burali-Forti

this entity surface form: Burali-Forti