von Neumann–Bernays–Gödel set theory

E15613

axiomatic set theory conservative extension of Zermelo–Fraenkel set theory set theory two-sorted first-order theory

Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.

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All labels observed (8)

Label	Occurrences
von Neumann–Bernays–Gödel set theory canonical	9
NBG	1
NBG set theory	1
Zermelo–Fraenkel set theory by treating the collection of all ordinals as a proper class	1
axiomatic set theory (von Neumann–Bernays–Gödel set theory)	1
von Neumann–Bernays–Gödel class theory	1
von Neumann–Bernays–Gödel set theory by class–set distinction	1
von Neumann–Gödel–Bernays set theory	1

How this entity was disambiguated

This entity first appeared as the object of triple T131677 — 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: von Neumann–Bernays–Gödel set theory
Context triple: [John von Neumann, notableConcept, von Neumann–Bernays–Gödel set theory]

A. 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.
B. von Neumann universe
The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.
C. von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
D. 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.
E. Cantor’s paradox
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.
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: von Neumann–Bernays–Gödel set theory
Target entity description: Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.

A. 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.
B. von Neumann universe
The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.
C. von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
D. 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.
E. Cantor’s paradox
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.
F. None of above. chosen

Statements (50)

Predicate	Object
instanceOf	axiomatic set theory ⓘ conservative extension of Zermelo–Fraenkel set theory ⓘ set theory ⓘ two-sorted first-order theory ⓘ
allows	quantification over sets ⓘ
distinguishesBetween	classes ⓘ sets ⓘ
extends	Zermelo–Fraenkel set theory ⓘ
formalizes	talk about proper classes such as the class of all sets ⓘ
hasAbbreviation	NBG ⓘ linked to: von Neumann–Bernays–Gödel set theory
hasAlternativeName	NBG set theory ⓘ linked to: von Neumann–Bernays–Gödel set theory von Neumann–Gödel–Bernays set theory ⓘ linked to: von Neumann–Bernays–Gödel set theory
hasAxiom	axiom of choice (for sets) ⓘ axiom of empty set ⓘ axiom of extensionality ⓘ axiom of foundation ⓘ axiom of infinity ⓘ axiom of pairing ⓘ axiom of replacement (for sets) ⓘ axiom of union ⓘ axiom schema of class comprehension (restricted) ⓘ
hasFeature	all sets are classes but not conversely ⓘ conservative over ZF for set-theoretic statements ⓘ finite axiomatizability (with global choice) ⓘ proper classes cannot be members of any class ⓘ treats classes as first-order objects ⓘ
hasHistoricalOrigin	von Neumann's work on axiomatizing set theory in the 1920s ⓘ
hasKeyConcept	definable classes ⓘ global choice ⓘ proper class ⓘ set-class distinction ⓘ
hasLanguage	first-order language with membership and class predicates ⓘ
hasSort	class ⓘ set ⓘ
hasVariant	NBG with global choice ⓘ NBG without global choice ⓘ
isConservativeOver	Zermelo–Fraenkel set theory ⓘ
isEquiconsistentWith	Zermelo–Fraenkel set theory with choice ⓘ linked to: Zermelo–Fraenkel set theory
isRelatedTo	Morse–Kelley set theory ⓘ linked to: Morse–Kelley set theory by class–set distinction
isUsedIn	axiomatic set theory ⓘ category theory foundations ⓘ class theory ⓘ foundations of mathematics ⓘ
isWeakerThan	Morse–Kelley set theory (in proof-theoretic strength) ⓘ linked to: Morse–Kelley set theory by class–set distinction
restricts	quantification over classes to formulas without class quantifiers (in standard formulation) ⓘ
wasDevelopedBy	John von Neumann ⓘ Kurt Gödel ⓘ Paul Bernays ⓘ
wasFurtherDevelopedBy	Kurt Gödel in the 1940s ⓘ
wasRefinedBy	Paul Bernays in the 1930s ⓘ

How these facts were elicited

Referenced by (16)

Full triples — surface form annotated when it differs from this entity's canonical label.

John von Neumann → notableConcept → von Neumann–Bernays–Gödel set theory ⓘ

Cantor’s paradox → avoidedIn → von Neumann–Bernays–Gödel set theory ⓘ

Burali-Forti paradox → resolvedIn → Zermelo–Fraenkel set theory by treating the collection of all ordinals as a proper class ⓘ