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.

Aliases (6)

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	von Neumann–Bernays–Gödel set theory → surface form: "NBG"
hasAlternativeName	von Neumann–Bernays–Gödel set theory → surface form: "NBG set theory" von Neumann–Bernays–Gödel set theory → surface form: "von Neumann–Gödel–Bernays 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 → surface form: "Zermelo–Fraenkel set theory with choice"
isRelatedTo	Morse–Kelley set theory by class–set distinction → surface form: "Morse–Kelley set theory"
isUsedIn	axiomatic set theory → category theory foundations → class theory → foundations of mathematics →
isWeakerThan	Morse–Kelley set theory by class–set distinction → surface form: "Morse–Kelley set theory (in proof-theoretic strength)"
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 →

Referenced by (11)

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

Grothendieck universe → alternativeTo → von Neumann–Bernays–Gödel set theory →

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

von Neumann–Bernays–Gödel set theory → hasAbbreviation → von Neumann–Bernays–Gödel set theory →

this entity surface form: "NBG"

von Neumann–Bernays–Gödel set theory → hasAlternativeName → von Neumann–Bernays–Gödel set theory →

this entity surface form: "von Neumann–Gödel–Bernays set theory"

von Neumann–Bernays–Gödel set theory → hasAlternativeName → von Neumann–Bernays–Gödel set theory →

this entity surface form: "NBG set theory"

set theory → hasAxiomSystem → von Neumann–Bernays–Gödel set theory →

Morse–Kelley set theory by class–set distinction → isRelatedTo → von Neumann–Bernays–Gödel set theory →

this entity surface form: "von Neumann–Bernays–Gödel class theory"

Morse–Kelley set theory by class–set distinction → isStrongerThan → von Neumann–Bernays–Gödel set theory →

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

Burali-Forti paradox → resolvedIn → von Neumann–Bernays–Gödel set theory →

this entity surface form: "Zermelo–Fraenkel set theory by treating the collection of all ordinals as a proper class"

Burali-Forti paradox → resolvedIn → von Neumann–Bernays–Gödel set theory →

this entity surface form: "von Neumann–Bernays–Gödel set theory by class–set distinction"