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Mathematical logic and foundations Definition

Von Neumann–Bernays–Gödel set theory

Builds on Set theoryClassZermelo–Fraenkel set theoryAxiom of extensionalityAxiom of choiceAxiom of regularity

Definition

Von Neumann–Bernays–Gödel set theory, written NBG, is the first-order theory in the language {∈}\set{\in} whose variables range over classes. A class XX is a set if X∈YX \in Y for some class YY, and a proper class otherwise. The axioms of NBG are these.

  1. Extensionality: classes with the same elements are equal, as in the axiom of extensionality.
  2. Pairing, union, power set and infinity: as in ZF, stated for sets.
  3. Foundation: the axiom of regularity, for all nonempty classes.
  4. Replacement: if FF is a class that is a function and aa is a set, then the image F[a]F[a] is a set.
  5. Class comprehension: for each formula φ(x)\varphi(x) in which all quantifiers range over sets, possibly with class parameters, there is a class {x:x is a set and φ(x)}\set{x : x \text{ is a set and } \varphi(x)}.
  6. Global choice: there is a class function that picks an element from every nonempty set.

Remarks

Class comprehension can be replaced by finitely many of its instances, so NBG has a finite list of axioms; ZF has no finite axiomatization if it is consistent. NBG is a conservative extension of ZFC: a statement about sets is provable in NBG if and only if it is provable in ZFC. Example: the class V={x:x=x}V = \set{x : x = x} of all sets exists by comprehension, with φ(x)\varphi(x) the formula x=xx = x. Non-example: VV is not a set. If it were, then comprehension would give the class R={x:x∉x}R = \set{x : x \notin x}, which would be a set as a subclass of VV, and then R∈R↔R∉RR \in R \leftrightarrow R \notin R. Morse–Kelley set theory is the stronger theory in which the formula in class comprehension may also quantify over classes. Some authors drop global choice, or replace it by the axiom of choice for sets; NBG without global choice is conservative over ZF.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first.

  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Negation definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Union definition
  13. Biconditional definition
  14. Truth table concept
  15. Modus ponens concept
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Propositional logic concept
  20. Subset definition
  21. Axiom of extensionality statement
  22. Union of a family of sets definition
  23. Mathematical proof concept
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Power set definition
  27. Predicate logic concept
  28. Tautology definition
  29. Antisymmetric relation definition
  30. Reflexive relation definition
  31. Transitive relation definition
  32. Zermelo–Fraenkel set theory concept
  33. Function definition
  34. Logical equivalence definition
  35. Preorder definition
  36. Choice function definition
  37. Codomain definition
  38. Injective function definition
  39. Double negation law statement
  40. Partial order definition
  41. Axiom of choice statement
  42. Surjective function definition
  43. Proof by contradiction concept
  44. Least element definition
  45. Minimal element definition
  46. Total order definition
  47. Russell's paradox statement
  48. Bijective function definition
  49. Well-founded relation definition
  50. Well-order definition
  51. Countably infinite set definition
  52. Equinumerous sets definition
  53. Ordinal number definition
  54. Axiom of regularity definition
  55. Cardinality definition
  56. Cardinal number definition
  57. Class definition
  58. Order of cardinals definition
  59. Aleph-null object
  60. Cardinality of the continuum object
  61. Continuum hypothesis statement
  62. Set theory definition
  63. Von Neumann–Bernays–Gödel set theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Von_Neumann–Bernays–Gödel_set_theory
About this page
TagTY89DBMJYQ
URIm2://TY89DBMJYQ
Kinddefinition
WritingW0 stub
MSC03E30, 03E70
DefinesVon Neumann–Bernays–Gödel set theory
UsesSet theory, Class, Zermelo–Fraenkel set theory, Axiom of extensionality, Axiom of choice, Axiom of regularity
Changed2 November 2026

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