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Mathematical logic and foundations Set theory: axioms and basics Concept

Von Neumann–Bernays–Gödel set theory

Builds on Set theoryClassZermelo–Fraenkel set theory

Summary

Von Neumann–Bernays–Gödel set theory, NBG, is an axiomatic set theory whose objects are classes. In it, sets are the classes that are members of other classes. It proves the same statements about sets as Zermelo–Fraenkel set theory with choice. Unlike ZFC, it can be given by finitely many axioms. It lets one speak directly about large collections, such as the class of all sets or of all groups.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionOrdinal number, concept, no evidence levelOrdinal numberconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionAxiom of extensionality, statement, no evidence levelAxiom of extensionali…statementMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionPredicate logic, concept, no evidence levelPredicate logicconceptLogical equivalence, definition, no evidence levelLogical equivalencedefinitionZermelo–Fraenkel set theory, concept, no evidence levelZermelo–Fraenkel set …conceptSet theory, concept, no evidence levelSet theoryconceptDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptRussell's paradox, statement, no evidence levelRussell's paradoxstatementClass, concept, no evidence levelClassconceptVon Neumann–Bernays–Gödel set theory, concept, no evidence levelVon Neumann–Bernays–G…concept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Ordinal number concept
  3. Proposition concept
  4. Set concept
  5. Set definition
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Negation definition
  10. Predicate definition
  11. Biconditional definition
  12. Truth table concept
  13. Modus ponens concept
  14. Universal quantifier definition
  15. Axiom of extensionality statement
  16. Mathematical proof concept
  17. Propositional logic concept
  18. Tautology definition
  19. Predicate logic concept
  20. Logical equivalence definition
  21. Zermelo–Fraenkel set theory concept
  22. Set theory concept
  23. Double negation law statement
  24. Proof by contradiction concept
  25. Russell's paradox statement
  26. Class concept
  27. Von Neumann–Bernays–Gödel set theory concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Von_Neumann–Bernays–Gödel_set_theory
About this page
TagJTPZ1HZ9KK
URIm2://JTPZ1HZ9KK
Kindconcept
WritingW0 stub
MSC03E30, 03E70
SpecializesSet theory
UsesSet theory, Class, Zermelo–Fraenkel set theory
Same as
Changed2 November 2026

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