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Mathematical logic and foundations Set theory: axioms and basics Object

Von Neumann universe

Builds on Ordinal numberPower setTransfinite inductionAxiom of regularityInaccessible cardinal

Description

The von Neumann universe VV is the class of all sets, arranged in stages indexed by the ordinal numbers. The stages VαV_\alpha are defined by transfinite recursion on the ordinal α\alpha:

  1. V0=∅V_0 = \varnothing.
  2. Vα+1=𝒫(Vα)V_{\alpha + 1} = \mathcal{P}(V_\alpha), the power set of VαV_\alpha.
  3. Vλ=⋃β<λVβV_\lambda = \bigcup_{\beta < \lambda} V_\beta for each limit ordinal λ\lambda.

Then V=⋃αVαV = \bigcup_{\alpha} V_\alpha, the union over all ordinals. The rank of a set xx in VV is the least ordinal α\alpha such that x⊆Vαx \subseteq V_\alpha.

Remarks

Each VαV_\alpha is a transitive set, and Vβ⊆VαV_\beta \subseteq V_\alpha when β≤α\beta \le \alpha. The first stages are V1={∅}V_1 = \{\varnothing\} and V2={∅,{∅}}V_2 = \{\varnothing, \{\varnothing\}\}, and in general ∣Vn+1∣=2∣Vn∣\abs{V_{n+1}} = 2^{\abs{V_n}} for finite nn. The stage VωV_\omega is the set of hereditarily finite sets, and it is countable. Over the other axioms of ZF, the axiom of regularity is equivalent to the statement that every set lies in some VαV_\alpha. If κ\kappa is an inaccessible cardinal, then VκV_\kappa is a model of ZFC.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionInaccessible cardinal, concept, no evidence levelInaccessible cardinalconceptOrdinal number, concept, no evidence levelOrdinal numberconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptWell-founded relation, concept, no evidence levelWell-founded relationconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionAxiom of extensionality, statement, no evidence levelAxiom of extensionali…statementSubset, definition, no evidence levelSubsetdefinitionPropositional logic, concept, no evidence levelPropositional logicconceptBinary relation, definition, no evidence levelBinary relationdefinitionPower set, definition, no evidence levelPower setdefinitionPredicate logic, concept, no evidence levelPredicate logicconceptAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionZermelo–Fraenkel set theory, concept, no evidence levelZermelo–Fraenkel set …conceptPreorder, definition, no evidence levelPreorderdefinitionAxiom of regularity, concept, no evidence levelAxiom of regularityconceptPartial order, definition, no evidence levelPartial orderdefinitionMinimal element, definition, no evidence levelMinimal elementdefinitionWell-founded relation, definition, no evidence levelWell-founded relationdefinitionTransfinite induction, statement, no evidence levelTransfinite inductionstatementVon Neumann universe, object, no evidence levelVon Neumann universeobject
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. Inaccessible cardinal concept
  3. Ordinal number concept
  4. Proposition concept
  5. Set concept
  6. Well-founded relation concept
  7. Set definition
  8. Conjunction definition
  9. Disjunction definition
  10. Implication definition
  11. Negation definition
  12. Predicate definition
  13. Ordered pair definition
  14. Biconditional definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Axiom of extensionality statement
  18. Subset definition
  19. Propositional logic concept
  20. Binary relation definition
  21. Power set definition
  22. Predicate logic concept
  23. Antisymmetric relation definition
  24. Reflexive relation definition
  25. Transitive relation definition
  26. Zermelo–Fraenkel set theory concept
  27. Preorder definition
  28. Axiom of regularity concept
  29. Partial order definition
  30. Minimal element definition
  31. Well-founded relation definition
  32. Transfinite induction statement
  33. Von Neumann universe object

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Von_Neumann_universe
About this page
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UsesOrdinal number, Power set, Transfinite induction, Axiom of regularity, Inaccessible cardinal
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Changed2 November 2026

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