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Mathematical logic and foundations Definition

Axiom of constructibility

Builds on Constructible universeZermelo–Fraenkel set theoryOrdinal number

Definition

Work in ZF, and let L=⋃αLαL = \bigcup_\alpha L_\alpha be the constructible universe, where α\alpha runs over the ordinals. The axiom of constructibility is the sentence ∀x ∃α (α is an ordinal∧x∈Lα).\forall x \, \exists \alpha \, (\alpha \text{ is an ordinal} \wedge x \in L_\alpha). It is written V=LV = L, where VV is the class of all sets.

Remarks

The sentence holds inside LL itself: the construction of LαL_\alpha is absolute for transitive models of ZF that contain α\alpha, so the class LL computed inside LL is again LL. This gives the relative consistency of V=LV = L with ZF. For a non-example, take a forcing extension L[c]L[c] of LL by a Cohen real cc; the real cc is not in LL, so V=LV = L fails in L[c]L[c]. Under V=LV = L there is a well-order of the reals that is Δ21\Delta^1_2 in the projective hierarchy. Scott proved that V=LV = L implies that there is no measurable cardinal.

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 …conceptDouble 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 levelClassconceptConstructible universe, object, no evidence levelConstructible universeobjectAxiom of constructibility, definition, no evidence levelAxiom of constructibi…definition
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. Double negation law statement
  23. Proof by contradiction concept
  24. Russell's paradox statement
  25. Class concept
  26. Constructible universe object
  27. Axiom of constructibility definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Axiom_of_constructibility
About this page
TagQ1VYK7362X
URIm2://Q1VYK7362X
Kinddefinition
WritingW0 stub
MSC03E45, 03E35
DefinesAxiom of constructibility
UsesConstructible universe, Zermelo–Fraenkel set theory, Ordinal number
Changed2 November 2026

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