Math 2.0

Mathematical logic and foundations Definition

Woodin cardinal

Builds on Cardinal numberElementary embeddingVon Neumann universe

Definition

Let κ\kappa be a cardinal. For an ordinal γ\gamma, let VγV_\gamma be the set of sets of rank less than γ\gamma in the von Neumann universe. The cardinal κ\kappa is a Woodin cardinal if for every function f ⁣:κ→κf \colon \kappa \to \kappa there is an ordinal α<κ\alpha < \kappa such that:

  1. f(β)<αf(\beta) < \alpha for every β<α\beta < \alpha.
  2. There is an elementary embedding j ⁣:V→Mj \colon V \to M, with MM a transitive class, whose critical point is α\alpha.
  3. Vj(f)(α)⊆MV_{j(f)(\alpha)} \subseteq M.

Remarks

If κ\kappa is a Woodin cardinal, then κ\kappa is Mahlo, and there are κ\kappa measurable cardinals below κ\kappa. Every supercompact cardinal is a Woodin cardinal. A non-example: a measurable cardinal κ\kappa that is the least measurable cardinal is not Woodin, since a Woodin cardinal has measurable cardinals below it. Martin and Steel proved that if there are nn Woodin cardinals with a measurable cardinal above them all, then every Πn+11\Pi^1_{n+1} set of reals is determined.

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first.

Show the path as a graph
Cardinal number, concept, no evidence levelCardinal numberconceptElement, definition, no evidence levelElementdefinitionElementary embedding, concept, no evidence levelElementary embeddingconceptInaccessible 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 universeobjectWoodin cardinal, definition, no evidence levelWoodin cardinaldefinition
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. Cardinal number concept
  2. Element definition
  3. Elementary embedding concept
  4. Inaccessible cardinal concept
  5. Ordinal number concept
  6. Proposition concept
  7. Set concept
  8. Well-founded relation concept
  9. Set definition
  10. Conjunction definition
  11. Disjunction definition
  12. Implication definition
  13. Negation definition
  14. Predicate definition
  15. Ordered pair definition
  16. Biconditional definition
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Axiom of extensionality statement
  20. Subset definition
  21. Propositional logic concept
  22. Binary relation definition
  23. Power set definition
  24. Predicate logic concept
  25. Antisymmetric relation definition
  26. Reflexive relation definition
  27. Transitive relation definition
  28. Zermelo–Fraenkel set theory concept
  29. Preorder definition
  30. Axiom of regularity concept
  31. Partial order definition
  32. Minimal element definition
  33. Well-founded relation definition
  34. Transfinite induction statement
  35. Von Neumann universe object
  36. Woodin cardinal definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Woodin_cardinal
About this page
Tag592QGNJCZC
URIm2://592QGNJCZC
Kinddefinition
WritingW0 stub
MSC03E55, 03E60
DefinesWoodin cardinal
UsesCardinal number, Elementary embedding, Von Neumann universe
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.