Math 2.0

Mathematical logic and foundations Definition

Inner model

Builds on ClassOrdinal numberZermelo–Fraenkel set theory

Definition

Work in ZF, and let MM be a class. The class MM is an inner model if:

  1. MM is transitive: if x∈Mx \in M and y∈xy \in x, then y∈My \in M.
  2. MM contains every ordinal.
  3. (M,∈)(M, \in) satisfies each axiom of ZF, that is, the relativization φM\varphi^M holds for each axiom φ\varphi of ZF.

Remarks

The universe VV and the constructible universe LL are inner models. For LL, clause 1 holds because each level LαL_\alpha is transitive, and clause 2 holds because α∈Lα+1\alpha \in L_{\alpha + 1}. The set Lω1L_{\omega_1} is a non-example: it is transitive, but it fails clause 2, since ω1∉Lω1\omega_1 \notin L_{\omega_1}. Clause 3 is a schema, one statement for each axiom, because satisfaction in a proper class is not expressed by a single formula. A transitive proper class that satisfies ZF contains all ordinals, so some authors drop clause 2 and ask instead that MM be a proper class. Some authors ask for the axioms of ZFC; the name "inner model of ZFC" makes this choice explicit.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first.

Show the path as a graph
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 levelClassconceptInner model, definition, no evidence levelInner modeldefinition
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. Inner model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Inner_model
About this page
TagWH1MR8VHE3
URIm2://WH1MR8VHE3
Kinddefinition
WritingW0 stub
MSC03E45
DefinesInner model
UsesClass, Ordinal number, Zermelo–Fraenkel set theory
Changed2 November 2026

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