Math 2.0

Mathematical logic and foundations Definition

Model theory

Builds on TheoryModelStructurePredicate logic

Definition

Model theory is the part of mathematical logic whose objects are the following.

  1. Formal languages, most often the language of first-order logic over a signature.
  2. Theories, that is, sets of sentences of such a language.
  3. Structures for the signature, and the relation of truth between structures and sentences, in particular the models of theories.

It studies the classes of models of theories and the sets definable in structures.

Remarks

Example: the statement that every field elementarily equivalent to ℂ\C is algebraically closed of characteristic 00 is a result of model theory. It concerns the class of models of the complete theory Th(ℂ)\mathrm{Th}(\C).

Non-example: the computation of the homology of a sphere is not a result of model theory, because it concerns no formal language and no relation of truth.

Classical model theory works with first-order logic; other parts study infinitary logic, finite structures or second-order logic.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Model, concept, no evidence levelModelconceptProposition, concept, no evidence levelPropositionconceptStructure, concept, no evidence levelStructureconceptTheory, concept, no evidence levelTheoryconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptModel theory, definition, no evidence levelModel theorydefinition
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. Model concept
  2. Proposition concept
  3. Structure concept
  4. Theory concept
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Negation definition
  9. Predicate definition
  10. Biconditional definition
  11. Propositional logic concept
  12. Predicate logic concept
  13. Model theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Model_theory
About this page
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Kinddefinition
WritingW0 stub
MSC03C07, 03C98
DefinesModel theory
UsesTheory, Model, Structure, Predicate logic
Changed2 November 2026

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