model theory


Model theoryMathworldPlanetmath is a branch of mathematical logic which deals with the general problem of classifying mathematical structures. While every branch of mathematics has its own classifying problem (such as the classification of finite simple groups (http://planetmath.org/ExamplesOfFiniteSimpleGroups) or the classification of smooth manifolds), the approach of model theory is unique in attempting to classify structuresMathworldPlanetmath in general by the sentencesMathworldPlanetmath which are true of those structures. Like most classification problems, this problem is almost certainly unsolvable when stated in full generality. However, many special cases are worthy of study.

The basic theoretical notions in model theory are the structure and the theory. A structure (http://planetmath.org/StructuresAndSatisfaction), sometimes called an L-structure, is a set with associated symbols (http://planetmath.org/SignaturePlanetmathPlanetmathPlanetmath) representing constants, relationsMathworldPlanetmath, and functions. A theory is a collectionMathworldPlanetmath of sentences in a formal languageMathworldPlanetmath. We say that a structure 𝔎 satisfies a sentence φ provided that the sentence is true no matter how its variables are interpreted in 𝔎. If 𝔎 satisfies all sentences of a theory T, then we say that 𝔎 is a model or T, or 𝔎 models T, and write 𝔎T. This allows us to talk about the class of models of a particular theory or the theory of a particular structure.

Basic constructions in model theory include elementary extensions, ultrapowers and ultraproductsMathworldPlanetmath, and the elimination of imaginaries. Some of the basic results of model theory include the Löwenheim-Skolem theorem (http://planetmath.org/DownwardLowenheimSkolemTheorem) (in downward and upward (http://planetmath.org/UpwardLowenheimSkolemTheorem) forms), the compactness theorem for first-order logic (http://planetmath.org/CompactnessTheoremForFirstOrderLogic), and Łoś’s theorem. Important conceptsMathworldPlanetmath include Morley rank, o-minimality, and quantifier eliminationMathworldPlanetmath.

Early pioneers of model theory include Löwenheim, Skolem, Gödel, Tarski, and Maltsev, and the work of Henkin, Robinson, and (again) Tarski helped distinguish model theory from the rest of mathematical logic. Ax and Kochen used model theory to prove a result on Diophantine problems over function fields, and Hrushovski used model theory to prove the Mordell–Lang conjecture for function fields.

References

  • 1 C. C. Chang and H. J. Keisler, Model theory, North-Holland Publishing Company, Amsterdam, 1973.
  • 2 W. Hodges, A shorter model theory, Cambridge University Press, 1997 (2000).
  • 3 M. Manzano, Model theory, Oxford Logic Guides 37, Clarendon Press, Oxford, 1999.
  • 4 B. Poizat, Course in model theory: an introduction in contemporary mathematical logic, Springer Verlag, 1999.
Title model theory
Canonical name ModelTheory
Date of creation 2013-03-22 16:27:30
Last modified on 2013-03-22 16:27:30
Owner mps (409)
Last modified by mps (409)
Numerical id 7
Author mps (409)
Entry type Topic
Classification msc 03C99
Related topic AxiomaticAndCategoricalFoundationsOfMathematicsII2