model theory

model theory

Model theory

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

structures

in general by the

sentences

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/

Signature

) representing

constants

,

relations

, and

functions

. A

theory

is a

collection

of sentences in a

formal language

. 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

𝔎

⊧

T

. 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

ultraproducts

, 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

concepts

include Morley rank,

o-minimality

, and

quantifier elimination

.

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

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