Model theory

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Model theory
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{{short description|subfield of mathematical logic}}{{About|the mathematical discipline|the informal notion in other parts of mathematics and science|Mathematical model}}In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is one of the components that form a theory. A model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory.Model theory recognizes and is intimately concerned with a duality: it examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language: in a summary definition, dating from 1973,
universal algebra + logic = model theory.Chang and Keisler, p. 1
Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997):
model theory = algebraic geometry − fields.
Other nearby areas of mathematics include combinatorics, number theory, arithmetic dynamics, analytic functions, and non-standard analysis.In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. The most prominent professional organization in the field of model theory is the Association for Symbolic Logic.

Branches of model theory

This article focuses on finitary first order model theory of infinite structures. Finite model theory, which concentrates on finite structures, diverges significantly from the study of infinite structures in both the problems studied and the techniques used. Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not in general hold for these logics. However, a great deal of study has also been done in such logics.Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic.Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of non-standard analysis. An important step in the evolution of classical model theory occurred with the birth of stability theory (through Morley's theorem on uncountably categorical theories and Shelah's classification program), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories.During the last several decades applied model theory has repeatedly merged with the more pure stability theory. The result of this synthesis is called geometric model theory in this article (which is taken to include o-minimality, for example, as well as classical geometric stability theory). An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields. The ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory.

Universal algebra

Fundamental concepts in universal algebra are signatures σ and σ-algebras. Since these concepts are formally defined in the article on structures, the present article is an informal introduction which consists of examples of the way these terms are used.
The standard signature of rings is σring = {×,+,−,0,1}, where × and + are binary, − is unary, and 0 and 1 are nullary. The standard signature of semirings is σsmr = {×,+,0,1}, where the arities are as above. The standard signature of groups (with multiplicative notation) is σgrp = {×,−1,1}, where × is binary, −1 is unary and 1 is nullary. The standard signature of monoids is σmnd = {×,1}. A ring is a σring-structure which satisfies the identities {{nowrap|u + (v + w) {{=}} (u + v) + w,}} {{nowrap|u + v {{=}} v + u,}} {{nowrap|u + 0 {{=}} u,}} {{nowrap|u + (−u) {{=}} 0,}} {{nowrap|u × (v × w) {{=}} (u × v) × w,}} {{nowrap|u × 1 {{=}} u,}} {{nowrap|1 × u {{=}} u,}} {{nowrap|u × (v + w) {{=}} (u × v) + (u × w)}} and {{nowrap|(v + w) × u {{=}} (v × u) + (w × u).}} A group is a σgrp-structure which satisfies the identities {{nowrap|u × (v × w) {{=}} (u × v) × w,}} {{nowrap|u × 1 {{=}} u,}} {{nowrap|1 × u {{=}} u,}} {{nowrap|u × u−1 {{=}} 1}} and {{nowrap|u−1 × u {{=}} 1.}} A monoid is a σmnd-structure which satisfies the identities {{nowrap|u × (v × w) {{=}} (u × v) × w,}} {{nowrap|u × 1 {{=}} u}} and {{nowrap|1 × u {{=}} u.}} A semigroup is a {×}-structure which satisfies the identity {{nowrap|u × (v × w) {{=}} (u × v) × w.}} A magma is just a {×}-structure.
This is a very efficient way to define most classes of algebraic structures, because there is also the concept of σ-homomorphism, which correctly specializes to the usual notions of homomorphism for groups, semigroups, magmas and rings. For this to work, the signature must be chosen well.Terms such as the σring-term t(u,v,w) given by {{nowrap|(u + (v × w)) + (−1)}} are used to define identities {{nowrap|t {{=}} t{{'}},}} but also to construct free algebras. An equational class is a class of structures which, like the examples above and many others, is defined as the class of all σ-structures which satisfy a certain set of identities. Birkhoff's theorem states:
A class of σ-structures is an equational class if and only if it is not empty and closed under subalgebras, homomorphic images, and direct products.
An important non-trivial tool in universal algebra are ultraproducts Pi_{iin I}A_i/U, where I is an infinite set indexing a system of σ-structures Ai, and U is an ultrafilter on I.While model theory is generally considered a part of mathematical logic, universal algebra, which grew out of Alfred North Whitehead's (1898) work on abstract algebra, is part of algebra. This is reflected by their respective MSC classifications. Nevertheless, model theory can be seen as an extension of universal algebra.

Finite model theory

Finite model theory is the area of model theory which has the closest ties to universal algebra. Like some parts of universal algebra, and in contrast with the other areas of model theory, it is mainly concerned with finite algebras, or more generally, with finite σ-structures for signatures σ which may contain relation symbols as in the following example:
The standard signature for graphs is σgrph={E}, where E is a binary relation symbol. A graph is a σgrph-structure satisfying the sentence forall u forall v(uEv rightarrow vEu).
A σ-homomorphism is a map that commutes with the operations and preserves the relations in σ. This definition gives rise to the usual notion of graph homomorphism, which has the interesting property that a bijective homomorphism need not be invertible. Structures are also a part of universal algebra; after all, some algebraic structures such as ordered groups have a binary relation }}

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