In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).^[1] The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954.^[2] Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.

Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics. This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane".^[3] The applications of model theory to algebraic and Diophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

The most prominent scholarly organization in the field of model theory is the Association for Symbolic Logic.

Overview

This page focuses on finitary first order model theory of infinite structures.

The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively:

model theory = universal algebra + logic^[4]

where universal algebra stands for mathematical structures and logic for logical theories; and

model theory = algebraic geometry − fields.

where logical formulas are to definable sets what equations are to varieties over a field.^[5]

Nonetheless, the interplay of classes of models and the sets definable in them has been crucial to the development of model theory throughout its history. For instance, while stability was originally introduced to classify theories by their numbers of models in a given cardinality, stability theory proved crucial to understanding the geometry of definable sets.

Fundamental notions of first-order model theory

First-order logic

A first-order formula is built out of atomic formulas such as ${\displaystyle R(f(x,y),z)}$ or ${\displaystyle y=x+1}$ by means of the Boolean connectives ${\displaystyle \neg ,\land ,\lor ,\rightarrow }$ and prefixing of quantifiers ${\displaystyle \forall v}$ or ${\displaystyle \exists v}$. A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are ${\displaystyle \varphi }$ (or ${\displaystyle \varphi (x)}$ to indicate ${\displaystyle x}$ is the unbound variable in ${\displaystyle \varphi }$) and ${\displaystyle \psi }$ (or ${\displaystyle \psi (x)}$), defined as follows:

${\displaystyle {\begin{array}{lcl}\varphi &=&\forall u\forall v(\exists w(x\times w=u\times v)\rightarrow (\exists w(x\times w=u)\lor \exists w(x\times w=v)))\land x\neq 0\land x\neq 1,\\\psi &=&\forall u\forall v((u\times v=x)\rightarrow (u=x)\lor (v=x))\land x\neq 0\land x\neq 1.\end{array}}}$

(Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the σ_smr-structure ${\displaystyle {\mathcal {N}}}$ of the natural numbers, for example, an element ${\displaystyle n}$ satisfies the formula ${\displaystyle \varphi }$ if and only if ${\displaystyle n}$ is a prime number. The formula ${\displaystyle \psi }$ similarly defines irreducibility. Tarski gave a rigorous definition, sometimes called "Tarski's definition of truth", for the satisfaction relation ${\displaystyle \models }$, so that one easily proves:

${\displaystyle {\mathcal {N}}\models \varphi (n)\iff n}$ is a prime number.

${\displaystyle {\mathcal {N}}\models \psi (n)\iff n}$ is irreducible.

A set ${\displaystyle T}$ of sentences is called a (first-order) theory, which takes the sentences in the set as its axioms. A theory is satisfiable if it has a model ${\displaystyle {\mathcal {M}}\models T}$, i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set ${\displaystyle T}$. A complete theory is a theory that contains every sentence or its negation. The complete theory of all sentences satisfied by a structure is also called the theory of that structure.

It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems) that a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory. Therefore, model theorists often use "consistent" as a synonym for "satisfiable".

Basic model-theoretic concepts

A signature or language is a set of non-logical symbols such that each symbol is either a constant symbol, or a function or relation symbol with a specified arity. Note that in some literature, constant symbols are considered as function symbols with zero arity, and hence are omitted. A structure is a set ${\displaystyle M}$ together with interpretations of each of the symbols of the signature as relations and functions on ${\displaystyle M}$ (not to be confused with the formal notion of an "interpretation" of one structure in another).

Example: A common signature for ordered rings is ${\displaystyle \sigma _{or}=(0,1,+,\times ,-)}$, where ${\displaystyle 0}$ and ${\displaystyle 1}$ are 0-ary function symbols (also known as constant symbols), ${\displaystyle +}$ and ${\displaystyle \times }$ are binary (= 2-ary) function symbols, ${\displaystyle -}$Zdroj:https://en.wikipedia.org?pojem=Model-theoretic
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