TheInfoList

In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...
, model theory is the study of the relationship between
formal theories Formal, formality, informal or informality imply the complying with, or not complying with, some set theory, set of requirements (substantial form, forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for forma ...
(a collection of
sentences ''The Four Books of Sentences'' (''Libri Quattuor Sententiarum'') is a book of theology Theology is the systematic study of the nature of the Divinity, divine and, more broadly, of religious belief. It is taught as an Discipline (academia), aca ...
in a
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...
mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
), and their models (those structures in which the statements of the theory hold). 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 Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. was a Polish-American logician a ...
, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by
Saharon Shelah Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical str ...

's
stability theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Compared to other areas of mathematical logic such as
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
, 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 Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
is about the sacred, then model theory is about the profane"''. The applications of model theory to and
diophantine geometry In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be wr ...
reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. The most prominent scholarly organization in the field of model theory is the
Association for Symbolic Logic The Association for Symbolic Logic (ASL) is an international organization ''International Organization'' is a quarterly peer-reviewed academic journal that covers the entire field of international relations, international affairs. It was establish ...
.

# Varied emphasis

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 algebraUniversal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...
+
logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit ...

where universal algebra stands for mathematical structures and logic for logical theories; and :model theory =
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
s. where logical formulas are to definable sets what equations are to varieties over a field. 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.

# Branches

finitary In mathematics and logic, an Operation (mathematics), operation is finitary if it has Finite cardinality, finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an Infinite set, infinite numbe ...
first order model theory of infinite structures.
Finite model theoryFinite model theory (FMT) is a subarea of model theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, 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 logic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s or
infinitary logicAn infinitary logic is a logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, though ...
s is hampered by the fact that completeness and
compactness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Gödel's completeness theorem is a fundamental theorem in mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...
, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem,
Scott Scott may refer to: Places Canada * Scott, Quebec, municipality in the Nouvelle-Beauce regional municipality in Quebec * Scott, Saskatchewan, a town in the Rural Municipality of Tramping Lake No. 380 * Rural Municipality of Scott No. 98, Saskat ...
'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 The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard ...
. An important step in the evolution of classical model theory occurred with the birth of
stability theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(through Morley's theorem on uncountably categorical theories and
ShelahShelah may refer to: * Shelah (son of Judah), a son of Judah according to the Bible * Shelah (name), a Hebrew personal name * Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading * Saleh, a prophet described in ...

'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 proof from geometric model theory is '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.

# Fundamental notions of first-order model theory

## First-order logic

A first-order ''formula'' is built out of
atomic formula In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...
s such as ''R''(''f''(''x'',''y''),''z'') or ''y'' = ''x'' + 1 by means of the Boolean connectives $\neg,\land,\lor,\rightarrow$ and prefixing of quantifiers $\forall v$ or $\exists v$. A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are φ (or φ(x) to mark the fact that at most x is an unbound variable in φ) and ψ defined as follows: : :$\psi \;=\; \forall u\forall v\left(\left(u\times v=x\right)\rightarrow \left(u=x\right)\lor\left(v=x\right)\right)\land x\ne 0\land x\ne1.$ (Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the σsmr-structure $\mathcal N$ of the natural numbers, for example, an element ''n'' ''satisfies'' the formula φ if and only if ''n'' is a prime number. The formula ψ similarly defines irreducibility. Tarski gave a rigorous definition, sometimes called "Tarski's definition of truth", for the satisfaction relation $\models$, so that one easily proves: :$\mathcal N\models\varphi\left(n\right) \iff n$ is a prime number. :$\mathcal N\models\psi\left(n\right) \iff n$ is irreducible. A set ''T'' of sentences is called a (first-order)
theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, G ...
. A theory is ''satisfiable'' if it has a ''model'' $\mathcal M\models T$, i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set ''T''. A complete theory is a theory that contains every
sentence Sentence(s) or The Sentence may refer to: Common uses * Sentence (law), the punishment a judge gives to a defendant found guilty of a crime * Sentence (linguistics), a grammatical unit of language * Sentence (mathematical logic), a formula not cont ...
or its negation. The complete theory of all sentences satisfied by a structure is also called the ''theory of that structure''. Gödel's completeness theorem (not to be confused with his
incompleteness theorems Complete may refer to: Logic * Completeness (logic) In mathematical logic and metalogic, a formal system is called complete with respect to a particular property (philosophy), property if every Well-formed formula, formula having the property can ...
) says that a theory has a model if and only if it is
consistent In classical deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditiona ...

, 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 A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...
or
language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the ...
is a set of non-logical symbols such that each symbol is either a function symbol or a relation symbol and has a specified
arity Arity () is the number of arguments In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...
. A
structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...

is a set $M$ together with interpretations of each of the symbols of the signature as relations and functions on $M$ (not to be confused with the
interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event i ...
of one structure in another). A common signature for ordered rings is $\sigma_=\$, where $0$ and $1$ are 0-ary function symbols (also known as constant symbols), $+$ and $\times$ are binary function symbols, $-$ is a unary function symbol, and $<$ is a binary relation symbol. Then, when these symbols are interpreted to correspond with their usual meaning on $\Q$ (so that e.g. $+$ is a function from $\Q^2$ to $\Q$ and $<$ is a subset of $\Q^2$), one obtains a structure $\left(\Q,\sigma_\right)$. A structure $\mathcal$ is said to model a set of first-order sentences $T$ in the given language if each sentence in $T$ is true in $\mathcal$ with respect to the interpretation of the signature previously specified for $\mathcal$. A substructure $\mathcal A$ of a σ-structure $\mathcal B$ is a subset of its domain, closed under all functions in its signature σ, which is regarded as a σ-structure by restricting all functions and relations in σ to the subset. This generalises the analogous concepts from algebra; For instance, a subgroup is a substructure in the signature with multiplication and inverse. A substructure is said to be ''elementary'' if for any first-order formula φ and any elements ''a''1, ..., ''a''''n'' of $\mathcal A$, :$\mathcal A\models \varphi\left(a_1, ...,a_n\right)$ if and only if $\mathcal B\models \varphi\left(a_1, ...,a_n\right)$. In particular, if ''φ'' is a sentence and $\mathcal A$ an elementary substructure of $\mathcal B$, then $\mathcal A\models \varphi$ if and only if $\mathcal B\models \varphi$. Thus, an elementary substructure is a model of a theory exactly when the superstructure is a model. Therefore, while the field of algebraic numbers $\overline$ is an elementary substructure of the field of complex numbers $\mathbb$, the rational field $\mathbb$ is not, as we can express "There is a square root of 2" as a first-order sentence satisfied by $\mathbb$ but not by $\mathbb$. An
embedding In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of a σ-structure $\mathcal A$ into another σ-structure $\mathcal B$ is a map ''f'': ''A'' → ''B'' between the domains which can be written as an isomorphism of $\mathcal A$ with a substructure of $\mathcal B$. If it can be written as an isomorphism with an elementary substructure, it is called an elementary embedding. Every embedding is an
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
homomorphism, but the converse holds only if the signature contains no relation symbols, such as in groups or fields. A field or a vector space can be regarded as a (commutative) group by simply ignoring some of its structure. The corresponding notion in model theory is that of a ''reduct'' of a structure to a subset of the original signature. The opposite relation is called an ''expansion'' - e.g. the (additive) group of the
rational numbers In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, regarded as a structure in the signature can be expanded to a field with the signature or to an ordered group with the signature . Similarly, if σ' is a signature that extends another signature σ, then a complete σ'-theory can be restricted to σ by intersecting the set of its sentences with the set of σ-formulas. Conversely, a complete σ-theory can be regarded as a σ'-theory, and one can extend it (in more than one way) to a complete σ'-theory. The terms reduct and expansion are sometimes applied to this relation as well.

## Compactness and the Löwenheim-Skolem theorem

The
compactness theorem In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. The analogous statement with ''consistent'' instead of ''satisfiable'' is trivial, since every proof can have only a finite number of antecedents used in the proof. The completeness theorem allows us to transfer this to satsifiability. However, there are also several direct (semantic) proofs of the compactness theorem. As a corollary (i.e., its contrapositive), the
compactness theorem In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
says that every unsatisfiable first-order theory has a finite unsatisfiable subset. This theorem is of central importance in model theory, where the words "by compactness" are commonplace. Another cornerstone of first-order model theory is the Löwenheim-Skolem theorem. According to the Löwenheim-Skolem Theorem, every infinite structure in a countable signature has a countable elementary substructure. Conversely, for any infinite cardinal κ every infinite structure in a countable signature that is of cardinality less than κ can be elementarily embedded in another structure of cardinality κ (There is a straightforward generalisation to uncountable signatures). In particular, the Löwenheim-Skolem Theorem implies that any theory in a countable signature with infinite models has a countable model as well as arbitrarily large models. In a certain sense made precise by Lindström's theorem, first-order logic is the most expressive logic for which both the Löwenheim–Skolem theorem and the compactness theorem hold.

# Definability

## Definable sets

In model theory, definable sets are important objects of study. For instance, in $\mathbb N$ the formula :$\forall u\forall v\left(\exists w \left(x\times w=u\times v\right)\rightarrow\left(\exists w\left(x\times w=u\right)\lor\exists w\left(x\times w=v\right)\right)\right)\land x\ne 0\land x\ne1$ defines the subset of prime numbers, while the formula :$\exists y \left(2\times y = x\right)$ defines the subset of even numbers. In a similar way, formulas with ''n'' free variables define subsets of $\mathcal^n$. For example, in a field, the formula :$y = x \times x$ defines the curve of all $\left(x,y\right)$ such that $y = x^2$. Both of the definitions mentioned here are ''parameter-free'', that is, the defining formulas don't mention any fixed domain elements. However, one can also consider definitions ''with parameters from the model''. For instance, in $\mathbb$, the formula :$y = x \times x + \pi$ uses the parameter $\pi$ from $\mathbb$ to define a curve.

## Eliminating quantifiers

In general, definable sets without quantifiers are easy to describe, while definable sets involving possibly nested quantifiers can be much more complicated. This makes quantifier elimination a crucial tool for analysing definable sets: A theory ''T'' has quantifier elimination if every first-order formula φ(''x''1, ..., ''x''''n'') over its signature is equivalent modulo ''T'' to a first-order formula ψ(''x''1, ..., ''x''''n'') without quantifiers, i.e. $\forall x_1\dots\forall x_n\left(\phi\left(x_1,\dots,x_n\right)\leftrightarrow \psi\left(x_1,\dots,x_n\right)\right)$ holds in all models of ''T''. If the theory of a structure has quantifier elimination, every set definable in a structure is definable by a quantifier-free formula over the same parameters as the original definition. For example, the theory of algebraically closed fields in the signature σring = (×,+,−,0,1) has quantifier elimination. This means that in an algebraically closed field, every formula is equivalent to a Boolean combination of equations between polynomials. If a theory does not have quantifier elimination, one can add additional symbols to its signature so that it does. Early model theory spent much effort on proving axiomatizability and quantifier elimination results for specific theories, especially in algebra. But often instead of quantifier elimination a weaker property suffices: A theory ''T'' is called model-complete if every substructure of a model of ''T'' which is itself a model of ''T'' is an elementary substructure. There is a useful criterion for testing whether a substructure is an elementary substructure, called the Tarski–Vaught test. It follows from this criterion that a theory ''T'' is model-complete if and only if every first-order formula φ(''x''1, ..., ''x''''n'') over its signature is equivalent modulo ''T'' to an existential first-order formula, i.e. a formula of the following form: :$\exists v_1\dots\exists v_m\psi\left(x_1,\dots,x_n,v_1,\dots,v_m\right)$, where ψ is quantifier free. A theory that is not model-complete may or may not have a ''model completion'', which is a related model-complete theory that is not, in general, an extension of the original theory. A more general notion is that of a ''model companion''.

## Minimality

In every structure, every finite subset $\$ is definable with parameters: Simply use the formula :$x = a_1 \vee \dots \vee a_n$. Since we can negate this formula, every cofinite subset (which includes all but finitely many elements of the domain) is also always definable. This leads to the concept of a ''minimal structure''. A structure $\mathcal$ is called minimal if every subset $A \subseteq \mathcal$ definable with parameters from $\mathcal$ is either finite or cofinite. The corresponding concept at the level of theories is called ''strong minimality'': A theory ''T'' is called strongly minimal if every model of ''T'' is minimal. A structure is called ''strongly minimal'' if the theory of that structure is strongly minimal. Equivalently, a structure is strongly minimal if every elementary extension is minimal. Since the theory of algebraically closed fields has quantifier elimination, every definable subset of an algebraically closed field is definable by a quantifier-free formula in one variable. Quantifier-free formulas in one variable express Boolean combinations of polynomial equations in one variable, and since a nontrivial polynomial equation in one variable has only a finite number of solutions, the theory of algebraically closed fields is strongly minimal. On the other hand, the field $\mathbb$ of real numbers is not minimal: Consider, for instance, the definable set :$\varphi \left(x\right) \;=\; \exists y \left(y \times y = x\right)$. This defines the subset of non-negative real numbers, which is neither finite nor cofinite. One can in fact use $\varphi$ to define arbitrary intervals on the real number line. It turns out that these suffice to represent every definable subset of $\mathbb$. This generalisation of minimality has been very useful in the model theory of ordered structures. A densely totally ordered structure $\mathcal$ in a signature including a symbol for the order relation is called
o-minimal In mathematical logic, and more specifically in model theory, an infinite structure (mathematical logic), structure (''M'',<,...) which is Total order, totally ordered by < is called an o-minimal structure if and only if every definable set, defi ...
if every subset $A \subseteq \mathcal$ definable with parameters from $\mathcal$ is a finite union of points and intervals.

## Definable and interpretable structures

Particularly important are those definable sets that are also substructures, i. e. contain all constants and are closed under function application. For instance, one can study the definable subgroups of a certain group. However, there is no need to limit oneself to substructures in the same signature. Since formulas with ''n'' free variables define subsets of $\mathcal^n$, ''n''-ary relations can also be definable. Functions are definable if the function graph is a definable relation, and constants $a \in \mathcal$ are definable if there is a formula $\varphi\left(x\right)$ such that ''a'' is the only element of $\mathcal$ such that $\varphi\left(a\right)$ is true. In this way, one can study definable groups and fields in general structures, for instance, which has been important in geometric stability theory. One can even go one step further, and move beyond immediate substructures. Given a mathematical structure, there are very often associated structures which can be constructed as a quotient of part of the original structure via an equivalence relation. An important example is a quotient group of a group. One might say that to understand the full structure one must understand these quotients. When the equivalence relation is definable, we can give the previous sentence a precise meaning. We say that these structures are ''interpretable''. A key fact is that one can translate sentences from the language of the interpreted structures to the language of the original structure. Thus one can show that if a structure $\mathcal$ interprets another whose theory is undecidable, then $\mathcal$ itself is undecidable.

# Types

## Basic notions

For a sequence of elements $a_1, \dots, a_n$ of a structure $\mathcal$ and a subset ''A'' of $\mathcal$, one can consider the set of all first-order formulas $\varphi\left(x_1, \dots, x_n\right)$ with parameters in ''A'' that are satisfied by $a_1, \dots, a_n$. This is called the ''complete (n-)type realised by'' $a_1, \dots, a_n$ ''over A''. If there is an
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of $\mathcal$ that is constant on ''A'' and sends $a_1, \dots, a_n$ to $b_1, \dots, b_n$ respectively, then $a_1, \dots, a_n$ and $b_1, \dots, b_n$ realise the same complete type over ''A''. The real number line $\mathbb$, viewed as a structure with only the order relation , will serve as a running example in this section. Every element $a \in \mathbb$ satisfies the same 1-type over the empty set. This is clear since any two real numbers ''a'' and ''b'' are connected by the order automorphism that shifts all numbers by ''b-a''. The complete 2-type over the empty set realised by a pair of numbers $a_1, a_2$ depends on their order: either $a_1 < a_2$, $a_1 = a_2$ or $a_2 < a_1$. Over the subset $\mathbb \subseteq \mathbb$ of integers, the 1-type of a non-integer real number ''a'' depends on its value rounded down to the nearest integer. More generally, whenever $\mathcal$ is a structure and ''A'' a subset of $\mathcal$, a (partial) ''n-type over A'' is a set of formulas ''p'' with at most ''n'' free variables that are realised in an elementary extension $\mathcal$ of $\mathcal$. If ''p'' contains every such formula or its negation, then ''p'' is ''complete''. The set of complete ''n''-types over ''A'' is often written as $S_n^\left(A\right)$. If ''A'' is the empty set, then the type space only depends on the theory ''T'' of $\mathcal$. The notation $S_n\left(T\right)$ is commonly used for the set of types over the empty set consistent with ''T''. If there is a single formula $\varphi$ such that the theory of $\mathcal$ implies $\varphi \rightarrow \psi$ for every formula $\psi$ in ''p'', then ''p'' is called ''isolated''. Since the real numbers $\mathbb$ are
Archimedean Archimedean means of or pertaining to or named in honor of the Greece, Greek mathematics, mathematician Archimedes and may refer to: In mathematics: *Absolute value (algebra), Archimedean absolute value *Archimedean circle *Archimedean constant *Ar ...
, there is no real number larger than every integer. However, a compactness argument shows that there is an elementary extension of the real number line in which there is an element larger than any integer. Therefore, the set of formulas $\$ is a 1-type over $\mathbb \subseteq \mathbb$ that is not realised in the real number line $\mathbb$. A subset of $\mathcal^n$ that can be expressed as exactly those elements of $\mathcal^n$ realising a certain type over ''A'' is called ''type-definable'' over ''A''. For an algebraic example, suppose $M$ is an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The theory has quantifier elimination . This allows us to show that a type is determined exactly by the polynomial equations it contains. Thus the set of complete $n$-types over a subfield $A$ corresponds to the set of
prime ideal In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
s of the
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

## Structures and types

While not every type is realised in every structure, every structure realises its isolated types. If the only types over the empty set that are realised in a structure are the isolated types, then the structure is called ''atomic''. On the other hand, no structure realises every type over every parameter set; if one takes all of $\mathcal$ as the parameter set, then every 1-type over $\mathcal$ realised in $\mathcal$ is isolated by a formula of the form ''a = x'' for an $a \in \mathcal$. However, any proper elementary extension of $\mathcal$ contains an element that is ''not'' in $\mathcal$. Therefore a weaker notion has been introduced that captures the idea of a structure realising all types it could be expected to realise. A structure is called ''saturated'' if it realises every type over a parameter set $A \subset \mathcal$ that is of smaller cardinality than $\mathcal$ itself. While an automorphism that is constant on ''A'' will always preserve types over ''A'', it is generally not true that any two sequences $a_1, \dots, a_n$ and $b_1, \dots, b_n$ that satisfy the same type over ''A'' can be mapped to each other by such an automorphism. A structure $\mathcal$ in which this converse does holds for all ''A'' of smaller cardinality than $\mathcal$ is called homogeneous. The real number line is atomic in the language that contains only the order $<$, since all ''n''-types over the empty set realised by $a_1, \dots, a_n$ in $\mathbb$ are isolated by the order relations between the $a_1, \dots, a_n$. It is not saturated, however, since it does not realise any 1-type over the countable set $\mathbb$ that implies ''x'' to be larger than any integer. The rational number line $\mathbb$ is saturated, in contrast, since $\mathbb$ is itself countable and therefore only has to realise types over finite subsets to be saturated.

## Stone Spaces

The set of definable subsets of $\mathcal^n$ over some parameters $A$ is a
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. By
Stone's representation theorem for Boolean algebras In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
there is a natural dual
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, which consists exactly of the complete $n$-types over $A$. The topology generated by sets of the form $\$ for single formulas $\varphi$. This is called the ''Stone space of n-types over A''. This topology explains some of the terminology used in model theory: The compactness theorem says that the Stone Space is a compact topological space, and a type ''p'' is isolated if and only if ''p'' is an isolated point in the Stone topology. While types in algebraically closed fields correspond to the spectrum of the polynomial ring, the topology on the type space is the constructible topology: a set of types is basic
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
iff it is of the form $\$ or of the form $\$. This is finer than the
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
.

# Categoricity

A theory was originally called ''categorical'' if it determines a structure up to isomorphism. It turns out that this definition is not useful, due to serious restrictions in the expressivity of first-order logic. The Löwenheim–Skolem theorem implies that if a theory ''T'' has an infinite model for some infinite
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
, then it has a model of size κ for any sufficiently large
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
κ. Since two models of different sizes cannot possibly be isomorphic, only finite structures can be described by a categorical theory. However, the weaker notion of κ-categoricity for a cardinal κ has become a key concept in model theory. A theory ''T'' is called ''κ-categorical'' if any two models of ''T'' that are of cardinality κ are isomorphic. It turns out that the question of κ-categoricity depends critically on whether κ is bigger than the cardinality of the language (i.e. $\aleph_0$ + , σ, , where , σ, is the cardinality of the signature). For finite or countable signatures this means that there is a fundamental difference between $\omega$-cardinality and κ-cardinality for uncountable κ.

## ω-categoricity

$\omega$-categorical theories can be characterised by properties of their type space: :For a complete first-order theory ''T'' in a finite or countable signature the following conditions are equivalent: :#''T'' is $\omega$-categorical. :#Every type in ''Sn''(''T'') is isolated. :#For every natural number ''n'', ''Sn''(''T'') is finite. :#For every natural number ''n'', the number of formulas φ(''x''1, ..., ''x''n) in ''n'' free variables, up to equivalence modulo ''T'', is finite. The theory of $\left(\mathbb,<\right)$, which is also the theory of $\left(\mathbb,<\right)$, is $\omega$-categorical, as every ''n''-type $p\left(x_1, \dots, x_n\right)$ over the empty set is isolated by the pairwise order relation between the $x_i$. This means that every countable dense linear order is order-isomorphic to the rational number line. On the other hand, the theories of $\mathbb$, $\mathbb$ and $\mathbb$ as fields are not $\omega$-categorical. This follows from the fact that in all those fields, any of the infinitely many natural numbers can be defined by a formula of the form $x = 1 + \dots + 1$. $\aleph_0$-categorical theories and their countable models also have strong ties with oligomorphic groups: :A complete first-order theory ''T'' in a finite or countable signature is $\omega$-categorical if and only if its automorphism group is oligomorphic. The equivalent charcaterisations of this subsection, due independently to Engeler, Ryll-Nardzewski and Svenonius, are sometimes referred to as the Ryll-Nardzewski theorem. In combinatorial signatures, a common source of $\omega$-categorical theories are Fraïssé limits, which are obtained as the limit of amalgamating all possible configurations of a class of finite relational structures.

## Uncountable categoricity

Michael Morley showed in 1963 that there is only one notion of ''uncountable categoricity'' for theories in countable languages. :
Morley's categoricity theorem In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...
:If a first-order theory ''T'' in a finite or countable signature is κ-categorical for some uncountable cardinal κ, then ''T'' is κ-categorical for all uncountable cardinals κ. Morley's proof revealed deep connections between uncountable categoricity and the internal structure of the models, which became the starting point of classification theory and stability theory. Uncountably categorical theories are from many points of view the most well-behaved theories. In particular, complete strongly minimal theories are uncountably categorical. This shows that the theory of algebraically closed fields of a given characteristic is uncountably categorical, with the transcendence degree of the field determining its isomorphism type. A theory that is both $\omega$-categorical and uncountably categorical is called ''totally categorical''.

# Selected applications

Among the early successes of model theory are Tarski's proofs of the decidability of various algebraically interesting classes, such as the
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s,
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s and
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s of a given
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. In the 1960s, considerations around
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s and the
ultraproduct The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ( ...
construction led to the development of
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by
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
. In 1965, James Ax and Simon B. Kochen showed a special case of Artin's conjecture on diophantine equations, the Ax-Kochen theorem, again using an ultraproduct construction. More recently, the connection between stability and the geometry of definable sets led to several applications from algebraic and diophantine geometry, including 's 1996 proof of the geometric Mordell-Lang conjecture in all characteristics In 2011,
Jonathan Pila Jonathan Solomon Pila (born 1962) Fellow of the Royal Society, FRS One or more of the preceding sentences incorporates text from the royalsociety.org website where: is an Australian mathematician at the University of Oxford. Education Pila earned ...
applied techniques around o-minimality to prove the André-Oort conjecture for products of Modular curves. In a separate strand of inquiries that also grew around stable theories, Laskowski showed in 1992 that NIP (model theory), NIP theories describe exactly those definable classes that are Probably approximately correct learning, PAC-learnable in machine learning theory.

# History

Model theory as a subject has existed since approximately the middle of the 20th century. However some earlier research, especially in
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...
, is often regarded as being of a model-theoretical nature in retrospect. The first significant result in what is now model theory was a special case of the downward Löwenheim–Skolem theorem, published by Leopold Löwenheim in 1915. The
compactness theorem In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
was implicit in work by Thoralf Skolem, but it was first published in 1930, as a lemma in Kurt Gödel's proof of his Gödel's completeness theorem, completeness theorem. The Löwenheim–Skolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 from Anatoly Maltsev. The development of model theory as an independent discipline was brought on by
Alfred Tarski Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. was a Polish-American logician a ...
, a member of the Lwów–Warsaw school during the Interwar period, interbellum. Tarski's work included logical consequence, deductive systems, the algebra of logic, the theory of definability, and the Semantic theory of truth, semantic definition of truth, among other topics. His semantic methods culminated in the model theory he and a number of his University of California, Berkeley, Berkeley students developed in the 1950s and '60s. In the further history of the discipline, different strands began to emerge, and the focus of the subject shifted. In the 1960s, techniques around ultraproducts became a popular tool in model theory. At the same time, researchers such as James Ax were investigating the first-order model theory of various algebraic classes, and others such as H. Jerome Keisler were extending the concepts and results of first-order model theory to other logical systems. Then,
Saharon Shelah Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical str ...

's work around categoricity and Morley's problem changed the complexion of model theory, giving rise to a whole new class of concepts. The
stability theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(classification theory) Shelah developed since the late 1960s aims to classify theories by the number of different models they have of any given cardinality. Over the next decades, it became clear that the resulting stability hierarchy is closely connected to the geometry of sets that are definable in those models; this gave rise to the subdiscipline now known as geometric stability theory.

# Connections to related branches of mathematical logic

## Finite model theory

Finite model theory (FMT) is the subarea of model theory (MT) that deals with its restriction to interpretations on finite structures, which have a finite universe. Since many central theorems of model theory do not hold when restricted to finite structures, FMT is quite different from MT in its methods of proof. Central results of classical model theory that fail for finite structures under FMT include the
compactness theorem In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
,
Gödel's completeness theorem Gödel's completeness theorem is a fundamental theorem in mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...
, and the method of ultraproducts for first-order logic. The main application areas of FMT are descriptive complexity theory, database theory and formal language theory.

## Set theory

Any Formal system, set theory (which is expressed in a countable language), if it is consistent, has a countable model; this is known as Skolem's paradox, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model. Particularly the proof of the independence of the continuum hypothesis requires considering sets in models which appear to be uncountable when viewed from ''within'' the model, but are countable to someone ''outside'' the model. The model-theoretic viewpoint has been useful in set theory; for example in Kurt Gödel's work on the constructible universe, which, along with the method of Forcing (mathematics), forcing developed by Paul Cohen (mathematician), Paul Cohen can be shown to prove the (again philosophically interesting) Independence (mathematical logic), independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory. In the other direction, model theory itself can be formalized within ZFC set theory. For instance, the formalization of Satisfaction relation, satisfaction in ZFC is done inductively, based on T-schema, Tarski's T-schema and observation of where the members of the range of variable assignments lie.Open Logic Project
First-order logic
(p.27). Accessed September 27, 2021.
The development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, or more exactly the Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC framework. For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality). Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension. Neither of these results are provable in ZFC alone. Finally, some questions arising from model theory (such as compactness for infinitary logics) have been shown to be equivalent to large cardinal axioms.

* Algebraic theory * Axiomatizable class * Compactness theorem * Descriptive complexity * Elementary equivalence * List of first-order theories, First-order theories * Hyperreal number * Institutional model theory * Kripke semantics * Löwenheim–Skolem theorem * Model-theoretic grammar * Proof theory * Saturated model * Skolem normal form * Web Ontology Language#Relation to description logics, Connection of Web Ontology Languages (OWLs) to description logics

# References

* * * *

## Other textbooks

* * * * * * * * * *

## Free online texts

* * * * Wilfrid Hodges, Hodges, Wilfrid,
Model theory
'. The Stanford Encyclopedia Of Philosophy, E. Zalta (ed.). * Wilfrid Hodges, Hodges, Wilfrid,
First-order Model theory
'. The Stanford Encyclopedia Of Philosophy, E. Zalta (ed.). * Simmons, Harold (2004),
An introduction to Good old fashioned model theory
'. Notes of an introductory course for postgraduates (with exercises). * Jon Barwise, J. Barwise and Solomon Feferman, S. Feferman (editors)
Model-Theoretic Logics
Perspectives in Mathematical Logic, Volume 8, New York: Springer-Verlag, 1985. {{Authority control Model theory, Mathematical logic, Model Metalogic