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Area of mathematical logic
the mathematical discipline

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). 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. 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". 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.

A first order formula is built out of atomic formulas such as or by means of the Boolean connectives and prefixing of quantifiers or A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are (or to indicate is the unbound variable in ) and (or ), defined as follows:

(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 of the natural numbers, for example, an element satisfies the formula if and only if is a prime number. The formula similarly defines irreducibility. Tarski gave a rigorous definition, sometimes called "Tarski's definition of truth", for the satisfaction relation , so that one easily proves:

is a prime number. is irreducible. A set 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 , i. e. a structure (of the appropriate signature) which satisfies all the sentences in the set 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".

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 together with interpretations of each of the symbols of the signature as relations and functions on (not to be confused with the formal notion of an "interpretation" of one structure in another). Example: A common signature for ordered rings is , where and are 0 ary function symbols (also known as constant symbols), and are binary (= 2 ary) function symbols, is a unary (= 1 ary) function symbol, and is a binary relation symbol. Then, when these symbols are interpreted to correspond with their usual meaning on (so that e. g. is a function from to and is a subset of ), one obtains a structure
A structure is said to model a set of first order sentences in the given language if each sentence in is true in with respect to the interpretation of the signature previously specified for (Again, not to be confused with the formal notion of an "interpretation" of one structure in another) A model of is a structure that models
A substructure of a σ structure 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 a1, , an of ,
if and only if In particular, if φ is a sentence and an elementary substructure of , then if and only if Thus, an elementary substructure is a model of a theory exactly when the superstructure is a model. Example: While the field of algebraic numbers is an elementary substructure of the field of complex numbers , the rational field is not, as we can express "There is a square root of 2" as a first order sentence satisfied by but not by
An embedding of a σ structure into another σ structure is a map f: A → B between the domains which can be written as an isomorphism of with a substructure of If it can be written as an isomorphism with an elementary substructure, it is called an elementary embedding. Every embedding is an injective 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, regarded as a structure in the signature {+,0} can be expanded to a field with the signature {×,+,1,0} or to an ordered group with the signature {+,0,<}. 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.

The compactness theorem 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 satisfiability. However, there are also several direct (semantic) proofs of the compactness theorem. As a corollary (i. e., its contrapositive), the compactness theorem 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.

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 , n ary relations can also be definable. Functions are definable if the function graph is a definable relation, and constants are definable if there is a formula such that a is the only element of such that 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 interprets another whose theory is undecidable, then itself is undecidable.

For a sequence of elements of a structure and a subset A of , one can consider the set of all first order formulas with parameters in A that are satisfied by This is called the complete (n )type realised by over A. If there is an automorphism of that is constant on A and sends to respectively, then and realise the same complete type over A. The real number line , viewed as a structure with only the order relation {<}, will serve as a running example in this section. Every element 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 depends on their order: either , or Over the subset 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 is a structure and A a subset of , 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 of
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 If A is the empty set, then the type space only depends on the theory of The notation is commonly used for the set of types over the empty set consistent with
If there is a single formula such that the theory of implies for every formula in p, then p is called isolated. Since the real numbers are Archimedean, 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 that is not realised in the real number line
A subset of that can be expressed as exactly those elements of realising a certain type over A is called type definable over A. For an algebraic example, suppose is an algebraically closed field. 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 types over a subfield corresponds to the set of prime ideals of the polynomial ring , and the type definable sets are exactly the affine varieties.

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 as the parameter set, then every 1 type over realised in is isolated by a formula of the form a = x for an However, any proper elementary extension of contains an element that is not in 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 that is of smaller cardinality than itself. While an automorphism that is constant on A will always preserve types over A, it is generally not true that any two sequences and that satisfy the same type over A can be mapped to each other by such an automorphism. A structure in which this converse does holds for all A of smaller cardinality than 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 in are isolated by the order relations between the It is not saturated, however, since it does not realise any 1 type over the countable set that implies x to be larger than any integer. The rational number line is saturated, in contrast, since is itself countable and therefore only has to realise types over finite subsets to be saturated.

The set of definable subsets of over some parameters is a Boolean algebra. By Stone's representation theorem for Boolean algebras there is a natural dual topological space, which consists exactly of the complete types over The topology generated by sets of the form for single formulas 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 iff it is of the form or of the form This is finer than the Zariski topology.

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, then it has a model of size κ for any sufficiently large cardinal number 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.  + |σ|, where |σ| is the cardinality of the signature). For finite or countable signatures this means that there is a fundamental difference between cardinality and κ cardinality for uncountable κ.

A key factor in the structure of the class of models of a first order theory is its place in the stability hierarchy. A complete theory T is called stable for a cardinal if for any model of T and any parameter set of cardinality not exceeding , there are at most complete T types over A. A theory is called stable if it is stable for some infinite cardinal Traditionally, theories that are stable are called stable.

The stability hierarchy is also crucial for analysing the geometry of definable sets within a model of a theory. In stable theories, Morley rank is an important dimension notion for definable sets S within a model. It is defined by transfinite induction:

The Morley rank is at least 0 if S is non empty. For α a successor ordinal, the Morley rank is at least α if in some elementary extension N of M, the set S has infinitely many disjoint definable subsets, each of rank at least α − 1. For α a non zero limit ordinal, the Morley rank is at least α if it is at least β for all β less than α. A theory T in which every definable set has well defined Morley Rank is called totally transcendental; if T is countable, then T is totally transcendental if and only if T is stable. Morley Rank can be extended to types by setting the Morley Rank of a type to be the minimum of the Morley ranks of the formulas in the type. Thus, one can also speak of the Morley rank of an element a over a parameter set A, defined as the Morley rank of the type of a over A. There are also analogues of Morley rank which are well defined if and only if a theory is superstable (U rank) or merely stable (Shelah's rank). Those dimension notions can be used to define notions of independence and of generic extensions. More recently, stability has been decomposed into simplicity and "not the independence property" (NIP). Simple theories are those theories in which a well behaved notion of independence can be defined, while NIP theories generalise o minimal structures. They are related to stability since a theory is stable if and only if it is NIP and simple, and various aspects of stability theory have been generalised to theories in one of these classes.

Model theoretic results have been generalised beyond elementary classes, that is, classes axiomatisable by a first order theory. 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. This is made concrete by Lindstrom's theorem, stating roughly that first order logic is essentially the strongest logic in which both the Löwenheim Skolem theorems and compactness hold. However, model theoretic techniques have been developed extensively for these logics too. It turns out, however, that much of the model theory of more expressive logical languages is independent of Zermelo Fraenkel set theory. More recently, alongside the shift in focus to complete stable and categorical theories, there has been work on classes of models defined semantically rather than axiomatised by a logical theory. One example is homogeneous model theory, which studies the class of substructures of arbitrarily large homogeneous models. Fundamental results of stability theory and geometric stability theory generalise to this setting. As a generalisation of strongly minimal theories, quasiminimally excellent classes are those in which every definable set is either countable or co countable. They are key to the model theory of the complex exponential function. The most general semantic framework in which stability is studied are abstract elementary classes, which are defined by a strong substructure relation generalising that of an elementary substructure. Even though its definition is purely semantic, every abstract elementary class can be presented as the models of a first order theory which omit certain types. Generalising stability theoretic notions to abstract elementary classes is an ongoing research program.

Among the early successes of model theory are Tarski's proofs of quantifier elimination for various algebraically interesting classes, such as the real closed fields, Boolean algebras and algebraically closed fields of a given characteristic. Quantifier elimination allowed Tarski to show that the first order theories of real closed and algebraically closed fields as well as the first order theory of Boolean algebras are decidable, classify the Boolean algebras up to elementary equivalence and show that the theories of real closed fields and algebraically closed fields of a given characteristic are unique. Furthermore, quantifier elimination provided a precise description of definable relations on algebraically closed fields as algebraic varieties and of the definable relations on real closed fields as semialgebraic sets

In the 1960s, the introduction of the ultraproduct construction led to new applications in algebra. This includes Ax's work on pseudofinite fields, proving that the theory of finite fields is decidable, and Ax and Kochen's proof of as special case of Artin's conjecture on diophantine equations, the Ax Kochen theorem. The ultraproduct construction also led to Abraham Robinson's development of nonstandard analysis, which aims to provide a rigorous calculus of infinitesimals. More recently, the connection between stability and the geometry of definable sets led to several applications from algebraic and diophantine geometry, including Ehud Hrushovski's 1996 proof of the geometric Mordell Lang conjecture in all characteristics In 2001, similar methods were used to prove a generalisation of the Manin Mumford conjecture. In 2011, Jonathan Pila 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 theories describe exactly those definable classes that are PAC learnable in machine learning theory. This has led to several interactions between these separate areas. In 2018, the correspondence was extended as Hunter and Chase showed that stable theories correspond to online learnable classes.

Model theory as a subject has existed since approximately the middle of the 20th century, and the name was coined by Alfred Tarski, a member of the Lwów–Warsaw school, in 1954. However some earlier research, especially in mathematical logic, 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 was implicit in work by Thoralf Skolem, but it was first published in 1930, as a lemma in Kurt Gödel's proof of his 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 during the interbellum. Tarski's work included logical consequence, deductive systems, the algebra of logic, the theory of definability, and the semantic definition of truth, among other topics. His semantic methods culminated in the model theory he and a number of his 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, inspired by Morley's problem, Shelah developed stability theory. His work around stability changed the complexion of model theory, giving rise to a whole new class of concepts. This is known as the paradigm shift. 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. An example of an influential proof from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields.

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. In particular, many central results of classical model theory that fail when restricted to finite structures. This includes the compactness theorem, Gödel's completeness theorem, and the method of ultraproducts for first order logic. At the interface of finite and infinite model theory are algorithmic or computable model theory and the study of 0 1 laws, where the infinite models of a generic theory of a class of structures provide information on the distribution of finite models. Prominent application areas of FMT are descriptive complexity theory, database theory and formal language theory.

Any 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 developed by Paul Cohen can be shown to prove the (again philosophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory. In the other direction, model theory is itself formalised within Zermelo Fraenkel set theory. For instance, the development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, and is in fact equivalent over Zermelo Fraenkel set theory without choice to 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.

Hodges, Wilfrid, Model theory. The Stanford Encyclopedia Of Philosophy, E. Zalta (ed.). 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). J. Barwise and S. Feferman (editors), Model Theoretic Logics, Perspectives in Mathematical Logic, Volume 8, New York: Springer Verlag, 1985.