Model theory

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.

This article focuses on finitary first order model theory.

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

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.

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.

A first-order formula is built out of atomic formulas 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 \(\varphi\) (or \(\varphi(x)\) to indicate \(x\) is the unbound variable in \(\varphi\)) and \(\psi\) (or \(\psi(x)\)), defined as follows:

\(\begin{array}{lcl} \varphi & = & \forall u\forall v(\exists w (x\times w=u\times v)\rightarrow(\exists w(x\times w=u)\lor\exists w(x\times w=v)))\land x\ne 0\land x\ne1, \\\psi & = & \forall u\forall v((u\times v=x)\rightarrow (u=x)\lor(v=x))\land x\ne 0\land x\ne1. \end{array}\)

(Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the semiring of natural numbers \(\mathcal N\), viewed as a structure with binary functions for addition and multiplication and constants for 0 and 1 of the natural numbers, for example, an element \(n\) satisfies the formula \(\varphi\) if and only if \(n\) is a prime number. The formula \(\psi\) 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:

\(\mathcal N\models\varphi(n) \iff n\) is a prime number.

\(\mathcal N\models\psi(n) \iff n\) is irreducible.

A set \(T\) 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 \(\mathcal M\models T\), 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 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 \(M\) together with interpretations of each of the symbols of the signature as relations and functions on \(M\) (not to be confused with the formal notion of an "interpretation" of one structure in another).

Example: A common signature for ordered rings is \(\sigma_{or}=(0,1,+,\times,-,<)\), where \(0\) and \(1\) are 0-ary function symbols (also known as constant symbols), \(+\) and \(\times\) 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 \(\Q\) (so that e.g. \(+\) is a function from \(\Q^2\) to \(\Q\) and \(<\) is a subset of \(\Q^2\)), one obtains a structure \((\Q,\sigma_{or})\).

A structure \(\mathcal{N}\) is said to model a set of first-order sentences \(T\) in the given language if each sentence in \(T\) is true in \(\mathcal{N}\) with respect to the interpretation of the signature previously specified for \(\mathcal{N}\). (Again, not to be confused with the formal notion of an "interpretation" of one structure in another) A model of \(T\) is a structure that models \(T\).

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 \(\varphi\) and any elements a₁, ..., a_n of \(\mathcal A\),

\(\mathcal A\models \varphi(a_1, ...,a_n)\) if and only if \(\mathcal B\models \varphi(a_1, ...,a_n)\).

In particular, if \(\varphi\) 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.

Example: While the field of algebraic numbers \(\overline{\mathbb{Q}}\) is an elementary substructure of the field of complex numbers \(\mathbb{C}\), the rational field \(\mathbb{Q}\) is not, as we can express "There is a square root of 2" as a first-order sentence satisfied by \(\mathbb{C}\) but not by \(\mathbb{Q}\).

Condensed: the full section is in Wikipedia.

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 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.

In model theory, definable sets are important objects of study. For instance, in \(\mathbb N\) the formula

\(\forall u\forall v(\exists w (x\times w=u\times v)\rightarrow(\exists w(x\times w=u)\lor\exists w(x\times w=v)))\land x\ne 0\land x\ne1\)

defines the subset of prime numbers, while the formula

\(\exists y (2\times y = x)\)

defines the subset of even numbers. In a similar way, formulas with n free variables define subsets of \(\mathcal{M}^n\). For example, in a field, the formula

\(y = x \times x\)

defines the curve of all \((x,y)\) 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 \(\mathbb{R}\), the formula

\(y = x \times x + \pi\)

uses the parameter \(\pi\) from \(\mathbb{R}\) to define a curve.

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(x₁, ..., x_n) over its signature is equivalent modulo T to a first-order formula ψ(x₁, ..., x_n) without quantifiers, i.e. \(\forall x_1\dots\forall x_n(\phi(x_1,\dots,x_n)\leftrightarrow \psi(x_1,\dots,x_n))\) 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 σ_ring = (×,+,−,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. Axiomatisability and quantifier elimination results for specific theories, especially in algebra, were among the early landmark results of model theory. 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(x₁, ..., x_n) over its signature is equivalent modulo T to an existential first-order formula, i.e. a formula of the following form:

\(\exists v_1\dots\exists v_m\psi(x_1,\dots,x_n,v_1,\dots,v_m)\),

where ψ is quantifier free. A theory that is not model-complete may 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.

In every structure, every finite subset \(\{a_1, \dots, a_n\}\) is definable with parameters: Simply use the formula

\(x = a_1 \vee \dots \vee x = a_n\).

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 \(\mathcal{M}\) is called minimal if every subset \(A \subseteq \mathcal{M}\) definable with parameters from \(\mathcal{M}\) 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 \(\mathbb{R}\) of real numbers is not minimal: Consider, for instance, the definable set

\(\varphi (x) \;=\; \exists y (y \times y = x)\).

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 \(\mathbb{R}\). This generalisation of minimality has been very useful in the model theory of ordered structures. A densely totally ordered structure \(\mathcal{M}\) in a signature including a symbol for the order relation is called o-minimal if every subset \(A \subseteq \mathcal{M}\) definable with parameters from \(\mathcal{M}\) is a finite union of points and intervals.

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 \(\mathcal{M}^n\), n-ary relations can also be definable. Functions are definable if the function graph is a definable relation, and constants \(a \in \mathcal{M}\) are definable if there is a formula \(\varphi(x)\) such that a is the only element of \(\mathcal{M}\) such that \(\varphi(a)\) 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 \(\mathcal{M}\) interprets another whose theory is undecidable, then \(\mathcal{M}\) itself is undecidable.

For a sequence of elements \(a_1, \dots, a_n\) of a structure \(\mathcal{M}\) and a subset A of \(\mathcal{M}\), one can consider the set of all first-order formulas \(\varphi(x_1, \dots, x_n)\) 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 of \(\mathcal{M}\) 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 \(\mathbb{R}\), viewed as a structure with only the order relation {<}, will serve as a running example in this section. Every element \(a \in \mathbb{R}\) 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 \(\mathbb{Z} \subseteq \mathbb{R}\) 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 \(\mathcal{M}\) is a structure and A a subset of \(\mathcal{M}\), 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 \(\mathcal{N}\) of \(\mathcal{M}\). 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 \(S_n^{\mathcal{M}}(A)\). If A is the empty set, then the type space only depends on the theory \(T\) of \(\mathcal{M}\). The notation \(S_n(T)\) 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 \(\mathcal{M}\) implies \(\varphi \rightarrow \psi\) for every formula \(\psi\) in p, then p is called isolated.

Since the real numbers \(\mathbb{R}\) 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 \{n < x | n \in \mathbb{Z} \}\) is a 1-type over \(\mathbb{Z} \subseteq \mathbb{R}\) that is not realised in the real number line \(\mathbb{R}\).

A subset of \(\mathcal{M}^n\) that can be expressed as exactly those elements of \(\mathcal{M}^n\) realising a certain type over A is called type-definable over A. For an algebraic example, suppose \(M\) 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 \(n\)-types over a subfield \(A\) corresponds to the set of prime ideals of the polynomial ring \(A[x_1,\ldots,x_n]\), 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 \(\mathcal{M}\) as the parameter set, then every 1-type over \(\mathcal{M}\) realised in \(\mathcal{M}\) is isolated by a formula of the form a = x for an \(a \in \mathcal{M}\). However, any proper elementary extension of \(\mathcal{M}\) contains an element that is not in \(\mathcal{M}\). 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 \(A \subset \mathcal{M}\) that is of smaller cardinality than \(\mathcal{M}\) 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 \(\mathcal{M}\) in which this converse does hold for all A of smaller cardinality than \(\mathcal{M}\) is called (strongly) 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 \(\mathbb{R}\) 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 \(\mathbb{Z}\) that implies x to be larger than any integer. The rational number line \(\mathbb{Q}\) is saturated, in contrast, since \(\mathbb{Q}\) is itself countable and therefore only has to realise types over finite subsets to be saturated.

The set of definable subsets of \(\mathcal{M}^n\) over some parameters \(A\) 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 \(n\)-types over \(A\). The topology generated by sets of the form \(\{p | \varphi \in p\}\) 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 iff it is of the form \(\{p: f(x) = 0 \in p\}\) or of the form \(\{p: f(x) \neq 0 \in p\}\). This is finer than the Zariski topology.

Constructing models that realise certain types and do not realise others is an important task in model theory. Not realising a type is referred to as omitting it, and is generally possible by the (Countable) Omitting types theorem:

Let \(\mathcal{T}\) be a theory in a countable signature and let \(\Phi\) be a countable set of non-isolated types over the empty set.

Then there is a model \(\mathcal{M}\) of \(\mathcal{T}\) which omits every type in \(\Phi\).

This implies that if a theory in a countable signature has only countably many types over the empty set, then this theory has an atomic model.

On the other hand, there is always an elementary extension in which any set of types over a fixed parameter set is realised:

Let \(\mathcal{M}\) be a structure and let \(\Phi\) be a set of complete types over a given parameter set \(A \subset \mathcal{M}.\)

Then there is an elementary extension \(\mathcal{N}\) of \(\mathcal{M}\) which realises every type in \(\Phi\).

However, since the parameter set is fixed and there is no mention here of the cardinality of \(\mathcal{N}\), this does not imply that every theory has a saturated model. In fact, whether every theory has a saturated model is independent of the axioms of Zermelo-Fraenkel set theory, and is true if the generalised continuum hypothesis holds.

Ultraproducts are used as a general technique for constructing models that realise certain types. An ultraproduct is obtained from the direct product of a set of structures over an index set I by identifying those tuples that agree on almost all entries, where almost all is made precise by an ultrafilter U on I. An ultraproduct of copies of the same structure is known as an ultrapower. The key to using ultraproducts in model theory is Łoś's theorem:

Let \(\mathcal{M}_i\) be a set of σ-structures indexed by an index set I and U an ultrafilter on I. Then any σ-formula \(\varphi([(a_i)_{i \in :I}])\) is true in the ultraproduct of the \(\mathcal{M}_i\) by \(U\) if the set of all \(i \in I\) for which \(\mathcal{M}_i \models \varphi(a_i)\) lies in U.

In particular, any ultraproduct of models of a theory is itself a model of that theory, and thus if two models have isomorphic ultrapowers, they are elementarily equivalent. The Keisler-Shelah theorem provides a converse:

If M and N are elementarily equivalent, then there is a set I and an ultrafilter U on I such that the ultrapowers by U of M and :N are isomorphic.

Therefore, ultraproducts provide a way to talk about elementary equivalence that avoids mentioning first-order theories at all. Basic theorems of model theory such as the compactness theorem have alternative proofs using ultraproducts, and they can be used to construct saturated elementary extensions if they exist.