Model theory

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This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model.

In mathematics, model theory is the study of (classes of) mathematical structures such as groups, fields, graphs or even models of set theory using tools from mathematical logic. Model theory has close ties to algebra and universal algebra.

This article focuses on finitary first order model theory of infinite structures. The model theoretic study of finite structures (for which see finite model theory) diverges significantly from the study of infinite structures in both the problems studied and the techniques used. Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness does not in general hold for these logics. However, a great deal of study has also been done in such languages.

The role of model theory

Model theory recognises and is intimately concerned with a duality: It examines semantical elements by means of syntactical elements of a corresponding language. To quote the first page of Chang and Keisler (1990):

universal algebra + logic = model theory.

In a similar way to proof theory, model theory is situated in an area of interdisciplinarity between mathematics, philosophy, and computer science. The most important professional organization in the field of model theory is the Association for Symbolic Logic.

The areas of model theory

An incomplete and somewhat arbitrary subdivision of model theory is into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic. Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of nonstandard analysis. An important step in the evolution of classical model theory occurred with the birth of stability theory (through Morley's theorem on totally categorical theories and Shelah's classification program), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. The result of this synthesis is called geometric model theory in this article (which is taken to include o-minimality, for example, as well as classical geometric stability theory). An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell-Lang conjecture for function fields. The ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory.

Universal algebra

Main article: universal algebra

Fundamental concepts in universal algebra are signatures σ and σ-algebras. Since these concepts are formally defined in the article on structures, the present article can content itself with an informal introduction which consists in examples of how these terms are used.

The standard signature of rings is σ_ring = {×,+,−,0,1}, where × and + are binary, − is unary, and 0 and 1 are nullary.

The standard signature of (multiplicative) groups is σ_grp = {×,⁻¹,1}, where × is binary, ⁻¹ is unary and 1 is nullary.

The standard signature of monoids is σ_mnd = {×,1}.

A ring is a σ_ring-structure which satisfies the identities u + (v+w) = (u+v) + w, u+0 = u, 0+u=u, u+(-u)=0, (-u)+u=0, u × (v×w) = (u×v) × w, u×1 = u, 1×u =u, u × (v+w) = (u×v) + (u×w) and (v+w) × u = (v × u) + (w × u).

A group is a σ_grp-structure which satisfies the identities u×(v×w)=(u×v)×w, u×1=u, 1×u=u, u×u⁻¹=1 and u⁻¹×u=1.

A monoid is a σ_mnd-structure which satisfies the identities u×(v×w)=(u×v)×w, u×1 = u and 1×u =u.

A semigroup is a σ_mnd-structure which satisfies the identity u×(v×w)=(u×v)×w.

A magma is just a {×}-structure.

This is a very efficient way to define most classes of algebraic structures, because there is also the concept of σ-homomorphism, which correctly specializes to the usual notions of homomorphism for groups, semigroups, magmas and rings. For this to work, the signature must be chosen well.

Using σ-congruences (equivalence relations that respect the operations of σ), which play the role of kernels of homomorphisms, universal algebra can state and prove the isomorphism theorems in great generality:

For every homomorphism h: A → B, Im(A) is isomorphic to A/ker(h).

$\delta\subseteq\epsilon$

Terms such as the σ_ring-term t=t(u,v,w) given by (u + (v×w)) − 1 are used to define identities t=t', but also to construct free algebras. An equational class is a class of structures which, like the examples above and many others, is defined as the class of all σ-structures which satisfy a certain set of identities.

$\Pi_{i\in I}A_i/U$

A class of σ-structures is an equational class iff it is not empty and closed under subalgebras, homomorphic images, and direct products.

While model theory is generally considered a part of mathematical logic, universal algebra, which grew out of Alfred North Whitehead's (1898) work on abstract algebra, is part of algebra. This is reflected by their respective MSC classifications. Nevertheless model theory can be seen as an extension of universal algebra.

Finite model theory

Main article: finite model theory

Finite model theory is the area of model theory which has the closest ties to universal algebra. Like some parts of universal algebra, and in contrast with the other areas of model theory, it is mainly concerned with finite algebras, or more generally, with finite σ-structures for signatures σ which may contain relation symbols as in the following example:

The standard signature for graphs is σ_grph={E}, where E is a binary relation symbol.

$\forall u \forall v(uEv \rightarrow vEu)$ $\forall u_1u_2\dots u_n(t=t')$

The logics employed in finite model theory are often substantially more expressive than first-order logic, the standard logic for model theory of infinite structures.

First-order logic

Main article: first-order logic

Whereas universal algebra provides the semantics for a signature, logic provides the syntax. With terms, identities and quasi-identities, even universal algebra has some limited syntactic tools; first-order logic is the result of making quantification explicit and adding negation into the picture.

$\neg,\land,\lor,\rightarrow$

$\phi \;=\; \forall u\forall v(\exists w (x\times w=u\times v)\rightarrow(\exists w(x\times w=u)\lor\exists w(x\times w=v)))$ $\psi \;=\; \forall u\forall v((u\times v=x)\rightarrow (u=x)\lor(v=x)).$ $\mathcal N$

$\mathcal N\models\phi(n) \iff n$ $\mathcal N\models\psi(n) \iff n$ $\mathcal M\models T$ $\mathcal M\models T$

Lack of expressivity (when compared to higher logics such as second-order logic) has its advantages, though. For model theorists the Löwenheim-Skolem theorem is an important practical tool rather than the source of Skolem's paradox. First-order logic is in some sense the most expressive logic for which both the Löwenheim-Skolem theorem and the compactness theorem hold:

Compactness theorem

Every inconsistent first-order theory has a finite inconsistent subset.

This important theorem, due to Gödel, is of central importance in infinite model theory, where the words "by compactness" are commonplace. One way to prove it is by means of ultraproducts.

An important question when one wants to apply model theory to a specific class of structures is whether this class is an elementary class, i.e. whether its members can be characterized as those structures which satisfy a first-order theory.

Axiomatizability, elimination of quantifiers, and model-completeness

The first step, often trivial, for applying the methods of model theory to a class of mathematical objects such as groups, or trees in the sense of graph theory, is to choose a signature σ and represent the objects as σ-structures. The next step is to show that the class is axiomatizable, i.e. there is a theory T such that a σ-structure is in the class if and only if it satisfies T. (This step fails for the trees, since connectedness cannot be expressed in first order.) Axiomatizability ensures that model theory can speak about the right objects. Quantifier elimination can be seen as a condition which ensures that model theory does not say too much about the objects.

$\forall x_1\dots\forall x_n(\phi(x_1,\dots,x_n)\leftrightarrow \psi(x_1,\dots,x_n))$ $\mathcal A$

If T has quantifier elimination, then every substructure of a model of T again satisfies T, and in fact something stronger holds: For every formula φ(x₁,...,x_n) and every tuple a₁,...,a_n in the substructure, φ(a₁,...,a_n) is true in the substructure if and only if it is true in the bigger structure. A substructure with this property is called an elementary substructure.

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(x₁,...,x_n) over its signature is equivalent modulo T to an existential first-order formula, i.e. a formula of the following form:

$\forall v_1\dots\forall v_m\psi(x_1,\dots,x_n,v_1,\dots,v_m)$

where ψ is quantifier free.

Model theory and set theory

Set theory (which is expressed in a countable language) 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.

Computable model theory

Main article: Computable model theory

Other basic notions of model theory

Maps between structures

Fix a language L, and let M and N be two L-structures. For symbols from the language, such as a constant c, let c^M be the interpretation of c in M and similarly for the other classes of symbols (functions and relations).

A map h from the domain of M to the domain of N is a homomorphism if the following conditions hold:

$c \in L$

$f \in L$

$R \in L$

If in addition, the map j is injective and the third condition is modified to read:

$R \in L$

then the map h is an embedding (of M into N).

Equivalent definitions of homomorphism and embedding are:

$\bar{a} = (a_1,a_2,\ldots,a_n)$

$M \models \phi [\bar{a}] \Rightarrow N \models \phi [\bar{b}]$ $\bar{b}$

$\bar{b} = (b_1,b_2,\ldots,b_n) = (h(a_1),h(a_2),\ldots,h(a_n)) = h(\bar{a})$

then h is a homomorphism. If instead:

$M \models \phi [\bar{a}] \Leftrightarrow N \models \phi [\bar{b}]$

then h is an embedding.

Formulae and definable sets

$\phi(x_1,\ldots,x_n)$

Thus for each formula in L, and each L-structure M we have the set defined by the formula. For any given M, the collection of definable sets is the important semantical notion corresponding to the collection of formulae.

Elimination of quantifiers and model completeness

A theory T is said to admit elimination of quantifiers if every formula is provably equivalent to a quantifier-free formula under T. The theory T is model complete if every formula is provably equivalent to an existential formula.

These definitions concerning the syntactics of T can be shown to be equivalent to the following statement concerning the models of T (i.e. the semantics of T):

T has quantifier elimination if and only if for any two models B and C of T and for any common substructure A of B and C, B and C are elementarily equivalent in the language of T augmented with constants from A. In fact, it is sufficient to show that any sentence with only existential quantifiers have the same truth value for B and C.

T is model complete if and only if for every A and B models of T, and L-embedding of A into B, we have that the embedding is elementary.

One can see from the definition that quantifier elimination is stronger than model completeness. This is because formulas in model complete theories are equivalent containing only existential quantifiers. Any formula in a theory that admits quantifier elimination is equivalent to a quantifier-free formula which can be viewed as a special kind of existential formula.

In early model theory, quantifier elimination was used to demonstrate that various theories possess certain model-theoretic properties like decidability and completeness. A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier-free formulas. This technique is used to show that Presburger arithmetic, i.e. the theory of the additive natural numbers, is decidable. The demonstration of the decidability of Presburger arithmetic already hints at the limitations of this technique. Theories could be decidable yet not admit quantifier elimination. Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Example: Nullstellensatz in ACF and DCF

Interpretability

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 M interprets another whose theory is undecidable, then M itself is undecidable.

Ultraproduct constructions

Main article: Ultraproduct

$\mathcal U$

$\vec x\sim_U\vec y \iff \{i\in I | x_i=y_i\}\in\mathcal U$ $\Pi_{i\in I}\mathbb A_i/\mathcal U$ $\mathcal U$ $\Pi_{i\in I}\mathbb A_i/\mathcal U$

Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson-Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.

Using the compactness and completeness theorems

Gödel's completeness theorem (not to be confused with his incompleteness theorems) says that a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory. This is the heart of model theory as it lets us answer questions about theories by looking at models and vice-versa. One should not confuse the completeness theorem with the notion of a complete theory. A complete theory is a theory that contains every sentence or its negation. Importantly, one can find a complete consistent theory extending any consistent theory. However, as shown by Gödel's incompleteness theorems only in relatively simple cases will it be possible to have a complete consistent theory that is also recursive, i.e. that can be described by a recursively enumerable set of axioms. In particular, the theory of natural numbers has no recursive complete and consistent theory. Non-recursive theories are of little practical use, since it is undecidable if a proposed axiom is indeed an axiom, making proof-checking practically impossible.

The compactness theorem states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. In the context of proof theory the analogous statement is trivial, since every proof can have only a finite number of antecedents used in the proof. In the context of model theory, however, this proof is somewhat more difficult. There are two well known proofs, one by Gödel (which goes via proofs) and one by Malcev (which is more direct and allows us to restrict the cardinality of the resulting model).

$\mathfrak{A}$

Types

Fix an L-structure M, and a natural number n. The set of definable subsets of Mⁿ over some parameters A is a Boolean algebra. By Stone's representation theorem for Boolean algebras there is a natural dual notion to this. One can consider this to be the topological space consisting of maximal consistent sets of formulae over A. We call this the space of (complete) n-types over A, and write S_n(A).

$m \in M^n$ $a \in N^n$

Many important properties in model theory can be expressed with types. Further many proofs go via constructing models with elements that contain elements with certain types and then using these elements.

$A[x_1,\ldots,x_n]$

Categoricity

If T is a first order theory in the language L and κ is a cardinal, then T is said to be κ-categorical iff any two models of T which are of cardinality κ are isomorphic. Categorical theories are from many points of view the most well behaved theories. The study of categoricity led on to the wider programme of stability. For more detail see Morley's categoricity theorem.

Model completion, model companions

Given first-order σ-theories T and T', T' is a model companion for T if

i) T' is model complete

ii) Every model of T has an extension that is a model of T'

iii) Every model of T' has an extension that is a model of T

$T' \cup Diag(\mathcal{M})$

from Marker page 106

Early history of model theory

Model theory as a subject exists since approximately the middle of the 20th century. 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-Skolems theorem and the compactness theorem received their respective general forms in 1936 and 1941 from Anatoly Maltsev.