, 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, taken as interpretations
that satisfy the sentences of that theory.
Model theory recognizes and is intimately concerned with a duality: it examines semantical
elements (meaning and truth) by means of syntactical
elements (formulas and proofs) of a corresponding language. In a summary definition, dating from 1973:
:model theory = universal algebra
Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges
:model theory = algebraic geometry
This is a clever slogan, implying that there are many commonalities: so, for example, an algebraic variety
can be informally described as the locus of points where a collection of polynomials are zero. Likewise, a model can be described as a locus of interpretations where a collection of sentences are true. There are further analogies extending to varying depths.
Another commonly recurring slogan states that ''"if proof theory
is about the sacred, then model theory is about the profane"'', indicating that these two topics are in a sense dual to each-other. Much like proof theory
, model theory is situated in an area of interdisciplinarity
, and computer science
. Model theory is used in a variety of settings, both academic and industrial. These include:
* Proving results on axiom system
s. For example, the proof that the continuum hypothesis
is independent from the other axioms of Zermelo–Fraenkel set theory
(ZFC) is done by building inside ZFC a model of ZFC where the continuum hypothesis is true, and another model where it is false (see ).
* Providing the foundation for satisfiability modulo theories
solvers, which are commonly deployed for functional verification
in electronic design automation
. Such solvers search for sentences that are satisfiable, up to equivalent statements in some specific theory, such as the theory of equality
or the theory of linear algebra
* Providing the foundation for relational model
s, which are the fragment consisting of structures
consist entirely of relations
. Well-known results include that SQL and noSQL are categorical dual
s to each other.
The most prominent scholarly organization in the field of model theory is the Association for Symbolic Logic
This page focuses on finitary first order
model theory of infinite structures. 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. Model theory in higher-order logic
s or infinitary logic
s is hampered by the fact that completeness
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
, the upward and downward Löwenheim–Skolem theorem
'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
's theorem on pseudo-finite field
s, and Robinson
's development of non-standard analysis
. An important step in the evolution of classical model theory occurred with the birth of stability theory
(through Morley's theorem
on uncountably 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 proof 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.
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
, Gödel's completeness theorem
, 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
Whereas universal algebra
provides the semantics
for a signature
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.
A first-order formula is built out of atomic formula
s such as ''R''(''f''(''x'',''y''),''z'') or ''y'' = ''x'' + 1 by means of the Boolean connectives
and prefixing of quantifiers
. 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:
(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
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
, so that one easily proves:
is a prime number.
A set ''T'' of sentences is called a (first-order) theory
. 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 ''T''. Consistency
of a theory is usually defined in a syntactical way, but in first-order logic by the completeness theorem
there is no need to distinguish between satisfiability and consistency. Therefore, model theorists often use "consistent" as a synonym for "satisfiable".
A theory is called categorical if it determines a structure up to isomorphism, but 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 for every theory ''T'' having a countable signature which has an infinite model for some infinite cardinal number
, then it has a model of size κ for any infinite cardinal number
κ. Since two models of different sizes cannot possibly be isomorphic, only finitary structures can be described by a categorical theory.
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
. 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.
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 infinite model theory, where the words "by compactness" are commonplace. One way to prove it is by means of ultraproduct
s. An alternative proof uses the completeness theorem, which is otherwise reduced to a marginal role in most of modern model 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 an elementary class
, i.e. axiomatizable in first-order logic (i.e. there is a theory ''T'' such that a σ-structure is in the class if and only if it satisfies ''T''). E.g. this step fails for the trees, since connectedness cannot be expressed in first-order logic. 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.
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.
holds in all models of ''T''. For example, the theory of algebraically closed fields in the signature σring
= (×,+,−,0,1) has quantifier elimination because every formula is equivalent to a Boolean combination of equations between polynomials.
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. 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
. Every embedding is an injective
homomorphism, but the converse holds only if the signature contains no relation symbols.
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:
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 model companions.
As observed in the section on first-order logic
, first-order theories cannot be categorical, i.e. they cannot describe a unique model up to isomorphism, unless that model is finite. But two famous model-theoretic theorems deal with the weaker notion of κ-categoricity for a cardinal
κ. 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 few characterizations of -categoricity
:For a complete first-order theory ''T'' in a finite or countable signature the following conditions are equivalent:
:#For every natural number ''n'', the Stone space
''(''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.
This result, due independently to Engeler
, is sometimes referred to as the Ryll-Nardzewski
-categorical theories and their countable models have strong ties with oligomorphic group
s. They are often constructed as Fraïssé limit
's highly non-trivial result that (for countable languages) there is only ''one'' notion of uncountable categoricity was the starting point for modern model theory, and in particular classification theory and stability theory:
:Morley's categoricity theorem
: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.
Uncountably categorical (i.e. κ-categorical for all uncountable cardinals κ) theories are from many points of view the most well-behaved theories. A theory that is both
-categorical and uncountably categorical is called totally categorical.
(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 itself can be formalized within ZFC set theory. 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.
Other basic notions
Reducts and expansions
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 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.
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.
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
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).
Model theory is usually concerned with first-order logic
, and many important results (such as the completeness and compactness theorems) fail in second-order logic
or other alternatives. In first-order logic all infinite cardinals look the same to a language which is countable
. This is expressed in the Löwenheim–Skolem theorem
s, which state that any countable theory with an infinite model
has models of all infinite cardinalities (at least that of the language) which agree with
on all sentences, i.e. they are 'elementarily equivalent
, and a natural number
. 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 notion to this. One can consider this to be the topological space
consisting of maximal consistent sets of formulae over
. We call this the space of (complete)
, and write
Now consider an element
. Then the set of all formulae
with parameters in
in free variables
is consistent and maximal such. It is called the ''type'' of
One can show that for any
, there exists some elementary extension
is the type of
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.
Illustrative 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 space of
-types over a subfield
with the set of prime ideal
s of the polynomial ring