Inmathematical logic,model theory is the study of the relationship betweenformal theories (a collection ofsentences in aformal language expressing statements about amathematical structure), and theirmodels (thosestructures in which the statements of the theory hold).[1] 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 bedefined in a model of a theory, and the relationship of such definable sets to each other.As a separate discipline, model theory goes back toAlfred Tarski, who first used the term "Theory of Models" in publication in 1954.[2]Since the 1970s, the subject has been shaped decisively bySaharon Shelah'sstability theory.
Compared to other areas of mathematical logic such asproof theory, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics.This has prompted the comment that"ifproof theory is about the sacred, then model theory is about the profane".[3]The applications of model theory toalgebraic andDiophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory issyntactic in nature, in contrast to model theory, which issemantic in nature.
The most prominent scholarly organization in the field of model theory is theAssociation for Symbolic Logic.
This page focuses onfinitaryfirst order model theory of infinite structures.
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:
where universal algebra stands for mathematical structures and logic for logical theories; and
where logical formulas are to definable sets what equations are to varieties over a field.[4]
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 givencardinality, stability theory proved crucial to understanding the geometry of definable sets.
A first-orderformula is built out ofatomic formulas such as or by means of theBoolean 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 semiring of natural numbers, viewed as a structure with binary functions for addition and multiplication and constants for 0 and 1 of the natural numbers, for example, an elementsatisfies the formula if and only if is a prime number. The formula similarly definesirreducibility. Tarski gave a rigorous definition, sometimes called"Tarski's definition of truth", for the satisfaction relation, so that one easily proves:
A set of sentences is called a (first-order)theory, which takes the sentences in the set as its axioms. A theory issatisfiable if it has amodel, i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set. A complete theory is a theory that contains everysentence or its negation.The complete theory of all sentences satisfied by a structure is also called thetheory of that structure.
It's a consequence of Gödel'scompleteness theorem (not to be confused with hisincompleteness theorems) that a theory has a model if and only if it isconsistent, i.e. no contradiction is proved by the theory.Therefore, model theorists often use "consistent" as a synonym for "satisfiable".
Asignature orlanguage is a set ofnon-logical symbols such that each symbol is either a constant symbol, or a function or relation symbol with a specifiedarity. Note that in some literature, constant symbols are considered as function symbols with zero arity, and hence are omitted. Astructure is a set together withinterpretations 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 theinterpretation of the signature previously specified for. (Again, not to be confused with the formal notion of an "interpretation" of one structure in another) Amodel of is a structure that models.
Asubstructure 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 beelementary if for any first-order formula and any elementsa1, ...,an of,
In particular, if 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.
Anembedding of a σ-structure into another σ-structure is a mapf: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 aninjective 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 areduct of a structure to a subset of the original signature. The opposite relation is called anexpansion - e.g. the (additive) group of therational 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.
Thecompactness theorem states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. The analogous statement withconsistent instead ofsatisfiable 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.[5]
Another cornerstone of first-order model theory is theLöwenheim–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.[6]
In a certain sense made precise byLindström's theorem, first-order logic is the most expressive logic for which both the Löwenheim–Skolem theorem and the compactness theorem hold.[7]
In model theory,definable sets are important objects of study. For instance, in the formula
defines the subset of prime numbers, while the formula
defines the subset of even numbers.In a similar way, formulas withn free variables define subsets of. For example, in a field, the formula
defines the curve of all such that.
Both of the definitions mentioned here areparameter-free, that is, the defining formulas don't mention any fixed domain elements. However, one can also consider definitionswith parameters from the model.For instance, in, the formula
uses the parameter from to define a curve.[8]
In general, definable sets without quantifiers are easy to describe, while definable sets involving possibly nested quantifiers can be much more complicated.[9]
This makesquantifier elimination a crucial tool for analysing definable sets: A theoryT has quantifier elimination if every first-order formulaφ(x1, ...,xn) over its signature is equivalent moduloT to a first-order formulaψ(x1, ...,xn) without quantifiers, i.e. holds in all models ofT.[10]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 signatureσring = (×,+,−,0,1) has quantifier elimination.[11] 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.[12] But often instead of quantifier elimination a weaker property suffices:
A theoryT is calledmodel-complete if every substructure of a model ofT which is itself a model ofT is an elementary substructure. There is a useful criterion for testing whether a substructure is an elementary substructure, called theTarski–Vaught test.[6] It follows from this criterion that a theoryT is model-complete if and only if every first-order formulaφ(x1, ...,xn) over its signature is equivalent moduloT 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 have amodel 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 amodel companion.[13]
In every structure, every finite subset is definable with parameters: Simply use the formula
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 aminimal structure. A structure is called minimal if every subset definable with parameters from is either finite or cofinite.The corresponding concept at the level of theories is calledstrong minimality:A theoryT is calledstrongly minimal if every model ofT is minimal.A structure is calledstrongly 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.[14]
On the other hand, the field of real numbers is not minimal: Consider, for instance, the definable set
This defines the subset of non-negative real numbers, which is neither finite nor cofinite.One can in fact use to define arbitrary intervals on the real number line.It turns out that these suffice to represent every definable subset of.[15]This generalisation of minimality has been very useful in the model theory of ordered structures.Adensely totally ordered structure in a signature including a symbol for the order relation is calledo-minimal if every subset definable with parameters from is a finite union of points and intervals.[16]
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 withn 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 thata 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 areinterpretable.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.[17]
For a sequence of elements of a structure and a subsetA of, one can consider the set of all first-order formulas with parameters inA that are satisfied by. This is called thecomplete (n-)type realised byover A.If there is anautomorphism of that is constant onA and sends to respectively, then and realise the same complete type overA.
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 numbersa andb are connected by the order automorphism that shifts all numbers byb-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 numbera depends on its value rounded down to the nearest integer.
More generally, whenever is a structure andA a subset of, a (partial)n-type over A is a set of formulasp with at mostn free variables that are realised in an elementary extension of. Ifp contains every such formula or its negation, thenp iscomplete. The set of completen-types overA is often written as. IfA 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 inp, thenp is calledisolated.
Since the real numbers areArchimedean, 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 overA is calledtype-definable overA.For an algebraic example, suppose is analgebraically 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 ofprime ideals of thepolynomial ring, and the type-definable sets are exactly the affine varieties.[18]
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 calledatomic.
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 forma = x for an. However, any proper elementary extension of contains an element that isnot 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 calledsaturated if it realises every type over a parameter set that is of smaller cardinality than itself.
While an automorphism that is constant onA will always preserve types overA, it is generally not true that any two sequences and that satisfy the same type overA can be mapped to each other by such an automorphism. A structure in which this converse does hold for allA of smaller cardinality than is calledhomogeneous.
The real number line is atomic in the language that contains only the order, since alln-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 impliesx 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.[19]
The set of definable subsets of over some parameters is aBoolean algebra. ByStone's representation theorem for Boolean algebras there is a natural dualtopological space, which consists exactly of the complete-types over. The topologygenerated by sets of the form for single formulas. This is called theStone space of n-types over A.[20]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 typep is isolated if and only ifp 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 theconstructible topology: a set of types is basicopen iff it is of the form or of the form. This is finer than theZariski topology.[21]
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 asomitting it, and is generally possible by the(Countable) Omitting types theorem:
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:
However, since the parameter set is fixed and there is no mention here of the cardinality of, 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 ofZermelo–Fraenkel set theory, and is true if thegeneralised continuum hypothesis holds.[24]
Ultraproducts are used as a general technique for constructing models that realise certain types.Anultraproduct is obtained from thedirect product of a set of structures over an index setI by identifying those tuples that agree on almost all entries, wherealmost all is made precise by anultrafilterU onI. An ultraproduct of copies of the same structure is known as anultrapower.The key to using ultraproducts in model theory isŁoś's theorem:
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.TheKeisler-Shelah theorem provides a converse:
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,[27] and they can be used to construct saturated elementary extensions if they exist.[24]
A theory was originally calledcategorical 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 theoryT has an infinite model for some infinitecardinal number, then it has a model of sizeκ for any sufficiently largecardinal numberκ. Since two models of different sizes cannot possibly be isomorphic, only finite structures can be described by a categorical theory.
However, the weaker notion ofκ-categoricity for a cardinalκ has become a key concept in model theory. A theoryT is calledκ-categorical if any two models ofT that are of cardinalityκ are isomorphic. It turns out that the question ofκ-categoricity depends critically on whetherκ is bigger than the cardinality of the language (i.e., where|σ| is the cardinality of the signature). For finite or countable signatures this means that there is a fundamental difference betweenω-cardinality andκ-cardinality for uncountableκ.
ω-categorical theories can be characterised by properties of their type space:
The theory of, which is also the theory of, isω-categorical, as everyn-type over the empty set is isolated by the pairwise order relation between the.This means that every countabledense linear order is order-isomorphic to the rational number line. On the other hand, the theories of ℚ, ℝ and ℂ as fields are not-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.
-categorical theories and their countable models also have strong ties witholigomorphic groups:
The equivalent characterisations of this subsection, due independently toEngeler,Ryll-Nardzewski andSvenonius, are sometimes referred to as the Ryll-Nardzewski theorem.
In combinatorial signatures, a common source ofω-categorical theories areFraïssé limits, which are obtained as the limit of amalgamating all possible configurations of a class of finite relational structures.
Michael Morley showed in 1963 that there is only one notion ofuncountable categoricity for theories in countable languages.[28]
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 bothω-categorical and uncountably categorical is calledtotally categorical.
A key factor in the structure of the class of models of a first-order theory is its place in thestability hierarchy.
A theory is calledstable if it is-stable for some infinite cardinal. Traditionally, theories that are-stable are called-stable.[29]
A fundamental result in stability theory is thestability spectrum theorem,[30] which implies that every complete theoryT in a countable signature falls in one of the following classes:
A theory of the first type is calledunstable, a theory of the second type is calledstrictly stable and a theory of the third type is calledsuperstable.Furthermore, if a theory is-stable, it is stable in every infinite cardinal,[31] so-stability is stronger than superstability.
Many construction in model theory are easier when restricted to stable theories; for instance, every model of a stable theory has a saturated elementary extension, regardless of whether the generalised continuum hypothesis is true.[32]
Shelah's original motivation for studying stable theories was to decide how many models a countable theory has of any uncountable cardinality.[33] If a theory is uncountably categorical, then it is-stable. More generally, theMain gap theorem implies that if there is an uncountable cardinal such that a theoryT has less than models of cardinality, thenT is superstable.
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 setsS within a model. It is defined bytransfinite induction:
A theoryT in which every definable set has well-defined Morley rank is calledtotally transcendental; ifT is countable, thenT is totally transcendental if and only ifT 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 elementa over a parameter setA, defined as the Morley rank of the type ofa overA. 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, whileNIP theories generalise o-minimal structures.They are related to stability since a theory is stable if and only if it is NIP and simple,[34] and various aspects of stability theory have been generalised to theories in one of these classes.
Model-theoretic results have been generalised beyondelementary classes, that is, classes axiomatisable by a first-order theory.
Model theory inhigher-order logics orinfinitary logics is hampered by the fact thatcompleteness andcompactness do not in general hold for these logics. This is made concrete byLindström'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.[35] It turns out, however, that much of the model theory of more expressive logical languages is independent ofZermelo–Fraenkel set theory.[36]
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 ishomogeneous 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.[37]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 thecomplex exponential function.[38]The most general semantic framework in which stability is studied areabstract elementary classes, which are defined by astrong 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.[39]
Among the early successes of model theory are Tarski's proofs of quantifier elimination for various algebraically interesting classes, such as thereal closed fields,Boolean algebras andalgebraically closed fields of a givencharacteristic. 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 asalgebraic varieties and of the definable relations on real-closed fields assemialgebraic sets[40][41]
In the 1960s, the introduction of theultraproduct construction led to new applications in algebra. This includesAx's work onpseudofinite fields, proving that the theory of finite fields is decidable,[42] and Ax andKochen's proof of as special case of Artin's conjecture on diophantine equations, theAx–Kochen theorem.[43] The ultraproduct construction also led toAbraham Robinson's development ofnonstandard analysis, which aims to provide a rigorous calculus ofinfinitesimals.[44]
More recently, the connection between stability and the geometry of definable sets led to several applications from algebraic and diophantine geometry, includingEhud Hrushovski's 1996 proof of the geometricMordell–Lang conjecture in all characteristics[45] In 2001, similar methods were used to prove a generalisation of the Manin-Mumford conjecture.In 2011,Jonathan Pila applied techniques aroundo-minimality to prove theAndré–Oort conjecture for products of Modular curves.[46]
In a separate strand of inquiries that also grew around stable theories, Laskowski showed in 1992 thatNIP theories describe exactly those definable classes that arePAC-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 toonline learnable classes.[47]
Model theory as a subject has existed since approximately the middle of the 20th century, and the name was coined byAlfred Tarski, a member of theLwów–Warsaw school, in 1954.[48] However some earlier research, especially inmathematical logic, is often regarded as being of a model-theoretical nature in retrospect.[49] The first significant result in what is now model theory was a special case of the downward Löwenheim–Skolem theorem, published byLeopold Löwenheim in 1915. Thecompactness theorem was implicit in work byThoralf Skolem,[50] but it was first published in 1930, as a lemma inKurt Gödel's proof of hiscompleteness theorem. The Löwenheim–Skolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 fromAnatoly Maltsev.The development of model theory as an independent discipline was brought on by Alfred Tarski during theinterbellum. Tarski's work includedlogical consequence,deductive systems, the algebra of logic, the theory of definability, and thesemantic definition of truth, among other topics. His semantic methods culminated in the model theory he and a number of hisBerkeley 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.[51] At the same time, researchers such asJames Ax were investigating the first-order model theory of various algebraic classes, and others such asH. Jerome Keisler were extending the concepts and results of first-order model theory to other logical systems. Then, inspired byMorley's problem, Shelah developedstability 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.[52] 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 isHrushovski's proof of theMordell–Lang conjecture for function fields.[53]
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.[54] In particular, many central results of classical model theory that fail when restricted to finite structures. This includes thecompactness theorem,Gödel's completeness theorem, and the method ofultraproducts forfirst-order logic. At the interface of finite and infinite model theory are algorithmic orcomputable model theory and the study of0-1 laws, where the infinite models of a generic theory of a class of structures provide information on the distribution of finite models.[55] Prominent application areas of FMT aredescriptive complexity theory,database theory andformal language theory.[56]
Anyset theory (which is expressed in acountable language), if it is consistent, has a countable model; this is known asSkolem'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 thecontinuum hypothesis requires considering sets in models which appear to be uncountable when viewed fromwithin the model, but are countable to someoneoutside the model.[57]
The model-theoretic viewpoint has been useful inset theory; for example inKurt Gödel's work on the constructible universe, which, along with the method offorcing developed byPaul Cohen can be shown to prove the (again philosophically interesting)independence of theaxiom of choice and the continuum hypothesis from the other axioms of set theory.[58]
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.[59] 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.[60]
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