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| {{Otheruses4|the mathematical discipline|the informal notion in other parts of mathematics and science|Mathematical model}}
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| In [[mathematics]], '''model theory''' is the study of (classes of) mathematical [[Structure (mathematical logic)|structures]] (e.g. [[Group (mathematics)|groups]], [[Field (mathematics)|fields]], [[graph (mathematics)|graphs]], universes of [[set theory]]) using tools from [[mathematical logic]]. It has close ties to [[abstract algebra]], particularly [[universal algebra]].
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| Objects of study in model theory are models of theories in a [[formal language]]. We call '''theory''' a set of sentences in a formal language, and '''model''' of a theory a structure that satisfies the sentences of that theory.
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| This article focuses on [[finitary]] [[First-order logic|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 [[Gödel's completeness theorem|completeness]] does not in general hold for these logics. However, a great deal of study has also been done in such languages.
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| Model theory recognises and is intimately concerned with a duality: It examines [[Semantics|semantical]] elements by means of [[Syntax|syntactical]] elements of a corresponding language. To quote the first page of [[Chen Chung Chang|Chang]] and [[Howard Jerome Keisler|Keisler]] (1990):<ref>Chang and Keisler, [http://books.google.com/books?id=uiHq0EmaFp0C&pg=PA1 p. 1].</ref>
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| :[[universal algebra]] + [[logic]] = '''model theory'''.
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| Model theory developed rapidly during the 1990s, and a more modern definition is provided by [[Wilfrid Hodges]] (1997):
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| :'''model theory''' = [[algebraic geometry]] − [[field (mathematics)|field]]s.
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| 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]].
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| 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 theorem]]s, [[Robert Lawson Vaught|Vaught]]'s two-cardinal theorem, [[Dana Scott|Scott]]'s isomorphism theorem, the [[omitting types theorem]], and the [[Ryll-Nardzewski theorem]]. Examples of early results from model theory applied to fields are [[Alfred Tarski|Tarski]]'s [[quantifier elimination|elimination of quantifiers]] for [[real closed fields]], [[James Ax|Ax]]'s theorem on pseudo-finite fields, and [[Abraham Robinson|Robinson]]'s development of [[non-standard analysis]]. An important step in the evolution of classical model theory occurred with the birth of [[stable theory|stability theory]] (through [[Morley's categoricity theorem|Morley's theorem]] on uncountably categorical theories and [[Saharon Shelah|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 [[Ehud Hrushovski|Hrushovski]]'s proof of the [[Mordell-Lang conjecture|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.
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| == Universal algebra ==
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| {{main|Universal algebra}}
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| Fundamental concepts in universal algebra are [[signature (logic)|signatures]] σ and σ-algebras. Since these concepts are formally defined in the article on [[structure (mathematical logic)|structure]]s, the present article can content itself with an informal introduction which consists in examples of how these terms are used.
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| :The standard signature of rings is σ<sub>ring</sub> = {×,+,−,0,1}, where × and + are [[binary operation|binary]], − is [[unary operation|unary]], and 0 and 1 are [[nullary]].
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| :The standard signature of semirings is σ<sub>smr</sub> = {×,+,0,1}, where the arities are as above.
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| :The standard signature of groups (with multiplicative notation) is σ<sub>grp</sub> = {×,<sup>−1</sup>,1}, where × is binary, <sup>−1</sup> is unary and 1 is nullary.
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| :The standard signature of monoids is σ<sub>mnd</sub> = {×,1}.
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| :A [[ring (mathematics)|ring]] is a σ<sub>ring</sub>-structure which satisfies the identities {{nowrap|''u'' + (''v'' + ''w'') {{=}} (''u'' + ''v'') + ''w'',}} {{nowrap|''u'' + ''v'' {{=}} ''v'' + ''u'',}} {{nowrap|''u'' + 0 {{=}} ''u'',}} {{nowrap|''u'' + (−''u'') {{=}} 0,}} {{nowrap|''u'' × (''v'' × ''w'') {{=}} (''u'' × ''v'') × ''w'',}} {{nowrap|''u'' × 1 {{=}} ''u'',}} {{nowrap|1 × ''u'' {{=}} ''u'',}} {{nowrap|''u'' × (''v'' + ''w'') {{=}} (''u'' × ''v'') + (''u'' × ''w'')}} and {{nowrap|(''v'' + ''w'') × ''u'' {{=}} (''v'' × ''u'') + (''w'' × ''u'').}}
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| :A [[Group (mathematics)|group]] is a σ<sub>grp</sub>-structure which satisfies the [[identity (mathematics)|identities]] {{nowrap|''u'' × (''v'' × ''w'') {{=}} (''u'' × ''v'') × ''w'',}} {{nowrap|''u'' × 1 {{=}} ''u'',}} {{nowrap|1 × ''u'' {{=}} ''u'',}} {{nowrap|''u'' × ''u''<sup>−1</sup> {{=}} 1}} and {{nowrap|''u''<sup>−1</sup> × ''u'' {{=}} 1.}}
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| :A [[monoid]] is a σ<sub>mnd</sub>-structure which satisfies the identities {{nowrap|''u'' × (''v'' × ''w'') {{=}} (''u'' × ''v'') × ''w'',}} {{nowrap|''u'' × 1 {{=}} ''u''}} and {{nowrap|1 × ''u'' {{=}} ''u''.}}
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| :A [[semigroup]] is a {×}-structure which satisfies the identity {{nowrap|''u'' × (''v'' × ''w'') {{=}} (''u'' × ''v'') × ''w''.}}
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| :A [[magma (mathematics)|magma]] is just a {×}-structure.
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| This is a very efficient way to define most classes of [[algebraic structure]]s, because there is also the concept of σ-[[Structure (mathematical logic)#Homomorphisms and embeddings|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.
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| Terms such as the σ<sub>ring</sub>-term ''t''(''u'',''v'',''w'') given by {{nowrap|(''u'' + (''v'' × ''w'')) + (−1)}} are used to define identities {{nowrap|''t'' {{=}} ''t{{'}}'',}} but also to construct [[Free object|free algebra]]s. An [[variety (universal algebra)|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. [[Birkhoff's HSP theorem|Birkhoff's theorem]] states:
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| :A class of σ-structures is an equational class if and only if it is not empty and closed under subalgebras, homomorphic images, and direct products.
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| An important non-trivial tool in universal algebra are [[ultraproduct]]s <math>\Pi_{i\in I}A_i/U</math>, where ''I'' is an infinite set indexing a system of σ-structures ''A<sub>i</sub>'', and ''U'' is an [[ultrafilter]] on ''I''.
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| 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 [[Mathematics Subject Classification|MSC]] classifications. Nevertheless model theory can be seen as an extension of universal algebra.
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| == Finite model theory ==
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| {{main|Finite model theory}}
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| Finite model theory is the area of model theory which has the closest ties to [[#Universal algebra|universal algebra]]. Like some parts of universal algebra, and in contrast with the other areas of model theory, it is mainly concerned with [[finite set|finite]] algebras, or more generally, with finite σ-[[structure (mathematical logic)|structure]]s for signatures σ which may contain relation symbols as in the following example:
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| :The standard signature for graphs is σ<sub>grph</sub>={''E''}, where ''E'' is a binary relation symbol.
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| :A [[Simple graph|graph]] is a σ<sub>grph</sub>-structure satisfying the sentences <math>\forall u \forall v(uEv \rightarrow vEu)</math> and <math>\forall u\neg(uEu)</math>.
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| A σ-homomorphism is a map that commutes with the operations and preserves the relations in σ. This definition gives rise to the usual notion of [[graph homomorphism]], which has the interesting property that a bijective homomorphism need not be invertible. Structures are also a part of universal algebra; after all, some [[algebraic structure]]s such as ordered groups have a binary relation <. What distinguishes finite model theory from universal algebra is its use of more general logical sentences (as in the example above) in place of identities. (In a model-theoretic context an identity ''t''=''t<nowiki>'</nowiki>'' is written as a sentence <math>\forall u_1u_2\dots u_n(t=t')</math>.)
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| The logics employed in finite model theory are often substantially more expressive than [[#First-order logic|first-order logic]], the standard logic for model theory of infinite structures.
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| == First-order logic ==
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| {{main|First-order logic}}
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| Whereas [[#Universal algebra|universal algebra]] provides the [[semantics]] for a [[signature (logic)|signature]], [[mathematical logic|logic]] provides the [[syntax]]. With terms, identities and [[quasiidentity|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.
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| 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 [[table of logic symbols|Boolean connectives]] <math>\neg,\land,\lor,\rightarrow</math> and prefixing of quantifiers <math>\forall v</math> or <math>\exists v</math>. 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:
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| :<math>{\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,}</math>
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| :<math>\psi \;=\; \forall u\forall v((u\times v=x)\rightarrow (u=x)\lor(v=x))\land x\ne 0\land x\ne1.</math>
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| (Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the σ<sub>smr</sub>-structure <math>\mathcal N</math> 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 [[T-schema|"Tarski's definition of truth"]], for the satisfaction relation <math>\models</math>, so that one easily proves:
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| :<math>\mathcal N\models\phi(n) \iff n</math> is a prime number.
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| :<math>\mathcal N\models\psi(n) \iff n</math> is irreducible.
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| A set ''T'' of sentences is called a (first-order) [[theory (mathematical logic)|theory]]. A theory is '''satisfiable''' if it has a '''model''' <math>\mathcal M\models T</math>, 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 [[Gödel's completeness theorem|completeness theorem]] there is no need to distinguish between satisfiability and consistency. Therefore model theorists often use "consistent" as a synonym for "satisfiable".
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| 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''<ref>In a countable signature. The theorem has a straightforward generalization to uncountable signatures.</ref> which has an infinite model and for every infinite [[cardinal number]] κ, there is a model <math>\mathcal M\models T</math> such that the number of elements of <math>\mathcal M</math> is exactly κ. Therefore only finitary structures can be described by a categorical theory.
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| 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.
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| Due to [[Kurt Gödel|Gödel]], 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.
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| ==Axiomatizability, elimination of quantifiers, and model-completeness==
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| 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.
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| A theory ''T'' has [[quantifier elimination]] if every first-order formula φ(x<sub>1</sub>,...,x<sub>''n''</sub>) over its signature is equivalent modulo ''T'' to a first-order formula ψ(x<sub>1</sub>,...,x<sub>''n''</sub>) without quantifiers, i.e. <math>\forall x_1\dots\forall x_n(\phi(x_1,\dots,x_n)\leftrightarrow \psi(x_1,\dots,x_n))</math> holds in all models of ''T''. For example the theory of algebraically closed fields in the signature σ<sub>ring</sub>=(×,+,−,0,1) has quantifier elimination because every formula is equivalent to a Boolean combination of equations between polynomials.
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| A [[substructure]] of a σ-structure is a subset of its domain, closed under all functions in its signature σ, which is regarded as a σ-structure by restricting all functions and relations in σ to the subset. An [[embedding]] of a σ-structure <math>\mathcal A</math> into another σ-structure <math>\mathcal B</math> is a map f: A → B between the domains which can be written as an isomorphism of <math>\mathcal A</math> with a substructure of <math>\mathcal B</math>. Every embedding is an [[injective]] homomorphism, but the converse holds only if the signature contains no relation symbols.
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| 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:
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| 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<sub>1</sub>,...,x<sub>''n''</sub>) over its signature is equivalent modulo ''T'' to an existential first-order formula, i.e. a formula of the following form:
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| :<math>\exists v_1\dots\exists v_m\psi(x_1,\dots,x_n,v_1,\dots,v_m)</math>,
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| 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'''.
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| == Categoricity ==
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| As observed in the section on [[#First-order logic|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 number|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. <math>\aleph_0</math> + |σ|, where |σ| is the cardinality of the signature). For finite or countable signatures this means that there is a fundamental difference between <math>\aleph_0</math>-cardinality and κ-cardinality for uncountable κ.
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| A few [[Omega-categorical theory|characterizations of <math>\aleph_0</math>-categoricity]] include:
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| :For a complete first-order theory ''T'' in a finite or countable signature the following conditions are equivalent:
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| :#''T'' is <math>\aleph_0</math>-categorical.
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| :#For every natural number ''n'', the [[Type (model theory)#Stone spaces|Stone space]] ''S<sub>n</sub>''(''T'') is finite.
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| :#For every natural number ''n'', the number of formulas φ(''x''<sub>1</sub>, ..., ''x''<sub>n</sub>) in ''n'' free variables, up to equivalence modulo ''T'', is finite.
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| This result, due independently to [[Erwin Engeler|Engeler]], [[Czesław Ryll-Nardzewski|Ryll-Nardzewski]] and [[Lars Svenonius|Svenonius]], is sometimes referred to as the [[Omega-categorical theory|Ryll-Nardzewski]] theorem.
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| Further, <math>\aleph_0</math>-categorical theories and their countable models have strong ties with [[oligomorphic group]]s. They are often constructed as [[Fraïssé limit]]s.
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| [[Michael D. Morley|Michael Morley]]'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:
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| :[[Morley's categoricity theorem]]
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| :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.
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| '''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 <math>\aleph_0</math>-categorical and uncountably categorical is called '''totally categorical'''.
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| == Model theory and set theory ==
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| [[Set theory]] (which is expressed in a [[countable]] language), if it is consistent, has a countable model; this is known as [[Skolem's paradox]], since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model. Particularly the proof of the independence of the [[continuum hypothesis]] requires considering sets in models which appear to be uncountable when viewed from ''within'' the model, but are countable to someone ''outside'' the model.
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| The model-theoretic viewpoint has been useful in [[set theory]]; for example in [[Kurt Gödel| Kurt Gödel's]] work on the constructible universe, which, along with the method of forcing developed by [[Paul Cohen (mathematician)|Paul Cohen]] can be shown to prove the (again philosophically interesting) [[Independence (mathematical logic)|independence]] of the [[axiom of choice]] and the continuum hypothesis from the other axioms of set theory.
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| 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.
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| == Other basic notions of model theory ==
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| === Reducts and expansions ===
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| {{main|Reduct}}
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| A field or a vector space can be regarded as a (commutative) group by simply ignoring some of its structure. The corresponding notion in model theory is that of a '''reduct''' of a structure to a subset of the original signature. The opposite relation is called an ''expansion'' - e.g. the (additive) group of the [[rational numbers]], regarded as a structure in the signature {+,0} can be expanded to a field with the signature {×,+,1,0} or to an ordered group with the signature {+,0,<}.
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| 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.
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| === Interpretability ===
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| {{main|Interpretation (model theory)}}
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| 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.
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| 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'''.
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| 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 [[Decidability (logic)|undecidable]], then ''M'' itself is undecidable.
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| === Using the compactness and completeness theorems ===
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| [[Gödel's completeness theorem]] (not to be confused with his [[Gödel's incompleteness theorems|incompleteness theorems]]) says that a theory has a model if and only if it is [[consistency|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 (mathematical logic)|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 language|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 [[Decidability (logic)|undecidable]] if a proposed axiom is indeed an axiom, making proof-checking a [[supertask]].
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| 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 [[Kurt Gödel|Gödel]] (which goes via proofs) and one by [[Anatoly Ivanovich Malcev|Malcev]] (which is more direct and allows us to restrict the cardinality of the resulting model).
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| 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 <math>\mathfrak{A}</math> has models of all infinite cardinalities (at least that of the language) which agree with <math>\mathfrak{A}</math> on all sentences, i.e. they are '[[elementarily equivalent]]'.
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| === Types ===
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| {{main|Type (model theory)}}
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| Fix an <math>L</math>-structure <math>M</math>, and a natural number <math>n</math>. The set of definable subsets of <math>M^n</math> over some parameters <math>A</math> is a [[Boolean algebra (structure)|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 [[Topology|topological space]] consisting of maximal consistent sets of formulae over <math>A</math>. We call this the space of (complete) <math>n</math>-[[Type (model theory)|types]] over <math>A</math>, and write <math>S_n(A)</math>.
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| Now consider an element <math>m \in M^n</math>. Then the set of all formulae <math>\phi</math> with parameters in <math>A</math> in free variables <math>x_1,\ldots,x_n</math> so that <math>M \models \phi(m)</math> is consistent and maximal such. It is called the ''type'' of <math>m</math> over <math>A</math>.
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| One can show that for any <math>n</math>-type <math>p</math>, there exists some elementary [[Extension (model theory)|extension]] <math>N</math> of <math>M</math> and some <math>a \in N^n</math> so that <math>p</math> is the type of <math>a</math> over <math>A</math>.
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| 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.
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| '''Illustrative Example:''' Suppose <math>M</math> 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 <math>n</math>-types over a subfield <math>A</math> is [[bijective]] with the set of [[prime ideal]]s of the [[polynomial ring]] <math>A[x_1,\ldots,x_n]</math>. This is the same set as the [[Spectrum of a ring|spectrum]] of <math>A[x_1,\ldots,x_n]</math>. Note however that the topology considered on the type space is the [[constructible topology]]: a set of types is basic [[Open set|open]] iff it is of the form <math>\{p: f(x)=0 \in p\}</math> or of the form <math>\{p: f(x) \neq 0 \in p\}</math>. This is finer than the [[Zariski topology]].
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| == History ==
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| Model theory as a subject has existed 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]],<ref>''All three commentators [i.e. Vaught, van Heijenoort and Dreben] agree that both the completeness and compactness theorems were implicit in Skolem 1923 [...],'' Dawson (1993).</ref> but it was first published in 1930, as a lemma in [[Kurt Gödel]]'s proof of his [[Gödel's completeness theorem|completeness theorem]]. The Löwenheim–Skolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 from [[Anatoly Maltsev]].
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| == See also ==
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| {{col-begin}}
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| {{col-break}}
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| * [[Axiomatizable class]]
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| * [[Compactness theorem]]
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| * [[Descriptive complexity]]
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| * [[Elementary equivalence]]
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| {{col-break}}
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| * [[List of first-order theories|First-order theories]]
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| * [[Forcing (mathematics)|Forcing]]
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| * [[Hyperreal number]]
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| * [[Institutional model theory]]
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| {{col-break}}
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| * [[Kripke semantics]]
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| * [[Löwenheim–Skolem theorem]]
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| * [[Proof theory]]
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| * [[Saturated model]]
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| {{col-end}}
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| == Notes ==
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| {{reflist}}
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| == References ==
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| === Canonical textbooks ===
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| * {{Cite book | last1=[[Chen Chung Chang|Chang]] | first1=Chen Chung | last2=Keisler | first2=H. Jerome | author2-link=Howard Jerome Keisler | title=Model Theory | origyear=1973 | publisher=Elsevier | edition=3rd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-88054-3 | year=1990 | postscript=<!--None-->}}
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| * {{Cite book | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher= [[Cambridge University Press]]| location=Cambridge | isbn=978-0-521-58713-6 | year=1997 | postscript=<!--None-->}}
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| * {{cite book | last=Marker | first=David | title= Model Theory: An Introduction | publisher=Springer | year=2002 | isbn=0-387-98760-6| series=[[Graduate Texts in Mathematics]] 217}}
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| === Other textbooks ===
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| * {{cite book | last=Bell | first=John L. | coauthors=Slomson, Alan B. | year=2006 | title=Models and Ultraproducts: An Introduction | edition=reprint of 1974 | origyear=1969 | publisher=[[Dover Publications]] | isbn=0-486-44979-3 }}
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| * {{cite book | last=Ebbinghaus | first=Heinz-Dieter | coauthors=Flum, Jörg; Thomas, Wolfgang | publisher=[[Springer Science+Business Media|Springer]] | title=Mathematical Logic | year=1994 | isbn= 0-387-94258-0}}
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| * {{cite book | last=Ziegler | first=Martin | coauthors=Tent, Katrin | publisher=[[Cambridge University Press]] | title=A Course in Model Theory | year=2012 | isbn= 9780521763240}}
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| * {{cite book | author = Hinman, Peter G. | title = Fundamentals of Mathematical Logic | publisher = [[A K Peters, Ltd.|A K Peters]] | year = 2005 | isbn = 1-56881-262-0}}
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| * {{cite book | last=Manzano | first=Maria | authorlink=Maria Manzano | publisher=[[Alianza editorial]] | title=Teoria de modelos | year=1989 | isbn=84-206-8126-1 }}
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| * {{cite book | last=Hodges | first=Wilfrid | authorlink=Wilfrid Hodges | publisher=[[Cambridge University Press]] | title=Model theory | year=1993 | isbn=0-521-30442-3 }}
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| * {{cite book | last=Manzano | first=Maria | authorlink=Maria Manzano | publisher=[[Oxford University Press]] | title=Model theory | year=1999 | isbn=0-19-853851-0 }}
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| * {{cite book | last=Poizat | first=Bruno | publisher=Springer | title=A Course in Model Theory | year=2000 | isbn=0-387-98655-3 }}
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| * {{cite book | last=Rautenberg | first=Wolfgang | authorlink=Wolfgang Rautenberg | doi=10.1007/978-1-4419-1221-3 | title=A Concise Introduction to Mathematical Logic | url=http://www.springerlink.com/content/978-1-4419-1220-6/ | publisher=[[Springer Science+Business Media]] | location=[[New York City|New York]] | edition=3rd | isbn=978-1-4419-1220-6 | year=2010 }}
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| * {{cite book | last=Rothmaler | first=Philipp | title=Introduction to Model Theory | publisher=[[Taylor and Francis|Taylor & Francis]] | year=2000 | edition=new | isbn=90-5699-313-5 }}
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| === Free online texts ===
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| * {{cite book | last=Chatzidakis | first=Zoe | year=2001 | title=Introduction to Model Theory | pages=26 pages in [[Device independent file format|DVI]] format | url=http://www.logique.jussieu.fr/~zoe/papiers/MTluminy.dvi }}
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| * {{cite book | last=Pillay | first=Anand | title=Lecture Notes – Model Theory | year=2002 | url=http://www.math.uiuc.edu/People/pillay/lecturenotes_modeltheory.pdf | pages=61 pages }}
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| * {{springer|title=Model theory|id=p/m064390}}
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| * [[Wilfrid Hodges|Hodges, Wilfrid]], ''[http://plato.stanford.edu/entries/modeltheory-fo/ First-order Model theory]''. The Stanford Encyclopedia Of Philosophy, E. Zalta (ed.).
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| * Simmons, Harold (2004), ''[http://www.cs.man.ac.uk/~hsimmons/BOOKS/ModelTheory.pdf An introduction to Good old fashioned model theory]''. Notes of an introductory course for postgraduates (with exercises).
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| * J. Barwise and S. Feferman (editors), [http://projecteuclid.org/euclid.pl/1235417263 Model-Theoretic Logics], Perspectives in Mathematical Logic, Volume 8, New York: Springer-Verlag, 1985.
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| {{Mathematics-footer}}
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| {{Logic}}
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| [[Category:Model theory| ]]
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