In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, model theory is the study of (classes of) mathematical structures In universal algebra and in model theory, a structure consists of a set along with a collection of finitary functions and relations which are defined on it such as groups In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from, fields In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic, graphs In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. Typically, a graph is depicted in diagrammatic form as a set, or even universes of set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, using tools from mathematical logic Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the. A structure that gives meaning to the sentences of a formal language is called a model for the language. If a model for a language moreover satisfies a particular sentence or theory (set of sentences), it is called a model of the sentence or theory. Model theory has close ties to algebra Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure and universal algebra Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study.

This article focuses on finitary first order First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic is distinguished from propositional logic by its use of quantifiers; each model theory of infinite structures. Finite model theory Finite model theory is a subfield of model theory that focuses on properties of logical languages, such as first-order logic, over finite structures, such as finite groups, graphs, databases, and most abstract machines. It focuses in particular on connections between logical languages and computation, and is closely associated with discrete, 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 logics One of these is the type of variables appearing in quantifications; in first-order logic, roughly speaking, it is forbidden to quantify over predicates. See second-order logic for systems in which this is permitted or infinitary logics An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and completeness that are equivalent in finitary logic is hampered by the fact that completeness Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929 does not in general hold for these logics. However, a great deal of study has also been done in such languages.

Introduction

Model theory recognises and is intimately concerned with a duality: It examines semantical Semantics is the study of meaning. It typically focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for elements by means of syntactical In linguistics, syntax is the study of the principles and rules for constructing sentences in natural languages elements of a corresponding language. To quote the first page of Chang and Keisler (1990):

universal algebra Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study + logic Logic is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic = model theory.

In a similar way to proof theory Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the, model theory is situated in an area of interdisciplinarity An interdisciplinary field is a field of study that crosses traditional boundaries between academic disciplines or schools of thought, as new needs and professions have emerged between mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, philosophy Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing fundamental questions by its critical, generally systematic approach and its reliance on rational argument. The word "philosophy" comes from the, and computer science Computer science or computing science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe, and transform information. Computer science. The most important professional organization in the field of model theory is the Association for Symbolic Logic The Association for Symbolic Logic is an international organization of specialists in mathematical logic and philosophical logic—the largest such organization in the world. The ASL was founded in 1936, a crucial year in the development of modern logic, and its first president was Alonzo Church. The current president of the ASL is Penelope Maddy.

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 Computable model theory is a branch of model theory which deals with questions of computability as they apply to model-theoretical structures. It was developed almost simultaneously by mathematicians in the West, primarily located in the United States and Australia, and Soviet Russia during the middle of the 20th century. Because of the Cold War, 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 Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem, Scott Dana Stewart Scott is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. His research career has spanned computer science, mathematics, and philosophy, and has been characterized by a marriage of a concern for's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death's elimination of quantifiers Quantifier elimination is a concept in mathematical logic, model theory, and theoretical computer science. One way of classifying formulas is by the amount of quantification. Formulae with less depth of quantifier alternation are thought of as simpler and the quantifier free formulae as the simplest. A theory has quantifier elimination if for for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics's development of nonstandard analysis Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number. An important step in the evolution of classical model theory occurred with the birth of stability theory In model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a 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 theorem from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields. The ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory.

Example

To illustrate the basic relationship involving syntax and semantics in the context of a non-trivial models, one can start, on the syntactic side, with suitable axioms for the natural numbers such as Peano axioms, and the associated theory. Going on to the semantic side, one has the usual counting numbers as a model. In the 1930s, Skolem developed alternative models satisfying the axioms. This illustrates what is meant by interpretating a language or theory in a particular model. A more traditional example is interpreting the axioms of a particular algebraic system such as a group, in the context of a model provided by a specific group.

Universal algebra

Fundamental concepts in universal algebra are signatures In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes σ and σ-algebras. Since these concepts are formally defined in the article on structures In universal algebra and in model theory, a structure consists of a set along with a collection of finitary functions and relations which are defined on it, the present article can content itself with an informal introduction which consists in examples of how these terms are used.

The standard signature of rings is σ_ring = {×,+,−,0,1}, where × and + are binary In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division, − is unary In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a function, and 0 and 1 are nullary In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the number of domains in the corresponding Cartesian product. The term springs from such words as unary, binary, ternary, etc.

The standard signature of semirings is σ_smr = {×,+,0,1}, where the arities are as above.

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

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

A ring In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions—namely, the set must be an abelian group under addition and a monoid under is a σ_ring-structure which satisfies the identities u + (v + w) = (u + v) + w, u + v = v + u, u + 0 = u, u + (−u) = 0, u × (v × w) = (u × v) × w, u × 1 = u, 1 × u = u, u × (v + w) = (u × v) + (u × w) and (v + w) × u = (v × u) + (w × u).

A group In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from is a σ_grp-structure which satisfies the identities The concepts of "additive identity" and "multiplicative identity" are central to the Peano axioms. The number 0 is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all u × (v × w) = (u × v) × w, u × 1 = u, 1 × u = u, u × u⁻¹ = 1 and u⁻¹ × u = 1.

A monoid In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single is a σ_mnd-structure which satisfies the identities u × (v × w) = (u × v) × w, u × 1 = u and 1 × u = u.

A semigroup A semigroup is an algebraic structure consisting of a nonempty set S together with an associative binary operation. In other words, a semigroup is an associative magma. The terminology is derived from the anterior notion of a group. A semigroup differs from a group in that for each of its elements there might not exist an inverse; further, there is a σ_mnd-structure which satisfies the identity u × (v × w) = (u × v) × w.

A magma is just a {×}-structure.

This is a very efficient way to define most classes of algebraic structures In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra, because there is also the concept of σ-homomorphism In universal algebra and in model theory, a structure consists of a set along with a collection of finitary functions and relations which are defined on it, 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.

Terms such as the σ_ring-term t(u,v,w) given by (u + (v × w)) + (−1) are used to define identities t = t', but also to construct free algebras In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms. Examples include free groups, tensor algebras, or free. An equational class In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic images, subalgebras and products. In the context 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 theorem states:

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.

An important non-trivial tool in universal algebra are ultraproducts The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are , where I is an infinite set indexing a system of σ-structures A_i, and U is an ultrafilter In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0). If A on I.

While model theory is generally considered a part of mathematical logic Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the, universal algebra, which grew out of Alfred North Whitehead's (1898) work on abstract algebra, is part of algebra. This is reflected by their respective MSC classifications. Nevertheless model theory can be seen as an extension of universal algebra.

Finite model theory

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

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

A graph is a σ_grph-structure satisfying the sentences and .

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 structures 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' is written as a sentence .)

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

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