Category: Model theory

Type (model theory)
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave. More pr
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic te
Tennenbaum's theorem
Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic
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. Computable model theory introduces the ideas of computabl
Interpretation (model theory)
In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the idea of representing M inside N. For example ev
Hereditary property
In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of subobject depends on the context. These properties are particularly co
Indiscernibles
In mathematical logic, indiscernibles are objects that cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered.
Łoś–Vaught test
In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic,
Prime model
In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model is prime if it admits an elementary embedding into any model to which it i
Löwenheim–Skolem theorem
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. I
Wilkie's theorem
In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
Reduct
In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The opposite of "reduct" is "expansion."
Saturated model
In mathematical logic, and particularly in its subfield model theory, a saturated model M is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ul
Ziegler spectrum
In mathematics, the (right) Ziegler spectrum of a ring R is a topological space whose points are (isomorphism classes of) indecomposable pure-injective right R-modules. Its closed subsets correspond t
Reduced product
In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let {Si | i ∈ I} be a family of structures
Strongly minimal theory
In model theory—a branch of mathematical logic—a minimal structure is an infinite one-sorted structure such that every subset of its domain that is definable with parameters is either finite or cofini
Existentially closed model
In model theory, a branch of mathematical logic, the notion of an existentially closed model (or existentially complete model) of a theory generalizes the notions of algebraically closed fields (for t
Forking extension
In model theory, a forking extension of a type is an extension of that type that is not free whereas a non-forking extension is an extension that is as free as possible. This can be used to extend the
Chang's conjecture
In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by , p. 309), states that every model of type (ω2,ω1) for a countable language has an elementary sub
Finite model theory
Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). Finite model
Potential isomorphism
In mathematical logic and in particular in model theory, a potential isomorphism is a collection of finite between two models which satisfies certain closure conditions. Existence of a partial isomorp
Valuation (logic)
In logic and model theory, a valuation can be: * In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional
Structural Ramsey theory
In mathematics, structural Ramsey theory is a categorical generalisation of Ramsey theory, rooted in the idea that many important results of Ramsey theory have "similar" logical structure. The key obs
Pregeometry (model theory)
Pregeometry, and in full combinatorial pregeometry, are essentially synonyms for "matroid". They were introduced by Gian-Carlo Rota with the intention of providing a less "ineffably cacophonous" alter
Quantifier rank
In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory. Notice that the quantifier rank is a property of the for
Hardy field
In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that are closed under differentiation. They are named after the English mathematician G. H. Hardy.
Joint embedding property
In universal algebra and model theory, a class of structures K is said to have the joint embedding property if for all structures A and B in K, there is a structure C in K such that both A and B have
Feferman–Vaught theorem
Feferman–Vaught theorem in model theory is a theorem by Solomon Feferman and Robert Lawson Vaught that shows how to reduce, in an algorithmic way, the first-order theory of a product of first-order st
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. The completeness th
Satisfiability
In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula is satisfiable because it is true when and , while the formula
Ehrenfeucht–Mostowski theorem
In model theory, a field within mathematical logic, the Ehrenfeucht–Mostowski theorem gives conditions for the existence of a model with indiscernibles.
Model complete theory
In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula
Soundness
In logic, more precisely in deductive reasoning, an argument is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical
First-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer
Ax–Kochen theorem
The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homog
Stable group
In model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley rank (see below).
Pseudoelementary class
In logic, a pseudoelementary class is a class of structures derived from an elementary class (one definable in first-order logic) by omitting some of its sorts and relations. It is the mathematical lo
Löwenheim number
In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds. They are named after Leopold Löwenheim, who pr
Tame abstract elementary class
In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tamene
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, a
Ehrenfeucht–Fraïssé game
In the mathematical discipline of model theory, the Ehrenfeucht–Fraïssé game (also called back-and-forth games)is a technique based on game semantics for determining whether two structures are element
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its
Spectrum of a theory
In model theory, a branch of mathematical logic, the spectrum of a theoryis given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in
Morley rank
In mathematical logic, Morley rank, introduced by Michael D. Morley, is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.
Weakly o-minimal structure
In model theory, a weakly o-minimal structure is a model theoretic structure whose definable sets in the domain are just finite unions of convex sets.
Non-standard model
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).
Real closed ring
In mathematics, a real closed ring (RCR) is a commutative ring A that is a subring of a product of real closed fields, which is closed under continuous semi-algebraic functions defined over the intege
Cantor's isomorphism theorem
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. It is named after Georg Cant
Back-and-forth method
In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In p
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object is said to be embedde
List of first-order theories
In first-order logic, a first-order theory is given by a set of axioms in somelanguage. This entry lists some of the more common examples used in model theory and some of their properties.
Ordered exponential field
In mathematics, an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers.
Büchi arithmetic
Büchi arithmetic of base k is the first-order theory of the natural numbers with addition and the function which is defined as the largest power of k dividing x, named in honor of the Swiss mathematic
Complete theory
In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence the
Semantics of logic
In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic n
Elementary diagram
In the mathematical field of model theory, the elementary diagram of a structure is the set of all sentences with parameters from the structure that are true in the structure. It is also called the co
Elementary class
In model theory, a branch of mathematical logic, an elementary class (or axiomatizable class) is a class consisting of all structures satisfying a fixed first-order theory.
Diagram (mathematical logic)
In model theory, a branch of mathematical logic, the diagram of a structure is a simple but powerful concept for proving useful properties of a theory, for example the amalgamation property and the jo
Strength (mathematical logic)
The relative strength of two systems of formal logic can be defined via model theory. Specifically, a logic is said to be as strong as a logic if every elementary class in is an elementary class in .
Vaught conjecture
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order comp
Model-theoretic grammar
Model-theoretic grammars, also known as constraint-based grammars, contrast with generative grammars in the way they define sets of sentences: they state constraints on syntactic structure rather than
Categorical theory
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing its structure. In first-order
Tarski's exponential function problem
In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theo
Non-standard model of arithmetic
In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard
Algebraically compact module
In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the mod
Institution (computer science)
The notion of institution was created by Joseph Goguen and Rod Burstall in the late 1970s, in order to deal with the "population explosion among the logical systems used in computer science". The noti
Hrushovski construction
In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure rather than . It can be thought of as a kind
Abstract model theory
In mathematical logic, abstract model theory is a generalization of model theory that studies the general properties of extensions of first-order logic and their models. Abstract model theory provides
Stability spectrum
In model theory, a branch of mathematical logic, a complete first-order theory T is called stable in λ (an infinite cardinal number), if the Stone space of every model of T of size ≤ λ has itself size
Compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model t
Stable model semantics
The concept of a stable model, or answer set, is used to define a declarative semantics for logic programs with negation as failure. This is one of several standard approaches to the meaning of negati
Atomic model (mathematical logic)
In model theory, a subfield of mathematical logic, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula. Such types are called principal types, and
O-minimal theory
In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) which is totally ordered by < is called an o-minimal structure if and only if every definable subset X ⊂ M
Presburger arithmetic
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains o
Fraïssé limit
In mathematical logic, specifically in the discipline of model theory, the Fraïssé limit (also called the Fraïssé construction or Fraïssé amalgamation) is a method used to construct (infinite) mathema
C-minimal theory
In model theory, a branch of mathematical logic, a C-minimal theory is a theory that is "minimal" with respect to a ternary relation C with certain properties. Algebraically closed fields with a (Krul
Institutional model theory
In mathematical logic, institutional model theory generalizes a large portion of first-order model theory to an arbitrary logical system.
Substructure (mathematics)
In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the
Signature (logic)
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 algeb
Equisatisfiability
In Mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either
Ax–Grothendieck theorem
In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck. The theorem is often giv
Imaginary element
In model theory, a branch of mathematics, an imaginary element of a structure is roughly a definable equivalence class. These were introduced by , and elimination of imaginaries was introduced by .
In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly c
Quantifier elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement " such that " can be viewed as a qu
General frame
In logic, general frames (or simply frames) are Kripke frames with an additional structure, which are used to model modal and intermediate logics. The general frame semantics combines the main virtues
NIP (model theory)
In model theory, a branch of mathematical logic, a complete theory T is said to satisfy NIP ("not the independence property") if none of its formulae satisfy the independence property—that is, if none
Zariski geometry
In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve,
Łoś–Tarski preservation theorem
The Łoś–Tarski theorem is a theorem in model theory, a branch of mathematics, that states that the set of formulas preserved under taking substructures is exactly the set of universal formulas. The th
Abstract elementary class
In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure
Functional predicate
In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term.Functional pre
Model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and
Elementary equivalence
In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences. If N is a substruct
Kripke semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 195
Omega-categorical theory
In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case κ = = ω of κ-categoricity, and om
Differentially closed field
In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. Thi
Exponential field
In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation.
Pseudo-finite field
In mathematics, a pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of
Conservative extension
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Simil
U-rank
In model theory, a branch of mathematical logic, U-rank is one measure of the complexity of a (complete) type, in the context of stable theories. As usual, higher U-rank indicates less restriction, an
Boolean-valued model
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not li
Stable theory
In the mathematical field of 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
True arithmetic
In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that g
Two-variable logic
In mathematical logic and computer science, two-variable logic is the fragment of first-order logic where formulae can be written using only two different variables. This fragment is usually studied w
Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direc
Decidable sublanguages of set theory
In mathematical logic, various sublanguages of set theory are decidable. These include: * Sets with Monotone, Additive, and Multiplicative Functions. * Sets with restricted quantifiers.
Definable set
In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements satisfy some formula in the first-order language of that structure. A set can be defined with or
Skolem normal form
In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers. Every first-order formula may be converted in
Original proof of Gödel's completeness theorem
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a shorter version of the proof, published as an article in 1930, titled "The completeness of the
Transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle
Amalgamation property
In the mathematical field of model theory, the amalgamation property is a property of collections of structures that guarantees, under certain conditions, that two structures in the collection can be
Tame group
In mathematical group theory, a tame group is a certain kind of group defined in model theory. Formally, we define a bad field as a structure of the form (K, T), where K is an algebraically closed fie