|
Minimality
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 cofinite. A strongly minimal theory is a complete theory all models of which are minimal. A strongly minimal structure is a structure whose theory is strongly minimal. Thus a structure is minimal only if the parametrically definable subsets of its domain cannot be avoided, because they are already parametrically definable in the pure language of equality. Strong minimality was one of the early notions in the new field of classification theory and stability theory that was opened up by Morley's theorem on totally categorical structures. The nontrivial standard examples of strongly minimal theories are the one-sorted theories of infinite-dimensional vector spaces, and the theories ACF''p'' of algebraically closed fields of characteristic ''p''. As the example ACF''p'' shows, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 their models (those structures in which the statements of the theory hold). 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 be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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'' (with parameters taken from ''M'') is a finite of intervals and points. O-minimality can be regarded as a weak form of . A structure ''M'' is o-minimal if and only if every formula with one free variable and parameters in ''M'' is equivalent to a quantifier-free formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 without parameters, which are elements of the domain that can be referenced in the formula defining the relation. Definition Let \mathcal be a first-order language, \mathcal an \mathcal-structure with domain M, X a fixed subset of M, and m a natural number. Then: * A set A\subseteq M^m is ''definable in \mathcal with parameters from X'' if and only if there exists a formula \varphi _1,\ldots,x_m,y_1,\ldots,y_n/math> and elements b_1,\ldots,b_n\in X such that for all a_1,\ldots,a_m\in M, :(a_1,\ldots,a_m)\in A if and only if \mathcal\models\varphi _1,\ldots,a_m,b_1,\ldots,b_n/math> :The bracket notation here indicates the semantic evaluation of the free variables in the formula. * A set ''A is definable in \mathcal without parameters'' if it is definable in \mathcal with paramet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (Krull) valuation are perhaps the most important example. This notion was defined in analogy to the o-minimal theories, which are "minimal" (in the same sense) with respect to a linear order. Definition A ''C''-relation is a ternary relation ''C''(''x'';''y,z'') that satisfies the following axioms. # \forall xyz\, C(x;y,z)\rightarrow C(x;z,y) # \forall xyz\, C(x;y,z)\rightarrow\neg C(y;x,z) # \forall xyzw\, C(x;y,z)\rightarrow (C(w;y,z)\vee C(x;w,z)) # \forall xy\, x\neq y \rightarrow \exists z\neq y\, C(x;y,z) A C-minimal structure is a structure ''M'', in a signature containing the symbol ''C'', such that ''C'' satisfies the above axioms and every set of elements of ''M'' that is definable with parameters in ''M'' is a Boolean combination of instances of ''C'', i.e. of formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols. Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a ''semantic model'' when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as " interpretations", whereas the term "interpretation" generally has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehud Hrushovski
Ehud Hrushovski ( he, אהוד הרושובסקי; born 30 September 1959) is a mathematical logician. He is a Merton Professor of Mathematical Logic at the University of Oxford and a Fellow of Merton College, Oxford. He was also Professor of Mathematics at the Hebrew University of Jerusalem. Early life and education Hrushovski's father, Benjamin Harshav (Hebrew: בנימין הרשב, né Hruszowski; 1928–2015), was a literary theorist, a Yiddish and Hebrew poet and a translator, professor at Yale University and Tel Aviv University in comparative literature. Ehud Hrushovski earned his PhD from the University of California, Berkeley in 1986 under Leo Harrington; his dissertation was titled ''Contributions to Stable Model Theory''. He was a professor of mathematics at the Massachusetts Institute of Technology until 1994, when he became a professor at the Hebrew University of Jerusalem. Hrushovski moved in 2017 to the University of Oxford, where he is the Merton Professor of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Zilber
Boris Zilber (russian: Борис Иосифович Зильбер, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a professor of mathematical logic at the University of Oxford. He obtained his doctorate (Candidate of Sciences) from the Novosibirsk State University in 1975 under the supervision of Mikhail Taitslin and his habilitation (Doctor of Sciences) from the Saint Petersburg State University in 1986. He received the Senior Berwick Prize (2004) and the Pólya Prize (2015) from the London Mathematical Society. He also gave the Tarski Lectures The Alfred Tarski Lectures are an annual distinction in mathematical logic and series of lectures held at the University of California, Berkeley. Established in tribute to Alfred Tarski on the fifth anniversary of his death, the award has been give ... in 2002. References External links Prof. Zilber's homepage 20th-century British mathematicians 21st-century B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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" alternative term. Also, the term combinatorial geometry, sometimes abbreviated to geometry, was intended to replace "simple matroid". These terms are now infrequently used in the study of matroids. It turns out that many fundamental concepts of linear algebra – closure, independence, subspace, basis, dimension – are available in the general framework of pregeometries. In the branch of mathematical logic called model theory, infinite finitary matroids, there called "pregeometries" (and "geometries" if they are simple matroids), are used in the discussion of independence phenomena. The study of how pregeometries, geometries, and abstract closure operators influence the structure of first-order models is called geometric stability theory. Motivation If V is a vector space over som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure Operator
In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are determined by their closed sets, i.e., by the sets of the form cl(''X''), since the closure cl(''X'') of a set ''X'' is the smallest closed set containing ''X''. Such families of "closed sets" are sometimes called closure systems or "Moore families", in honor of E. H. Moore who studied closure operators in his 1910 ''Introduction to a form of general analysis'', whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekind and Georg Cantor. Closure operators are also called "hull operators", which prevents confusion with the "c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Extension
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 substructure of ''M'', one often needs a stronger condition. In this case ''N'' is called an elementary substructure of ''M'' if every first-order ''σ''-formula ''φ''(''a''1, …, ''a''''n'') with parameters ''a''1, …, ''a''''n'' from ''N'' is true in ''N'' if and only if it is true in ''M''. If ''N'' is an elementary substructure of ''M'', then ''M'' is called an elementary extension of ''N''. An embedding ''h'': ''N'' → ''M'' is called an elementary embedding of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of ''M''. A substructure ''N'' of ''M'' is elementary if and only if it passes the Tarski–Vaught test: every first-order formula ''φ''(''x'', ''b''1, …, ''b''''n'') with p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (field)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
