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O-minimal
In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) that 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 union of intervals and points. O-minimality can be regarded as a weak form of quantifier elimination. A structure ''M'' is o-minimal if and only if every with one [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Definition A linearly ordered structure, ''M'', with language ''L'' including an ordering relation <, is called weakly o-minimal if every parametrically definable subset of ''M'' is a finite union of convex (definable) subsets. A is weakly o-minimal if all its models are weakly o-minimal. Note that, in contrast to o-minimality, it is possible for a theory to have models that are weakly o-minimal and to have other models that are not weakly o-minimal. Difference from o-minimality In an o-minimal structure the definable sets in are finite unions of points ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), 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 definable set, 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 Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Closed Field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Definition A real closed field is a field ''F'' in which any of the following equivalent conditions is true: #''F'' is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in ''F'' if and only if it is true in the reals. #There is a total order on ''F'' making it an ordered field such that, in this ordering, every positive element of ''F'' has a square root in ''F'' and any polynomial of odd degree with coefficients in ''F'' has at least one root in ''F''. #''F'' is a formally real field such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Algebraic Geometry
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets. Terminology Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem. Related fields are o-minimal theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Formulations In terms of model theory, Wilkie's theorem deals with the language ''L''exp = (+, −, ·, ''m''. Gabrielov's theorem states that any formula in this language is equivalent to an existential one, as above. Hence the theory of the real ordered field with restricted analytic functions is model complete. Intermediate results Gabrielov's theorem applies to the real field with all restricted analytic functions adjoined, whereas Wilkie's theorem removes the need to restrict the function, but only allows one to add the exponential function. As an intermediate result Wilkie asked when the complement of a sub-analytic set could be defined using the same analytic functions that described the original set. It turns out the required functions are the Pfaffian functions. In part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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] |
Semialgebraic Set
In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers. Definition Let \mathbb be a real closed field (For example \mathbb could be the field of real numbers \mathbb). A subset S of \mathbb^n is a ''semialgebraic set'' if it is a finite union of sets defined by polynomial equalities of the form \ and of sets defined by polynomial inequalities of the form \. Properties Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pfaffian Function
In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s, but are named after German mathematician Johann Pfaff. Basic definition Some functions, when differentiated, give a result which can be written in terms of the original function. Perhaps the simplest example is the exponential function, ''f''(''x'') = ''e''''x''. If we differentiate this function we get ''ex'' again, that is :f^\prime(x) = f(x). Another example of a function like this is the reciprocal function, ''g''(''x'') = 1/''x''. If we differentiate this function we will see that :g^\prime(x) = -g(x)^2. Other functions may not have the above property, but their derivative may be written in terms of functions like those above. For example, if we take the function ''h''(''x'') = ''e''''x'' log ''x'' then we see :h^\prime(x) = e^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |