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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 :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 parameters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Analytical Hierarchy
Analytic or analytical may refer to: Chemistry * Analytical chemistry, the analysis of material samples to learn their chemical composition and structure * Analytical technique, a method that is used to determine the concentration of a chemical compound or chemical element * Analytical concentration Mathematics * Abstract analytic number theory, the application of ideas and techniques from analytic number theory to other mathematical fields * Analytic combinatorics, a branch of combinatorics that describes combinatorial classes using generating functions * Analytic element method, a numerical method used to solve partial differential equations * Analytic expression or analytic solution, a mathematical expression using well-known operations that lend themselves readily to calculation * Analytic geometry, the study of geometry based on numerical coordinates rather than axioms * Analytic number theory, a branch of number theory that uses methods from mathematical analysis M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Walter Rudin
Walter Rudin (May 2, 1921 – May 20, 2010) was an Austrian- American mathematician and professor of mathematics at the University of Wisconsin–Madison. In addition to his contributions to complex and harmonic analysis, Rudin was known for his mathematical analysis textbooks: '' Principles of Mathematical Analysis'', ''Real and Complex Analysis'', and ''Functional Analysis.'' Rudin wrote ''Principles of Mathematical Analysis'' only two years after obtaining his Ph.D. from Duke University, while he was a C. L. E. Moore Instructor at MIT. ''Principles'', acclaimed for its elegance and clarity, has since become a standard textbook for introductory real analysis courses in the United States. Rudin's analysis textbooks have also been influential in mathematical education worldwide, having been translated into 13 languages, including Russian, Chinese, and Spanish. Biography Rudin was born into a Jewish family in Austria in 1921. He was enrolled for a period of time at a Swiss bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Elementary Substructure
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] [Amazon] |