2010, ISBN: 9781155771144

[ED: Pappeinband], [PU: Bertrams Print On Demand], - Source: Wikipedia. Pages: 32. Chapters: Hyperreal number, Ultrafilter, Infinitesimal, Criticism of non-standard analysis, Non-standard… More...

[ED: Pappeinband], [PU: Bertrams Print On Demand], - Source: Wikipedia. Pages: 32. Chapters: Hyperreal number, Ultrafilter, Infinitesimal, Criticism of non-standard analysis, Non-standard calculus, Transfer principle, Internal set theory, Ultraproduct, Leibniz's notation, Saturated model, Non-standard model of arithmetic, Elementary Calculus: An Infinitesimal Approach, Smooth infinitesimal analysis, Influence of non-standard analysis, Infinitely near point, Hyperfinite set, Overspill, Hyperinteger, Non-classical analysis, Non-Archimedean ordered field, Standard part function, Increment theorem, Constructive non-standard analysis, Monad. Excerpt: The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form Such a number is infinite, and its inverse is infinitesimal. According to Keisler (1994), the term hyperreal was introduced by Edwin Hewitt (1948, p. 74), who spelled it with a hyphen: hyper-real. The hyperreal numbers satisfy the transfer principle, which is a mathematical implementation of Leibniz's heuristic Law of Continuity. The transfer principle states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals since R is a real closed field, so is *R. Since sin n = 0 for all integers n, one also has sin H = 0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Los' theorem of 1955. Concerns about the logical soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Euclid replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules which Robinson delineated. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis some find it more intuitive than standard real analysis. One immediate application is the definition of the basic concepts of analysis such as derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes for an infinitesimal . The idea of the hyperreal system is - Besorgungstitel - vorauss. Lieferzeit 3-5 Tage.., [SC: 0.00]<

ISBN: 9781155771144

[ED: Taschenbuch], [PU: Books LLC, Reference Series], Source: Wikipedia. Pages: 32. Chapters: Hyperreal number, Ultrafilter, Infinitesimal, Criticism of non-standard analysis, Non-standar… More...

[ED: Taschenbuch], [PU: Books LLC, Reference Series], Source: Wikipedia. Pages: 32. Chapters: Hyperreal number, Ultrafilter, Infinitesimal, Criticism of non-standard analysis, Non-standard calculus, Transfer principle, Internal set theory, Ultraproduct, Leibniz's notation, Saturated model, Non-standard model of arithmetic, Elementary Calculus: An Infinitesimal Approach, Smooth infinitesimal analysis, Influence of non-standard analysis, Infinitely near point, Hyperfinite set, Overspill, Hyperinteger, Non-classical analysis, Non-Archimedean ordered field, Standard part function, Increment theorem, Constructive non-standard analysis, Monad. Excerpt: The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form Such a number is infinite, and its inverse is infinitesimal. According to Keisler (1994), the term "hyperreal" was introduced by Edwin Hewitt (1948, p. 74), who spelled it with a hyphen: "hyper-real". The hyperreal numbers satisfy the transfer principle, which is a mathematical implementation of Leibniz's heuristic Law of Continuity. The transfer principle states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals since R is a real closed field, so is *R. Since sin n = 0 for all integers n, one also has sin H = 0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Los' theorem of 1955. Concerns about the logical soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Euclid replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules which Robinson delineated. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis some find it more intuitive than standard real analysis. One immediate application is the definition of the basic concepts of analysis such as derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes for an infinitesimal . The idea of the hyperreal system isVersandfertig in 3-5 Tagen, [SC: 0.00]<

2010, ISBN: 9781155771144

[ED: Pappeinband], [PU: Bertrams Print On Demand], - Source: Wikipedia. Pages: 32. Chapters: Hyperreal number, Ultrafilter, Infinitesimal, Criticism of non-standard analysis, Non-standard… More...

[ED: Pappeinband], [PU: Bertrams Print On Demand], - Source: Wikipedia. Pages: 32. Chapters: Hyperreal number, Ultrafilter, Infinitesimal, Criticism of non-standard analysis, Non-standard calculus, Transfer principle, Internal set theory, Ultraproduct, Leibniz's notation, Saturated model, Non-standard model of arithmetic, Elementary Calculus: An Infinitesimal Approach, Smooth infinitesimal analysis, Influence of non-standard analysis, Infinitely near point, Hyperfinite set, Overspill, Hyperinteger, Non-classical analysis, Non-Archimedean ordered field, Standard part function, Increment theorem, Constructive non-standard analysis, Monad. Excerpt: The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form Such a number is infinite, and its inverse is infinitesimal. According to Keisler (1994), the term hyperreal was introduced by Edwin Hewitt (1948, p. 74), who spelled it with a hyphen: hyper-real. The hyperreal numbers satisfy the transfer principle, which is a mathematical implementation of Leibniz's heuristic Law of Continuity. The transfer principle states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals since R is a real closed field, so is *R. Since sin n = 0 for all integers n, one also has sin H = 0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Los' theorem of 1955. Concerns about the logical soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Euclid replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules which Robinson delineated. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis some find it more intuitive than standard real analysis. One immediate application is the definition of the basic concepts of analysis such as derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes for an infinitesimal . The idea of the hyperreal system is - Besorgungstitel - vorauss. Lieferzeit 3-5 Tage.., [SC: 0.00]<

2012, ISBN: 1155771141

Kartoniert / Broschiert MATHEMATICS / Logic, mit Schutzumschlag neu, [PU:Books LLC, Reference Series]