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Non-Standard Analysis: Hyperreal Number, Ultrafilter, Infinitesimal, Criticism of Non-Standard Analysis, Non-Standard Calculus

Non-Standard Analysis: Hyperreal Number, Ultrafilter, Infinitesimal, Criticism of Non-Standard Analysis, Non-Standard Calculus


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About the Book

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 o ' 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 lo...


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Product Details
  • ISBN-13: 9781155771144
  • Publisher: Books LLC, Wiki Series
  • Publisher Imprint: Books LLC, Wiki Series
  • Height: 246 mm
  • No of Pages: 34
  • Spine Width: 2 mm
  • Weight: 82 gr
  • ISBN-10: 1155771141
  • Publisher Date: 30 Aug 2011
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Sub Title: Hyperreal Number, Ultrafilter, Infinitesimal, Criticism of Non-Standard Analysis, Non-Standard Calculus
  • Width: 189 mm


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