Lectures on the Hyperreals: An Introduction to Nonstandard Analysis / Edition 1

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This is an introduction to nonstandard analysis based on a course of lectures given several times by the author. It is suitable for use as a text at the beginning graduate or upper undergraduate level, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions; a source of new ideas, objects and proofs; and a wellspring of powerful new principles of reasoning (transfer, overflow, saturation, enlargement, hyperfinite approximation etc.). The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective, emphasizing the role of the transfer principle as a working tool of mathematical practice. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line, Ramsey's Theorem, nonstandard constructions of p-adic numbers and power series, and nonstandard proofs of the Stone representation theorem for Boolean algebras and the Hahn-Banach theorem. Features of the text include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set- theoretic approach to enlargements than the usual one based on superstructures.

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Editorial Reviews

From the Publisher

R. Goldblatt

Lectures on the Hyperreals

An Introduction to Nonstandard Analysis

"Suitable for a graduate course . . . could be covered in an advanced undergraduate course . . . The author’s ideas on how to achieve both intelligibility and rigor . . . will be useful reading for anyone intending to teach nonstandard analysis."—AMERICAN MATHEMATICAL SOCIETY

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Product Details

  • ISBN-13: 9780387984643
  • Publisher: Springer New York
  • Publication date: 10/1/1998
  • Series: Graduate Texts in Mathematics Series, #188
  • Edition description: 1998
  • Edition number: 1
  • Pages: 293
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.81 (d)

Table of Contents

What are the Hyperreals?- Large Sets.- Ultrapower Construction of the Hyperreals.- The Transfer Principle.- Hyperreals Great and Small.- Convergence of Sequences and Series.- Continuous Functions.- Differentiation.- The Riemann Integral.- Topology of the Reals.- Internal and External Sets.- Internal Functions and Hyperfinite Sets.- Universes and Frameworks.- The Existence of Nonstandard Entities.- Permanence, Comprehensiveness, Saturation.- Loeb Measure.- Ramsey Theory.- Completion by Enlargement.- Hyperfinite Approximation.- Books on Nonstandard Analysis.

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  • Anonymous

    Posted March 10, 2000

    I like it so far

    I've read through the first three chapters. So far the text is very good. He motivates the use of the ultrapower construction of the hyperreals very well. There is a minimum of logic and the compactness theorem is avoided (as far as I can tell); which is good for people like me who don't understand compactness very well. But having an inro to logic and set theory has been helpful when the author mentions Zorn's lemma and formulas.

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