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Lectures on the Hyperreals

An Introduction to Nonstandard Analysis

  • Textbook
  • © 1998

Overview

Authors:
  1. Robert Goldblatt

    1. School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 188)

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Table of contents (20 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Foundations

    1. Front Matter

      Pages 1-1
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    2. What Are the Hyperreals?

      • Robert Goldblatt
      Pages 3-14

    3. Large Sets

      • Robert Goldblatt
      Pages 15-21

    4. Ultrapower Construction of the Hyperreals

      • Robert Goldblatt
      Pages 23-33

    5. The Transfer Principle

      • Robert Goldblatt
      Pages 35-48

    6. Hyperreals Great and Small

      • Robert Goldblatt
      Pages 49-58

  3. Basic Analysis

    1. Front Matter

      Pages 59-59
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    2. Convergence of Sequences and Series

      • Robert Goldblatt
      Pages 61-73

    3. Continuous Functions

      • Robert Goldblatt
      Pages 75-90

    4. Differentiation

      • Robert Goldblatt
      Pages 91-104

    5. The Riemann Integral

      • Robert Goldblatt
      Pages 105-112

    6. Topology of the Reals

      • Robert Goldblatt
      Pages 113-122

  4. Internal and External Entities

    1. Front Matter

      Pages 123-123
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    2. Internal and External Sets

      • Robert Goldblatt
      Pages 125-145

    3. Internal Functions and Hyperfinite Sets

      • Robert Goldblatt
      Pages 147-154

  5. Nonstandard Frameworks

    1. Front Matter

      Pages 155-155
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    2. Universes and Frameworks

      • Robert Goldblatt
      Pages 157-181

    3. The Existence of Nonstandard Entities

      • Robert Goldblatt
      Pages 183-189

    4. Permanence, Comprehensiveness, Saturation

      • Robert Goldblatt
      Pages 191-199
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Keywords

About this book

There are good reasons to believe that nonstandard analysis, in some ver­ sion or other, will be the analysis of the future. KURT GODEL This book is a compilation and development of lecture notes written for a course on nonstandard analysis that I have now taught several times. Students taking the course have typically received previous introductions to standard real analysis and abstract algebra, but few have studied formal logic. Most of the notes have been used several times in class and revised in the light of that experience. The earlier chapters could be used as the basis of a course at the upper undergraduate level, but the work as a whole, including the later applications, may be more suited to a beginning graduate course. This prefacedescribes my motivationsand objectives in writingthe book. For the most part, these remarks are addressed to the potential instructor. Mathematical understanding develops by a mysterious interplay between intuitive insight and symbolic manipulation. Nonstandard analysis requires an enhanced sensitivity to the particular symbolic form that is used to ex­ press our intuitions, and so the subject poses some unique and challenging pedagogical issues. The most fundamental ofthese is how to turn the trans­ fer principle into a working tool of mathematical practice. I have found it vi Preface unproductive to try to give a proof of this principle by introducing the formal Tarskian semantics for first-order languages and working through the proofofLos's theorem.

Reviews

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

Authors and Affiliations

  • School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand

    Robert Goldblatt

Bibliographic Information

  • Book Title: Lectures on the Hyperreals

  • Book Subtitle: An Introduction to Nonstandard Analysis

  • Authors: Robert Goldblatt

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-0615-6

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1998

  • Hardcover ISBN: 978-0-387-98464-3Published: 01 October 1998

  • Softcover ISBN: 978-1-4612-6841-3Published: 03 October 2012

  • eBook ISBN: 978-1-4612-0615-6Published: 06 December 2012

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XIV, 293

  • Topics: Real Functions

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