# Lectures on the Hyperreals

## An Introduction to Nonstandard Analysis

Textbook

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

1. Front Matter
Pages i-xiv
2. ### Foundations

1. Front Matter
Pages 1-1
2. Robert Goldblatt
Pages 3-14
3. Robert Goldblatt
Pages 15-21
4. Robert Goldblatt
Pages 23-33
5. Robert Goldblatt
Pages 35-48
6. Robert Goldblatt
Pages 49-58
3. ### Basic Analysis

1. Front Matter
Pages 59-59
2. Robert Goldblatt
Pages 61-73
3. Robert Goldblatt
Pages 75-90
4. Robert Goldblatt
Pages 91-104
5. Robert Goldblatt
Pages 105-112
6. Robert Goldblatt
Pages 113-122
4. ### Internal and External Entities

1. Front Matter
Pages 123-123
2. Robert Goldblatt
Pages 125-145
3. Robert Goldblatt
Pages 147-154
5. ### Nonstandard Frameworks

1. Front Matter
Pages 155-155
2. Robert Goldblatt
Pages 157-181
3. Robert Goldblatt
Pages 183-189
4. Robert Goldblatt
Pages 191-199

### Introduction

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.

### Keywords

Boolean algebra Lebesgue measure Riemann integral calculus construction differential equation eXist functional measure measure theory proof real analysis tool topology ultrapower

#### Authors and affiliations

1. 1.School of Mathematical and Computing SciencesVictoria UniversityWellingtonNew Zealand

### 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
• Copyright Information Springer-Verlag New York, Inc. 1998
• Publisher Name Springer, New York, NY
• eBook Packages
• Hardcover ISBN 978-0-387-98464-3
• Softcover ISBN 978-1-4612-6841-3
• eBook ISBN 978-1-4612-0615-6
• Series ISSN 0072-5285
• Edition Number 1
• Number of Pages XIV, 293
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## 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