The Real Numbers

An Introduction to Set Theory and Analysis

• John Stillwell
Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xvi
2. John Stillwell
Pages 1-24
3. John Stillwell
Pages 25-55
4. John Stillwell
Pages 57-83
5. John Stillwell
Pages 85-110
6. John Stillwell
Pages 111-126
7. John Stillwell
Pages 127-147
8. John Stillwell
Pages 149-173
9. John Stillwell
Pages 175-191
10. John Stillwell
Pages 193-211
11. John Stillwell
Pages 213-223
12. Back Matter
Pages 225-244

Introduction

While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself.

By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics.

Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets,  countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

Keywords

Axiom of choice Continuous functions Infinity Measure theory Real numbers

Authors and affiliations

• John Stillwell
• 1
1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-01577-4
• Copyright Information Springer International Publishing Switzerland 2013
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-01576-7
• Online ISBN 978-3-319-01577-4
• Series Print ISSN 0172-6056
• Series Online ISSN 2197-5604
• Buy this book on publisher's site