The real numbers : an introduction to set theory and analysis / John Stillwell.
Stillwell, John, author.
2013
QA248 .S777 2013 (Mapit)
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Title The real numbers : an introduction to set theory and analysis / John Stillwell.
Author Stillwell, John, author.
ISBN 9783319015767 hardcover alkaline paper
3319015761 hardcover alkaline paper
Published Cham [Switzerland] ; New York : Springer, [2013]
Copyright ©2013
Language English
Description xvi, 244 pages : illustrations ; 25 cm.
Call Number QA248 .S777 2013
Summary 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.
Bibliography, etc. Note Includes bibliographical references (pages 225-229) and index.
Series Undergraduate texts in mathematics.
Record Appears in On-Campus Resources > Books
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The fundamental questions
From discrete to continuous
Infinite sets
Functions and limits
Open sets and continuity
Ordinals
The axiom of choice
Borel sets
Measure theory
Reflections.

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