The Real Numbers

An Introduction to Set Theory and Analysis

Nonfiction, Science & Nature, Mathematics, Mathematical Analysis, Logic
Cover of the book The Real Numbers by John Stillwell, Springer International Publishing
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: John Stillwell ISBN: 9783319015774
Publisher: Springer International Publishing Publication: October 16, 2013
Imprint: Springer Language: English
Author: John Stillwell
ISBN: 9783319015774
Publisher: Springer International Publishing
Publication: October 16, 2013
Imprint: Springer
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

More books from Springer International Publishing

Cover of the book Interactive Mobile Communication Technologies and Learning by John Stillwell
Cover of the book Solar Light Harvesting with Nanocrystalline Semiconductors by John Stillwell
Cover of the book Shale Gas: Ecology, Politics, Economy by John Stillwell
Cover of the book Augmented Reality by John Stillwell
Cover of the book HCI International 2015 - Posters’ Extended Abstracts by John Stillwell
Cover of the book KI 2016: Advances in Artificial Intelligence by John Stillwell
Cover of the book FinTech Revolution by John Stillwell
Cover of the book Complex Networks and Their Applications VII by John Stillwell
Cover of the book Efflux-Mediated Antimicrobial Resistance in Bacteria by John Stillwell
Cover of the book Demographic Dividends: Emerging Challenges and Policy Implications by John Stillwell
Cover of the book Cartilage Regeneration by John Stillwell
Cover of the book The Laplace Equation by John Stillwell
Cover of the book CSR in Private Enterprises in Developing Countries by John Stillwell
Cover of the book Medical Writing and Research Methodology for the Orthopaedic Surgeon by John Stillwell
Cover of the book Business Governance and Society by John Stillwell
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy