The Real Numbers: An Introduction to Set Theory and Analysis (Undergraduate Texts in Mathematics) 2013th Edition by John Stillwell (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The devotee of Landau’s “Foundations of Analysis” should get a copy of Elliot Mendelson’s “Number Systems and the Foundations of Analysis”. After 1973, Landau is the stone age, a rigorous treatment using very bad notation. Now, nobody needs to be subjected to Landau’s treatment of natural numbers which is antique and mathematically poor by today’s standards. People interested in the natural numbers should consult Leon Henkin’s 1960 article “On mathematical induction”, American Mathematical Monthly, 67. Any college math library should have it. After reading Mendelson’s book, there are two excellent enrichment books. One is “Retracing Elementary Mathematics” by Leon Henkin and 3 others. The other is the book in question, John Stillwell’s “The Real Numbers, An Introduction to Set Theory and Analysis”. Everyone interested in arithmetic and analysis should read this book! It describes the historical sequence from ancient times of theoretical problems and how they were solved. One sees the real numbers from a new angle, one that is enlightening and charming. And two angles are far better than one, especially in mathematics. I cannot recommend this book too highly.

⭐Very well done. A good balance of a concise mathematical treatment combined with both philosophical and historical context, which makes it highly readable. Considering that in just over 200 pages it spans from “What Are Numbers?” to Dedekind cuts to cardinalities to Borel sets to ZF axioms to AC & CH and more, this book is more about the overall fabric that reaches across these concepts rather than an intense treatment of the details. This may leave some wanting more in various areas, but to do that in one book would turn this into an 800-page tome. This book actually delivers on the back cover description.

⭐Definitely, you get what a Dedekind Cut is visually that is why I purchased the book. Doing the problem exercises shall put the meat on the bones of the exposition to be a completely filled in body of work. This is a book of in need of Elementary Number Theory to go with it. I recommend Burton’s Elementary Number Theory to accompany this book then some other books to fill in between the lines because the author relies more on the exercise problems to fill in the details than had I wished for a more detailed exposition and more clarity, or more compacted amounts of information per page, then this is not the case here. There is space to be filled in here, on some pages, there are jumps that makes it awkward to see where the author is trying to say.

⭐This is Stillwells sequel to his book Roads to Infinity. It provides an in depth look at how the investigations into infinity has influenced developments in Set theory, the construction of the real numbers and analysis, the engine behind calculus.. it places mathematical thinking into a timeline context where one is able to see how the great mathematicians were influenced by the greatest problem in mathematics, modelling tbe continuum a problem that remains unsolved today. It has to be read with the understanding that the world of mathematics has moved on somewhat since this was written. However tbe material remains relevant today as it covers in detail the drive to understand the connection between the discrete nature of number and the concept of the continuous curve. Today much of the work done by Cauchy, and Dedekind to explain irrational numbers is being questioned with the ideas of computable numbers and computing science moving away from the classical understanding of the real numbers and axiomatix set theory as the basis of all mathematics in particular the failure of the Axiom of Choice to provide a basic arithmetic of real numbers. Nevertheless tnis is a must have read for all students who may have or are struggling with analysis at undergraduate level. It would be interesting to read an updated account of where we are now by this very skilled author.

⭐Excellent

⭐A good introduction to set theoretic numbers theory – and how to develop these through to the reals. Very interesting read

⭐A great re-introduction. Really gets to the heart of the matter in a way accessible to Graduate … students … or not students anymore, like me.

⭐Packing as well as covering of the book is well.

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