Basic Set Theory (Student Mathematical Library, V. 17) by S. Shen (PDF)

Description

The main notions of set theory (cardinals, ordinals, transfinite induction) are fundamental to all mathematicians, not only to those who specialize in mathematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own leisurely treatment.

This book provides just that: a leisurely exposition for a diversified audience. It is suitable for a broad range of readers, from undergraduate students to professional mathematicians who want to finally find out what transfinite induction is and why it is always replaced by Zorn’s Lemma.

The text introduces all main subjects of “naive” (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor’s diagonal method, Zorn’s Lemma, Zermelo’s Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.

User’s Reviews

Editorial Reviews: Review “Lovely little book … does a truly marvelous job in covering what every one in the game should know, whether he be an analyst, geometer, algebraist or number theorist–or anything else, for that matter. It’s all there, from Cantor’s theory of cardinals to transfinite induction, from Zermelo to Zorn … it is a terrific book and does everything right: its selection of topics is not only logical, it is elegant, and the coverage is superb … The problems are very nice: interesting and non-trivial … and they supplement the main body of the text very well … the book is a pedagogical marvel … would be perfect for self-study … would also be a marvelous experience … to use the book in a first course on set theory … a very nice bit of work … I very recently used the book’s proof of the existence of a Hamel basis for any vector space in my course on Advanced Linear Algebra. It is an extremely slick and quick argument … And the discussion given in the book is typical of the entire book: to the point, elegant, and complete … I highly recommend this book … it covers the basic set-theoretic tool-kit every mathematician should carry around at all times, and does so with style. And then there are all the beautiful applications, challenging and elegant problems, and even a lot of surprises.” —- MAA Online”Well-written with excellent exercises both elementary and advanced … It would serve nicely either as a text or as independent reading.” —- Mathematical Reviews

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

⭐The book offers problems to solve but no answers.

⭐This book is excellent for self study and a very good introduction to the topic of non-axiomatic set theory. The book covers a large amount of material for its size, and manages to present it in a well structured and coherent fashio.n The problems it includes are not merely trivial exercises. They are sometimes hard, but they provide good insights to the material.

⭐The actual corpus occupies 108 pages of the text. Problems are scattered throughout at comfortable intervals, as they appear when the particular problem has been motivated by other results in the exposition. That is, proofs are provided for theorems (of which there are 47,) but results that would qualify only as propositions, lemmas, and corollaries are left as exercises for the motivated reader. Perhaps too scant for a standard, expository course, but a fantastic resource for self-study or a Moore-type course.

⭐This math text seems almost intentionally obscure, as if the writers take pride in their knowledge of the esoteric and wish to impress the reader with their mastery of abstractions.It omits motivation of the content it presents; instead it simply presents axioms, then proves theorems. I’d like to know *why* anybody ever concocted the theorems. What made them think, “this theorem would be neato if it were true, so I’ll try to prove it”? What is the thought process?I also read Halmos’ “Naïve Set Theory”. It tried to be approachable but fails.I also have Schaum’s Set Theory book to work through exercises, but of course those books are meant to be companions to proper textbooks and courses.Bill Shillito produced a series of lectures, “Introduction to Higher Mathematics”, which is the most accessible presentation of this kind of material I have seen. See the YouTube video CMWFmjlB8v0. They’re a great start, but lack sufficient depth to master the subject (which is fine; lectures should accompany textbooks, and students need to work through exercises to digest the material). I could only imagine that if somebody with Shillito’s values wrote a textbook, it would be the one I’d want to read. Bill Shillito is like the Bill Nye or Mister Wizard of higher mathematics. I hope he writes a textbook to accompany his videos.

⭐The low review is completely mistaken. Ideas are, in fact, presented as clearly as I’ve seen.

⭐Untypically for a book on set theory, this one makes for delightful reading. With excellent common sense discussions and proofs and a light touch this book sheds much light on topics even after one has waded through heavier texts.Highly recommended.

⭐This book is amazing! very accurate, short and dense!Fully recomended to every single mathematician in the planet.

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