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. <P>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. <P>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.
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Editorial Reviews
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"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
"Well-written with excellent exercises both elementary and advanced ... It would serve nicely either as a text or as independent reading." ---- Mathematical Reviews
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Most Helpful Customer Reviews
2 of 2 people found the following review helpful:
5.0 out of 5 stars
Excellent book covering the basics of non axiomatic set theory,
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This review is from: Basic Set Theory (Student Mathematical Library, V. 17) (Paperback)
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.
4 of 5 people found the following review helpful:
5.0 out of 5 stars
Excellent,
By A reader (South-Africa) - See all my reviews
This review is from: Basic Set Theory (Student Mathematical Library, V. 17) (Paperback)
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.
1 of 4 people found the following review helpful:
5.0 out of 5 stars
basic set theory,
This review is from: Basic Set Theory (Student Mathematical Library, V. 17) (Paperback)
This book is amazing! very accurate, short and dense!
Fully recomended to every single mathematician in the planet.
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Inside This Book
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First Sentence:
Let us recall several operations on sets and notation for them: A set consists of elements. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
continuum cardinality, same cardinality, sound fragment, smaller ordinals, proper initial segment, smaller cardinality, transfinite recursion, unused element, axiomatic set theory, countable ordinals, transfinite induction, linearly ordered set Key Phrases - Capitalized Phrases (CAPs): (learn more)
Cantor-Bernstein Theorem, Zorn's Lemma, Zermelo's Theorem, Cantor's Theorem, Koenig's Theorem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Let us recall several operations on sets and notation for them: A set consists of elements. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
continuum cardinality, same cardinality, sound fragment, smaller ordinals, proper initial segment, smaller cardinality, transfinite recursion, unused element, axiomatic set theory, countable ordinals, transfinite induction, linearly ordered set Key Phrases - Capitalized Phrases (CAPs): (learn more)
Cantor-Bernstein Theorem, Zorn's Lemma, Zermelo's Theorem, Cantor's Theorem, Koenig's Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
2
books
cite this book:
- Ordered Sets (Advances in Mathematics) by Egbert Harzheim in Back Matter
- A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 3) by Toshikazu Sunada Koju Shiga on page 21
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