Basic Set Theory
About this Title
A. Shen, Independent University of Moscow, Russia and N. K. Vereshchagin, Moscow State Lomonosov University, Moscow, Russia. Translated by Alexander Shen, Independent University of Moscow, Russia
Publication: The Student Mathematical Library
Publication Year 2002: Volume 17
ISBNs: 978-0-8218-2731-4 (print); 978-1-4704-1822-9 (online)
MathSciNet review: MR1915128
MSC: Primary 03-01; Secondary 03E10, 03E25
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 dedicated treatment.
This book provides just that in the form of 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.
Advanced undergraduates, graduate students, and research mathematicians.
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