Basic Set Theory

by Shen, S.; Vereshchagin, Nikolai Konstantinovich; Shen, A.

• ISBN13: 9780821827314
• ISBN10: 0821827316
• Format: Paperback
• Copyright: 2002-06-01
• Publisher: Amer Mathematical Society

Summary

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.

Table of Contents

Preface	vii
Sets and Their Cardinalities	1	(40)
Sets	1	(3)
Cardinality	4	(3)
Equal cardinalities	7	(2)
Countable sets	9	(7)
Cantor-Bernstein Theorem	16	(8)
Cantor's Theorem	24	(6)
Functions	30	(5)
Operations on cardinals	35	(6)
Ordered Sets	41	(68)
Equivalence relations and orderings	41	(6)
Isomorphisms	47	(5)
Well-founded orderings	52	(4)
Well-ordered sets	56	(3)
Transfinite induction	59	(7)
Zermelo's Theorem	66	(3)
Transfinite induction and Hamel basis	69	(5)
Zorn's Lemma and its application	74	(4)
Operations on cardinals revisited	78	(5)
Ordinals	83	(4)
Ordinal arithmetic	87	(4)
Recursive definitions and exponentiation	91	(8)
Application of ordinals	99	(10)
Bibliography	109	(2)
Glossary	111	(2)
Index	113