This is the second volume of a two-volume graduate text in set theory. The first volume covered the basics of modern set theory and was addressed primarily to beginning graduate students. This second volume is intended as a bridge between introductory set theory courses and advanced monographs that cover selected branches of set theory, such as forcing or large cardinals. The authors give short but rigorous introductions to set-theoretic concepts and techniques such as trees, partition calculus, cardinal invariants of the continuum, Martin's Axiom, closed unbounded and stationary sets, the Diamond Principle ($\diamond$), and the use of elementary submodels. Great care has been taken to motivate the concepts and theorems presented. The book is written as a dialogue with the reader. The presentation is interspersed with numerous exercises. The authors wish to entice readers into active participation in discovering the mathematics presented, making the book particularly suitable for self-study. Each topic is presented rigorously and in considerable detail. Carefully planned exercises lead the reader to active mastery of the techniques presented. Suggestions for further reading are given. Volume II can be read independently of Volume I.
Author(s): Martin Weese Winfried Just
Series: Graduate Studies in Mathematics 18
Publisher: American Mathematical Society
Year: 1997
Language: English
Commentary: pp 210-211 (part of Appendix) missing.
Pages: 224
Contents......Page 6
Preface......Page 8
Notation......Page 10
13.1. The general concept of a filter......Page 14
13.2. Ultraproducts......Page 21
13.3. A first look at Boolean algebras......Page 25
Mathographical Remarks......Page 37
14 Trees......Page 40
Mathographical Remarks......Page 61
15 A Little Ramsey Theory......Page 62
Mathographical Remarks......Page 78
16 The Delta-System Lemma......Page 80
17.1. Applications to Lebesgue measure and Baire category......Page 84
17.2. Miscellaneous applications of CH......Page 92
Mathographical Remarks......Page 98
18 From the Rasiowa-Sikorski Lemma to Martin's Axiom......Page 100
Mathographical Remarks......Page 107
19.1. MA essentials......Page 108
19.2. MA and cardinal invariants of the continuum......Page 115
19.3. Ultrafilters on w......Page 123
Mathographical Remarks......Page 129
20 Hausdorff Gaps......Page 130
Mathographical Remarks......Page 135
21.1. Closed unbounded and stationary sets of ordinals......Page 136
21.2. Closed unbounded and stationary subsets of ........Page 144
22 The <>-principle......Page 152
Mathographical Remarks......Page 159
23 Measurable Cardinals......Page 160
Mathographical Remarks......Page 170
24.1. Elementary facts about elementary submodels......Page 172
24.2. Applications of elementary submodels in set theory......Page 180
Mathographical Remarks......Page 198
25 Boolean Algebras......Page 200
Mathographical Remark......Page 218
26 Appendix: Some General Topology......Page 220
Index......Page 230
Index of Symbols......Page 236
