**Graduate Studies in Mathematics**

Volume: 18;
1997;
224 pp;
Hardcover

MSC: Primary 03;

Print ISBN: 978-0-8218-0528-2

Product Code: GSM/18

List Price: $48.00

Individual Member Price: $38.40

**Electronic ISBN: 978-1-4704-2075-8
Product Code: GSM/18.E**

List Price: $48.00

Individual Member Price: $38.40

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# Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician

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*Winfried Just; Martin Weese*

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.

#### Table of Contents

# Table of Contents

## Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Notation xi12 free
- Chapter 13. Filters and Ideals in Partial Orders 116 free
- Chapter 14. Trees 2742
- Chapter 15. A Little Ramsey Theory 4964
- Chapter 16. The Δ-System Lemma 6782
- Chapter 17. Applications of the Continuum Hypothesis 7186
- Chapter 18. From the Rasiowa-Sikorski Lemma to Martin's Axiom 87102
- Chapter 19. Martin's Axiom 95110
- Chapter 20. Hausdorff Gaps 117132
- Chapter 21. Closed Unbounded Sets and Stationary Sets 123138
- Chapter 22. The ◊-principle 139154
- Chapter 23. Measurable Cardinals 147162
- Chapter 24. Elementary Submodels 159174
- Chapter 25. Boolean Algebras 187202
- Chapter 26. Appendix: Some General Topology 207222
- Index 217232
- Index of Symbols 223238 free
- Back Cover Back Cover1240

#### Readership

Graduate students and research mathematicians interested in set theory; researchers who want to learn how to use set-theoretic tools, such as Martin's Axiom, the Diamond Principle, and closed unbounded and stationary sets.

#### Reviews

[The book is] thoughtfully written, and offers a large number of exercises. [It] can serve as an appetizer for many subfields of set theory; the authors not only give proofs of many of the classical theorems, but also do an excellent job of motivating theorems and definitions. The “mathographical remarks” at the end of each chapter contain points to textbooks, monographs and surveys, or sometimes even to research papers that are accessible to all who have read and understood [this book]. Through the exercises placed in the text, the authors have done an excellent job of synchronizing the students' thoughts with their own–after finishing an exercise the student is ready for the next definition, theorem, or example. [The book is] highly informative, a pleasure to read, and can be warmly recommended.

-- Journal of Symbolic Logic

As a text for a second semester in set theory, it is excellent …

-- Mathematical Reviews