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Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician
Winfried Just, Ohio University, Athens, and Martin Weese, University of Potsdam, Germany
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Graduate Studies in Mathematics
1997; 224 pp; hardcover
Volume: 18
ISBN-10: 0-8218-0528-2
ISBN-13: 978-0-8218-0528-2
List Price: US$45
Member Price: US$36
Order Code: GSM/18
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Descriptive Set Theory: Second Edition - Yiannis N Moschovakis

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.

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

Table of Contents

  • Filters and ideals in partial orders
  • Trees
  • A little Ramsey theory
  • The \(\Delta\)-system lemma
  • Applications of the continuum hypothesis
  • From the Rasiowa-Sikorski lemma to Martin's Axiom
  • Martin's Axiom
  • Hausdorff gaps
  • Closed unbounded sets and stationary sets
  • The \(\diamond\)-principle
  • Measurable cardinals
  • Elementary submodels
  • Boolean algebras
  • Appendix: Some general topology
  • Index
  • Index of symbols
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