Set Theory

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This monograph covers the recent major advances in various areas of set theory.

From the reviews:

“One of the classical textbooks and reference books in set theory….The present ‘Third Millennium’ edition…is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master….This well-written book promises to influence the next generation of set theorists, much as its predecessor has done.” –MATHEMATICAL REVIEWS

Contents

Part I. Basic Set Theory

1. Axioms of Set Theory

2. Ordinal Numbers

3. Cardinal Numbers

4. Real Numbers

5. The Axiom of Choice and Cardinal Arithmetic

6. The Axiom of Regularity

7. Filters, Ultrafilters and Boolean Algebras

8. Stationary Sets

9. Combinatorial Set Theory

10. Measurable Cardinals

11. Borel and Analytic Sets

12. Models of Set Theory

Part II. Advanced Set Theory

13. Constructible Sets

14. Forcing

15. Applications of Forcing

16. Iterated Forcing and Martin’s Axiom

17. Large Cardinals

18. Large Cardinals and L

19. Iterated Ultrapowers and L[U]

20. Very Large Cardinals

21. Large Cardinals and Forcing

22. Saturated Ideals

23. The Nonstationary Ideal

24. The Singular Cardinal Problem

25. Descriptive Set Theory

26. The Real Line

Part III. Selected Topics

27. Combinatorial Principles in L

28. More Applications of Forcing

29. More Combinatorial Set Theory

30. Complete Boolean Algebras

31. Proper Forcing

32. More Descriptive Set Theory

33. Determinacy

34. Supercompact Cardinals and the Real Line

35. Inner Models for Large Cardinals

36. Forcing and Large Cardinals

37. Martin’s Maximum

38. More on Stationary Sets

Product description

From the reviews of the third edition:

“Thomas Jech’s text has long been considered a classic survey of the state of the set theory … . As every logician will know, this is a work of extraordinary scholarship, essential for any graduate logician who needs to know where the current boundaries of research are situated. Each chapter ends with a valuable historical survey and there is an extensive bibliography. This will continue to be the bible for set theorists in the new century.” (Gerry Leversha, The Mathematical Gazette, March, 2005)

“The book does masterly what it is supposed to do. … every mathematician who wishes to refresh his knowledge of set theory will read it with pleasure. … They will also find historical notes, and precise references … . A very comprehensive bibliography, and detailed indexes complete the work. This book fills a serious gap in the literature and there is no doubt that it will become a standard reference … . One can strongly recommend its acquisition for any mathematical library.” (Jean-Roger Roisin, Bulletin of the Belgian Mathematical Society, Vol. 11 (3), 2004)

“One of the classical textbooks and reference books in set theory is Jech’s Set Theory. … The present ‘Third Millennium’ edition … is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master. … This well-written book promises to influence the next generation of set theorists, much as its predecessor has done over the last quarter of a century.” (Eva Coplakova, Mathematical Reviews, 2004 g)

“Jech’s book, ‘Set Theory’ has been a standard reference for over 25 years. This ‘Third Millennium Edition’, not only includes all the materials in the first two editions, but also covers recent developments of set theory during the last 25 years. We believe that this new version will become a standard reference on set theory for the next few years.” (Guohua Wu, New Zealand Mathematical Society Newsletter, April, 2004)

“Jech’s classic monograph has been a standard reference for a generation of set theorists. Though … labeled ‘The Third Millennium Edition’, the present work is in fact a new book. … Even sections presenting older results have been rewritten and modernized. Exercises have been moved to the end of each section. The bibliography, the section on notation, and the index have been considerably expanded as well. This new edition will certainly become a standard reference on set theory for years to come.” (Jörg D. Brendle, Zentralblatt MATH, Vol. 1007, 2003)

“Thomas Jech’s Set Theory contains the most comprehensive treatment of the subject in any one volume. The present third edition is a revised and expanded version … . The third edition has three parts. The first, Jech says, every student of set theory should learn, the second every set theorist should master and the third consists of various results reflecting ‘the state of the art of set theory at the turn of the new millennium’. This last part especially contains a lot of new material.” (Martin Bunder, The Australian Mathematical Society Gazette, Vol. 30 (2), 2003)

About the Author

Thomas J. Jech is a mathematician specializing in set theory who was at Penn State for more than 25 years.