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Softcover ISBN:  9780821844533 
Product Code:  ULECT/44 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
eBook ISBN:  9781470421885 
Product Code:  ULECT/44.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Softcover ISBN:  9780821844533 
eBook ISBN:  9781470421885 
Product Code:  ULECT/44.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $107.20 $81.20 

Book DetailsUniversity Lecture SeriesVolume: 44; 2008; 240 ppMSC: Primary 03
Over the last 20 years, the theory of Borel equivalence relations and related topics have been very active areas of research in set theory and have important interactions with other fields of mathematics, like ergodic theory and topological dynamics, group theory, combinatorics, functional analysis, and model theory. The book presents, for the first time in mathematical literature, all major aspects of this theory and its applications.
This book should be of interest to a wide spectrum of mathematicians working in set theory as well as the other areas mentioned. It provides a systematic exposition of results that so far have been only available in journals or are even unpublished. The book presents unified and in some cases significantly streamlined proofs of several difficult results, especially dichotomy theorems. It has rather minimal overlap with other books published in this subject.
ReadershipGraduate students and research mathematicians interested in logic, set theory, and applications.

Table of Contents

Chapters

Introduction

Chapter 1. Descriptive set theoretic background

Chapter 2. Some theorems of descriptive set theory

Chapter 3. Borel ideals

Chapter 4. Introduction to equivalence relations

Chapter 5. Borel reducibility of equivalence relations

Chapter 6. “Elementary” results

Chapter 7. Introduction to countable equivalence relations

Chapter 8. Hyperfinite equivalence relations

Chapter 9. More on countable equivalence relations

Chapter 10. The 1st and 2nd dichotomy theorems

Chapter 11. Ideal $\mathcal {I}_1$ and the equivalence relation $\mathsf {E}_1$

Chapter 12. Actions of the infinite symmetric group

Chapter 13. Turbulent group actions

Chapter 14. The ideal $\mathcal {I}_3$ and the equivalence relation $\mathsf {E}_3$

Chapter 15. Summable equivalence relations

Chapter 16. $\mathsf {c}_0$equalities

Chapter 17. Pinned equivalence relations

Chapter 18. Reduction of Borel equivalence relations to Borel ideals

Appendix A. On Cohen and Gandy–Harrington forcing over countable models


Additional Material

Reviews

The book is rather selfcontained, starting with a quick but detailed presentation of basic results from descriptive set theory needed in the sequel, and ending with a brief addendum on forcing, and more particularly on Gandy–Harrington forcing, which is extensively used in the proofs.
Mathematical Reviews


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Over the last 20 years, the theory of Borel equivalence relations and related topics have been very active areas of research in set theory and have important interactions with other fields of mathematics, like ergodic theory and topological dynamics, group theory, combinatorics, functional analysis, and model theory. The book presents, for the first time in mathematical literature, all major aspects of this theory and its applications.
This book should be of interest to a wide spectrum of mathematicians working in set theory as well as the other areas mentioned. It provides a systematic exposition of results that so far have been only available in journals or are even unpublished. The book presents unified and in some cases significantly streamlined proofs of several difficult results, especially dichotomy theorems. It has rather minimal overlap with other books published in this subject.
Graduate students and research mathematicians interested in logic, set theory, and applications.

Chapters

Introduction

Chapter 1. Descriptive set theoretic background

Chapter 2. Some theorems of descriptive set theory

Chapter 3. Borel ideals

Chapter 4. Introduction to equivalence relations

Chapter 5. Borel reducibility of equivalence relations

Chapter 6. “Elementary” results

Chapter 7. Introduction to countable equivalence relations

Chapter 8. Hyperfinite equivalence relations

Chapter 9. More on countable equivalence relations

Chapter 10. The 1st and 2nd dichotomy theorems

Chapter 11. Ideal $\mathcal {I}_1$ and the equivalence relation $\mathsf {E}_1$

Chapter 12. Actions of the infinite symmetric group

Chapter 13. Turbulent group actions

Chapter 14. The ideal $\mathcal {I}_3$ and the equivalence relation $\mathsf {E}_3$

Chapter 15. Summable equivalence relations

Chapter 16. $\mathsf {c}_0$equalities

Chapter 17. Pinned equivalence relations

Chapter 18. Reduction of Borel equivalence relations to Borel ideals

Appendix A. On Cohen and Gandy–Harrington forcing over countable models

The book is rather selfcontained, starting with a quick but detailed presentation of basic results from descriptive set theory needed in the sequel, and ending with a brief addendum on forcing, and more particularly on Gandy–Harrington forcing, which is extensively used in the proofs.
Mathematical Reviews