# Set Theory

• Thomas Jech
Book

Part of the Perspectives in Mathematical Logic book series (PML)

1. Front Matter
Pages I-XIV
2. ### Sets

1. Thomas Jech
Pages 1-77
2. Thomas Jech
Pages 78-136
3. ### More Sets

1. Thomas Jech
Pages 137-215
2. Thomas Jech
Pages 216-294
4. ### Large Sets

1. Thomas Jech
Pages 295-397
2. Thomas Jech
Pages 398-491
5. ### Sets of Reals

1. Thomas Jech
Pages 493-578
6. Back Matter
Pages 579-634

### Introduction

The main body of this book consists of 106 numbered theorems and a dozen of examples of models of set theory. A large number of additional results is given in the exercises, which are scattered throughout the text. Most exer­ cises are provided with an outline of proof in square brackets [ ], and the more difficult ones are indicated by an asterisk. I am greatly indebted to all those mathematicians, too numerous to men­ tion by name, who in their letters, preprints, handwritten notes, lectures, seminars, and many conversations over the past decade shared with me their insight into this exciting subject. XI CONTENTS Preface xi PART I SETS Chapter 1 AXIOMATIC SET THEORY I. Axioms of Set Theory I 2. Ordinal Numbers 12 3. Cardinal Numbers 22 4. Real Numbers 29 5. The Axiom of Choice 38 6. Cardinal Arithmetic 42 7. Filters and Ideals. Closed Unbounded Sets 52 8. Singular Cardinals 61 9. The Axiom of Regularity 70 Appendix: Bernays-Godel Axiomatic Set Theory 76 Chapter 2 TRANSITIVE MODELS OF SET THEORY 10. Models of Set Theory 78 II. Transitive Models of ZF 87 12. Constructible Sets 99 13. Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis 108 14. The In Hierarchy of Classes, Relations, and Functions 114 15. Relative Constructibility and Ordinal Definability 126 PART II MORE SETS Chapter 3 FORCING AND GENERIC MODELS 16. Generic Models 137 17. Complete Boolean Algebras 144 18.

### Keywords

Cardinal number Mengenlehre cardinals combinatorics forcing grosse Kardinalzahlen large cardinals mathematics proof set theory

#### Authors and affiliations

• Thomas Jech
• 1
1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-22400-7
• Copyright Information Springer-Verlag Berlin Heidelberg 1997
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-662-22402-1
• Online ISBN 978-3-662-22400-7
• Series Print ISSN 0172-6641
• Buy this book on publisher's site
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