Set Theory

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. 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 exerƯ 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 menƯ 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 This Introduction To Modern Set Theory Covers All Aspects Of Its Two Main General Areas: Classical Set Theory Including Large Cardinals, Infinitary Combinatorics, Desriptive Set Theory, And Independence Proofs Starting With Goedel's Proof Around 1938 Followed By Cohen's Proof In 1963, Whereby Cohen's Method Of Forcing Probably Had A Greater Influence On Mathematics. The Author's Primary Emphasis Is On Forcing And Large Cardinals (on Which He Has Collected An Enormous Amount Of Material Which Had Previously Been Available Only Through Scattered Journal Articles Or Private Communication) But There Is A Substantial Discussion Of Descriptive Set Theory And Infinitary Combinatorics As Well. The Author's Presentation Is Very Well-organized And Carefully Worked Out And Has Become A Standard Reference. By Thomas Jech. Front Matter....Pages I-XIV Axiomatic Set Theory....Pages 1-77 Transitive Models of Set Theory....Pages 78-136 Forcing and Generic Models....Pages 137-215 Some Applications of Forcing....Pages 216-294 Measurable Cardinals....Pages 295-397 Other Large Cardinals....Pages 398-491 Descriptive Set Theory....Pages 493-578 Back Matter....Pages 579-634 I Sets 1 Axiomatic Set Theory 2 Transitive Models of Set Theory II More Sets 3 Forcing and Generic Models 4 Some Applications of Forcing III Large Sets 5 Measurable Cardinals 6 Other Large Cardinals IV Sets of Reals 7 Descriptive Set Theory Historical Notes and Guide to the Bibliography Notation Name Index List of Corrections.