Simplified Independence Proofs (eBook)

Front Cover 1
Simplified Independence Proofs: Boolean Valued Models of Set Theory 4
Copyright Page 5
Contents 10
Preface 6
Glossary of Symbols 13
Location of Theorems 14
Locations of Definations 16
Chapter 1. General Remarks 18
A. Background Materials 18
B. Assumptions 19
C. Summary of Results 19
D. A Paradigm for the Proofs 20
E. A Guide for the Casual Reader 24
Chapter 2. Boolean Algebra 29
A. Topological Spaces 29
B. Properties of Boolean Algebras 37
C. Automorphisms 44
D. The Countable Chain Condition 48
Chapter 3. The Basic Model 51
A. Axioms of the Restricted Predicate Calculus 51
B. Axioms of Equality and Extensionality 58
C. Methods for Defining Members of V 68
D. The Basic Set Theory Axioms 82
E. Ordinals and Cardinals in the Model 92
F. The Axiom of Choice 98
Chapter 4. The Independence of V = L 106
A. Preliminaries 106
B. Proof of GCH 107
C. Subsets of . 109
Chapter 5. Analogies with Forcing 116
A. Comparison of Specific Proofs 116
B. Replacing Boolean Algebra by Forcing in Proofs? 119
Chapter 6. The Independence of AxC 122
A. The Key Idea of the Proof 122
B. The Choice of G 123
C. Subsets of . 124
D. The Real Numbers Are Not Well-ordered 130
Chapter 7. The Independence of the Continuum Hypothesis 133
A. The Key Result 133
B. Additional Cardinality Results 136
Chapter 8. The Generalized GCH—The Bounded Case 144
A. Statement of Easton’s Theorem 144
B. Specification of the Boolean Algebra 148
C. Substitutes for the Cohen Combinatorial Lemma 149
D. Cardinality Relations 157
E. Proof of Easton’s Theorem 162
F. A Note on the Proof 172
Chapter 9. The Generalized GCH–The Unbounded Case 173
A. Preliminary Considerations 173
B. Specification of the Boolean Algebra 175
C. Definition of the Universe 182
D. Definition of the Boolean Value of a Statement 186
E. Proof of the Axioms of Set Theory, Except the Power Set Axiom 191
F. Cardinality Relations 199
G. The Axiom of the Power Set 199
H. Proof of Easton’s Theorem 218
Chapter 10. Resolution of Conceptual Difficulties 219
A. What Is Truth? 219
B. Appeal to Strong Axioms 225
Bibliography 231
Subject Index 232