Intuitionistic Logic Model Theory and Forcing (eBook)

Front Cover 1
Intuitionistic Logic Model Theory and Forcing 4
Copyright Page 5
Contents 6
Acknowledgments 12
Introduction 14
PART I: LOGIC 18
Chapter 1. Propositional intuitionistic logic. semantics 20
§ 1. Formulas 20
§ 2. Models and validity 21
§ 3. Motivation 22
§ 4. Some properties of models 22
§ 5. Algebraic models 24
§ 6. Equivalence of algebraic and Kripke validity 24
Chapter 2. Propositional intuitionistic logic. proof theory 29
§ 1. Beth tableaus 29
§ 2. Correctness of Beth tableaus 31
§ 3. Hintikka collections 32
§ 4. Completeness of Beth tableaus 34
§ 5. Examples 36
Chapter 3. Related systems of logic 39
§ 1. ƒ-primitive intuitionistic logic. semantics 39
§ 2. ƒ-primitive intuitionistic logic. proof theory 40
§ 3. Minimal logic 41
§ 4. Classical logic 41
§ 5. Modal logic, S4 semantics
§ 6. Modal logic. S4 proof theory
§ 7. S4 and intuitionistic logic 44
Chapter 4. First order intuitionistic logic. semantics 46
§ 1. Formulas 46
§ 2. Models and validity 47
§ 3. Motivation 47
§ 4. Some properties of models 48
§ 5. Examples 49
§ 6. Truth and almost-truth sets 51
§ 7. Complete sequences 51
§ 8. A connection with classical logic 52
Chapter 5. First order intuitionistic logic. proof theory 54
§ 1. Beth tableaus 54
§ 2. Correctness of Beth tableaus 55
§ 3. Hintikka collections 57
§ 4. Hintikka elements 58
§ 5. Completeness of Beth tableaus 60
§ 6. Second completeness proof for Beth tableaus 61
§ 7. An axiom system, A1 64
§ 8. A second axiom system. A2 66
§ 9. Correctness of the system A2 67
§ 10. Completeness of the system A1 68
Chapter 6. Additional fist order results 72
§ 1. Compactness 72
§ 2. Concerning the excluded middle law 75
§ 3. Skolem–Löwenheim 76
§ 4. Kleene tableaus 77
§ 5. Craig interpolation lemma 79
§ 6. Models with constant B function 85
PART II: SET THEORY 90
Chapter 7. Intuitionistic Ma generalizations 92
§ 1. Introduction 92
§ 2. The classical Ma sequence 94
§ 3. The intuitionistic Ma sequence 95
§ 4. Dominance 97
§ 5. A little about equality 98
§ 6. Weak substitutivity of equality 99
§ 7. More on dominance 101
§ 8. Axiom of extensionality 102
§ 9. Null set axiom 103
§ 10. Unordered pairs axiom 103
§ 11. Union axiom 104
§ 12. Axiom of infinity 105
§ 13. Axiom of regularity 107
§ 14. Definability of the models 108
§ 15. Power set axiom 109
§ 16. X-equivalence 112
§ 17. Axiom of substitution 114
Chapter 8. Independence of the axiom of choice 117
§ 1. The specific model 117
§ 2. Symmetries 119
§ 3. Functions 120
§ 4. Axiom of choice 120
Chapter 9. Ordinals and cardinals 123
§ 1. Definitions 123
§ 2. Some properties of ordinals 123
§ 3. General ordinal representatives 124
§ 4. Canonical ordinal representatives 126
§ 5. Ordinalized models 126
§ 6. Properties of ordinal representatives 129
§ 7. Types of ordinals 129
§ 8. Cardinalized models 130
§ 9. Countably incompatible G 131
Chapter 10. Independence of the continuum hypothesis 134
§ 1. The specific model 134
§ 2. Countable incompatibility of W 135
§ 3. Cardinals and W 136
§ 4. Continuum hypothesis 137
Chapter 11. Definability and constructability 138
§ 1.Definitions 138
§ 2. Adequacy of the definability formula 140
§ 3. .-dominance 141
§ 4. The Ma sequence 142
§ 5 . Representatives of constructable sets 144
§ 6. Properties of constructable set representatives 145
§ 7. The principal result 148
Chapter 12. Independence of the axiom of constructability 151
§ 1. The specific model 151
§ 2. Axiom of constructability 152
Chapter 13. Additional results 153
§ 1. La representatives 153
§ 2. Definition functions 154
§ 3. Restriction on ordinals representable 155
§ 4. A classical connection 156
§ 5. Sets which are models 157
§ 6. Restriction on cardinals representable 158
§ 7. Axiom of choice 159
§ 8. Continuum hypothesis 160
§ 9. Classical counter models 162
Chapter 14. Additional classical model generalizations 164
§ 1. Introduction 164
§ 2. Boolean valued logics 164
§ 3. Boolean valued Ra generalizations 165
§ 4. Intuitionistic Ra generalizations 166
§ 5. < G, R , ., Rg>
§ 6. Equivalence of the Ra generalizations 174
§ 7. Boolean valued Ma generalizations 176
§ 8. Equivalence of the Ma generalizations 178
Appendix. to ch. 11 § 2 180
§ 1. Corresponding formulas 180
§ 2. Completeness of the definability formula 186
§ 3. Adequacy of the definability formula 187
Bibliography 189
Subject Index 191