Constructivism in Mathematics, Vol 1 (eBook)

Front Cover 1
Constructivism in Mathematics: An Introduction 4
Copyright Page 5
CONTENTS OF VOLUME I 20
Preface 8
Preliminaries 12
Chapter 1 Introduction 22
1. Constructivism 22
2. Constructivity 26
3. Weak counterexamples 29
4. A brief history of constructivism 37
5. Notes 51
Exercises 53
Chapter 2 Logic 56
1. Natural deduction 57
2. Logic with existence predicate 71
3. Relationships between classical and intuitionistic logic 77
4. Hilbert-type systems 89
5. Kripke semantics 96
6. Completeness for Kripke semantics 108
7. Definitional extensions 114
8. Notes 121
Exercises 124
Chapter 3 Arithmetic 134
1. Informal arithmetic and primitive recursive functions 135
2. Primitive recursive arithmetic PRA 141
3. Intuitionistic first-order arithmetic HA 147
4. Algorithms 152
5. Some metamathematics of HA 157
6. Elementary analysis and elementary inductive definitions 165
7. Formalization of elementary recursion theory 173
8. Intuitionistic second-order logic and arithmetic 181
9. Higher-order logic and arithmetic 189
10. Notes 195
Exercises 198
Chapter 4 Non-classical axioms 206
1. Preliminaries 207
2. Choice axioms 210
3. Church's thesis 213
4. Realizability 216
5. Markov's principle 224
6. Choice sequences and continuity axioms 227
7. The fan theorem 238
8. Bar induction and generalized inductive definitions 244
9. Uniformity principles and Kripke's schema 255
10. Notes 259
Exercises 263
Chapter 5 Real numbers 272
1. Introduction 272
2. Cauchy reals and their ordering 274
3. Arithmetic on R 281
4. Completeness properties and relativization 287
5. Dedekind reals 291
6. Arithmetic of Dedekind reals and extended reals 298
7. Two metamathematical digressions 301
8. Notes 305
Exercises 307
Chapter 6 Some elementary analysis 312
1. Intermediate-value and supremum theorems 312
2. Differentiation and integration 316
3. Consequences of WC-N and FAN 324
4. Analysis in CRM: consequences of CT0, ECT0, MP 328
5. Notes 341
Exercises 343
Bibliography 348
Index 364
List of symbols 374