Foundational Theories of Classical and Constructive Mathematics (eBook)

Preface 6
Contents 7
Introduction 12
References 60
Part I Senses of `Foundations of Mathematics' 61
Foundational Frameworks 62
1 Introduction: Questions of Justification and Rational Reconstruction (Between Hermeneutics and Cultural Revolution) 62
2 Desiderata 65
3 Implications: Set Theory and Category Theory 66
4 Modal-Structural Mathematics and Foundations 72
References 77
The Problem of Mathematical Objects 79
1 Parsons on Mathematical Intuition 80
1.1 Intuition of and Intuition That 80
1.2 Pure Abstract and Quasi-concrete Objects 80
1.3 The Language of Stroke Strings 81
2 Frege's Proof 83
3 Dummett's Objections 84
4 Dummett's Objection Refurbished 88
References 92
Set Theory as a Foundation 93
References 103
Foundations: Structures, Sets, and Categories 105
1 Ontology, Maybe Even Metaphysics 105
2 Epistemology: What We Know and How We (Can) Know 109
3 Organizing Things 113
References 118
Part II Foundations of Classical Mathematics 119
From Sets to Types, to Categories, to Sets 120
1 Sets to Types 120
1.1 IHOL 121
1.2 Semantics 122
2 Types to Categories 123
2.1 Topoi 124
2.2 Syntactic Topos 125
3 Categories to Sets 126
3.1 Category of Ideals 127
3.2 Basic Intuitionistic Set Theory 127
4 Composites 129
4.1 Sets to Categories 129
4.2 Types to Sets 129
4.3 Categories to Types 130
5 Conclusions 130
References 132
Enriched Stratified Systems for the Foundations of Category Theory 133
1 Introduction 133
2 What the Various Proposals Do and Don't Do 134
3 The System NFU With Stratified Pairing 136
4 First-Order Structures in NFUP 138
5 Meeting Requirements (R1) and (R2) in NFUP 140
6 The Requirement (R3) Type-Shifting Problems in NFUP
7 The Requirement (R3), Continued Building in ZFC
8 Cantorian Classes and Extension of NFU in ZFC 145
References 148
Recent Debate over Categorical Foundations 150
1 The Founding Ideas 151
2 Feferman and Rao 155
3 The Differences 156
References 158
Part III Between Foundations of Classical and Foundations of Constructive Mathematics 160
The Axiom of Choice in the Foundations of Mathematics 161
References 172
Reflections on the Categorical Foundations of Mathematics 174
1 Introduction 174
2 Type Theory 175
3 Elementary Toposes 176
4 Comparing Type Theories and Toposes 177
5 Models and Completeness 178
6 Gödel's Incompleteness Theorem 180
7 Reconciling Foundations 182
7.1 Constructive Nominalism 182
7.2 What Is the Category of Sets? 183
8 What Is Truth? 184
9 Continuously Variable Sets 185
10 Some Intuitionistic Principles 186
11 Concluding Remarks 187
References 188
Part IV Foundations of Constructive Mathematics 190
Local Constructive Set Theory and Inductive Definitions 191
1 Introduction 191
2 Inductive Definitions in CST 194
2.1 Inductive Definitions in CZF 194
2.2 Inductive Definitions in CZF+ 197
3 The Free Version of CST 198
3.1 A Free Logic 198
3.2 The Axiom System CZFf 198
3.3 The Axiom Systems CZFf-, CZFfI and CZFf* 201
4 Local Intuitionistic Zermelo Set Theory 202
5 Some Axiom Systems for Local CST 204
5.1 Many-Sorted Free Logic 204
5.2 The Axiom System LCZFf- 205
5.3 The Axiom System LCZFfI 206
5.4 The Axiom System LCZFf* 206
6 Well-Founded Trees in Local CST 207
References 209
Proofs and Constructions 210
1 Preamble 210
2 Brouwer, Hilbert and Mathematical Practice 210
3 Internal and External Negations 213
4 There Is Only One Negation 214
5 Intuitionism and Meaning 217
6 Fatally Weak Counterexamples 217
7 Proofs and Constructions 220
8 A Realizability Theory of Constructions 222
References 226
Euclidean Arithmetic: The Finitary Theory of Finite Sets 227
1 The Sorites Fallacy 227
2 The Ancient Concept of Number 228
3 Euclidean Arithmetic 229
4 Induction and Recursion 232
5 Arithmetical Functions and Relations 234
6 Natural Number Systems 235
7 Binary Expansions 240
8 Conclusions 241
References 242
Intentionality, Intuition, and Proof in Mathematics 244
1 Intentionality 245
2 Intuition as Fulfillment of Meaning-Intention 246
3 A General Conception of Proofs as Fulfillments of Mathematical Meaning-Intentions 247
4 Proofs and Purely Formal Proofs 249
5 Proofs, Practice, and Axioms 252
6 Frustrated Meaning-Intentions 254
7 Multiple Proofs for the Same Meaning-Intention 255
8 Proofs That Exceed Meaning-Intention, and Mismatches Between Proofs and Meaning-Intentions 256
9 Internal and External Proofs for Meaning-Intentions 256
10 Mistaken Proofs (Intuitions) 258
11 Constructive Proof 259
12 Conclusion 261
References 261
Foundations for Computable Topology 263
1 Foundations for Mathematics 264
2 Category Theory and Type Theory 267
3 Method and Critique 274
4 Stone Duality 279
5 Always Topologize 282
6 The Monadic Framework 287
7 The Sierpinski Space 293
8 Topology Using the Phoa Principle 296
9 Conclusion 303
References 305
Conclusion: A Perspective on Future Research in FOM 309
References 312