Sommaruga Foundational Theories of Classical and Constructive Mathematics

1. Auflage 2011
ISBN: 978-94-007-0431-2
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 316 Seiten

The book 'Foundational Theories of Classical and Constructive Mathematics' is a book on the classical topic of foundations of mathematics. Its originality resides mainly in its treating at the same time foundations of classical and foundations of constructive mathematics. This confrontation of two kinds of foundations contributes to answering questions such as: Are foundations/foundational theories of classical mathematics of a different nature compared to those of constructive mathematics? Do they play the same role for the resp. mathematics? Are there connections between the two kinds of foundational theories? etc. The confrontation and comparison is often implicit and sometimes explicit. Its great advantage is to extend the traditional discussion of the foundations of mathematics and to render it at the same time more subtle and more differentiated. Another important aspect of the book is that some of its contributions are of a more philosophical, others of a more technical nature. This double face is emphasized, since foundations of mathematics is an eminent topic in the philosophy of mathematics: hence both sides of this discipline ought to be and are being paid due to.

Autoren/Hrsg.

Sommaruga, Giovanni

Inhaltsverzeichnis

1;Preface;6
2;Contents;7
3;Introduction;12
3.1;References;60
4;Part I Senses of `Foundations of Mathematics';61
4.1;Foundational Frameworks;62
4.1.1;1 Introduction: Questions of Justification and Rational Reconstruction (Between Hermeneutics and Cultural Revolution);62
4.1.2;2 Desiderata;65
4.1.3;3 Implications: Set Theory and Category Theory;66
4.1.4;4 Modal-Structural Mathematics and Foundations;72
4.1.5;References;77
4.2;The Problem of Mathematical Objects;79
4.2.1;1 Parsons on Mathematical Intuition;80
4.2.1.1;1.1 Intuition of and Intuition That;80
4.2.1.2;1.2 Pure Abstract and Quasi-concrete Objects;80
4.2.1.3;1.3 The Language of Stroke Strings;81
4.2.2;2 Frege's Proof;83
4.2.3;3 Dummett's Objections;84
4.2.4;4 Dummett's Objection Refurbished;88
4.2.5;References;92
4.3;Set Theory as a Foundation;93
4.3.1;References;103
4.4;Foundations: Structures, Sets, and Categories;105
4.4.1;1 Ontology, Maybe Even Metaphysics;105
4.4.2;2 Epistemology: What We Know and How We (Can) Know;109
4.4.3;3 Organizing Things;113
4.4.4;References;118
5;Part II Foundations of Classical Mathematics;119
5.1;From Sets to Types, to Categories, to Sets;120
5.1.1;1 Sets to Types;120
5.1.1.1;1.1 IHOL;121
5.1.1.2;1.2 Semantics;122
5.1.2;2 Types to Categories;123
5.1.2.1;2.1 Topoi;124
5.1.2.2;2.2 Syntactic Topos;125
5.1.3;3 Categories to Sets;126
5.1.3.1;3.1 Category of Ideals;127
5.1.3.2;3.2 Basic Intuitionistic Set Theory;127
5.1.4;4 Composites;129
5.1.4.1;4.1 Sets to Categories;129
5.1.4.2;4.2 Types to Sets;129
5.1.4.3;4.3 Categories to Types;130
5.1.5;5 Conclusions;130
5.1.6;References;132
5.2;Enriched Stratified Systems for the Foundations of Category Theory;133
5.2.1;1 Introduction;133
5.2.2;2 What the Various Proposals Do and Don't Do;134
5.2.3;3 The System NFU With Stratified Pairing;136
5.2.4;4 First-Order Structures in NFUP;138
5.2.5;5 Meeting Requirements (R1) and (R2) in NFUP;140
5.2.6;6 The Requirement (R3); Type-Shifting Problems in NFUP;141
5.2.7;7 The Requirement (R3), Continued; Building in ZFC;143
5.2.8;8 Cantorian Classes and Extension of NFU in ZFC;145
5.2.9;References;148
5.3;Recent Debate over Categorical Foundations;150
5.3.1;1 The Founding Ideas;151
5.3.2;2 Feferman and Rao;155
5.3.3;3 The Differences;156
5.3.4;References;158
6;Part III Between Foundations of Classical and Foundations of Constructive Mathematics;160
6.1;The Axiom of Choice in the Foundations of Mathematics;161
6.1.1;References;172
6.2;Reflections on the Categorical Foundations of Mathematics;174
6.2.1;1 Introduction;174
6.2.2;2 Type Theory;175
6.2.3;3 Elementary Toposes;176
6.2.4;4 Comparing Type Theories and Toposes;177
6.2.5;5 Models and Completeness;178
6.2.6;6 Gödel's Incompleteness Theorem;180
6.2.7;7 Reconciling Foundations;182
6.2.7.1;7.1 Constructive Nominalism;182
6.2.7.2;7.2 What Is the Category of Sets?;183
6.2.8;8 What Is Truth?;184
6.2.9;9 Continuously Variable Sets;185
6.2.10;10 Some Intuitionistic Principles;186
6.2.11;11 Concluding Remarks;187
6.2.12;References;188
7;Part IV Foundations of Constructive Mathematics;190
7.1;Local Constructive Set Theory and Inductive Definitions;191
7.1.1;1 Introduction;191
7.1.2;2 Inductive Definitions in CST;194
7.1.2.1;2.1 Inductive Definitions in CZF;194
7.1.2.2;2.2 Inductive Definitions in CZF+;197
7.1.3;3 The Free Version of CST;198
7.1.3.1;3.1 A Free Logic;198
7.1.3.2;3.2 The Axiom System CZFf;198
7.1.3.3;3.3 The Axiom Systems CZFf-, CZFfI and CZFf*;201
7.1.4;4 Local Intuitionistic Zermelo Set Theory;202
7.1.5;5 Some Axiom Systems for Local CST;204
7.1.5.1;5.1 Many-Sorted Free Logic;204
7.1.5.2;5.2 The Axiom System LCZFf-;205
7.1.5.3;5.3 The Axiom System LCZFfI;206
7.1.5.4;5.4 The Axiom System LCZFf*;206
7.1.6;6 Well-Founded Trees in Local CST;207
7.1.7;References;209
7.2;Proofs and Constructions;210
7.2.1;1 Preamble;210
7.2.2;2 Brouwer, Hilbert and Mathematical Practice;210
7.2.3;3 Internal and External Negations;213
7.2.4;4 There Is Only One Negation;214
7.2.5;5 Intuitionism and Meaning;217
7.2.6;6 Fatally Weak Counterexamples;217
7.2.7;7 Proofs and Constructions;220
7.2.8;8 A Realizability Theory of Constructions;222
7.2.9;References;226
7.3;Euclidean Arithmetic: The Finitary Theory of Finite Sets;227
7.3.1;1 The Sorites Fallacy;227
7.3.2;2 The Ancient Concept of Number;228
7.3.3;3 Euclidean Arithmetic;229
7.3.4;4 Induction and Recursion;232
7.3.5;5 Arithmetical Functions and Relations;234
7.3.6;6 Natural Number Systems;235
7.3.7;7 Binary Expansions;240
7.3.8;8 Conclusions;241
7.3.9;References;242
7.4;Intentionality, Intuition, and Proof in Mathematics;244
7.4.1;1 Intentionality;245
7.4.2;2 Intuition as Fulfillment of Meaning-Intention;246
7.4.3;3 A General Conception of Proofs as Fulfillments of Mathematical Meaning-Intentions;247
7.4.4;4 Proofs and Purely Formal Proofs;249
7.4.5;5 Proofs, Practice, and Axioms;252
7.4.6;6 Frustrated Meaning-Intentions;254
7.4.7;7 Multiple Proofs for the Same Meaning-Intention;255
7.4.8;8 Proofs That Exceed Meaning-Intention, and Mismatches Between Proofs and Meaning-Intentions;256
7.4.9;9 Internal and External Proofs for Meaning-Intentions;256
7.4.10;10 Mistaken Proofs (Intuitions);258
7.4.11;11 Constructive Proof;259
7.4.12;12 Conclusion;261
7.4.13;References;261
7.5;Foundations for Computable Topology;263
7.5.1;1 Foundations for Mathematics;264
7.5.2;2 Category Theory and Type Theory;267
7.5.3;3 Method and Critique;274
7.5.4;4 Stone Duality;279
7.5.5;5 Always Topologize;282
7.5.6;6 The Monadic Framework;287
7.5.7;7 The Sierpinski Space;293
7.5.8;8 Topology Using the Phoa Principle;296
7.5.9;9 Conclusion;303
7.5.10;References;305
7.6;Conclusion: A Perspective on Future Research in FOM;309
7.6.1;References;312