Gödel's disjunction: the scope and limits of mathematical knowledge
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford University Press
2016
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| Edition: | First edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | x, 277 Seiten |
| ISBN: | 9780198759591 |
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CONTENTS
1 Introduction .
Leon Horsten and Philip Welch
1.1 Gödel’ s Disjunction and Beyond 1
1.1.1 The Disjunctive Thesis 2
1.1.2 The First Disjunct 2
1.1.3 The Second Disjunct 3
1.2 Formal Frameworks 4
1.2.1 Mathematical Philosophy to the Rescue? 4
1.2.2 Epistemic Mathematics 5
1.2.3 Computation and the Nature of Algorithm 6
1.3 Organisation of the Contributions 7
1.3.1 Algorithm; Consistency, and Epistemic Randomness 7
1.3.1.1 Dean 7
1.3.1.2 Visser 8
1.3.1.3 Moschovakis 9
1.3.1.4 Achourioti 9
1.3.2 Minds and Machines 10
1.3.2.1 Carlson 10
1.3.2.2 Koellner 10
1.3.2.3 Shapiro 12
1.3.3 Absolute Un decidability 12
1.3.3.1 Leach-Krouse 12
1.3.3.2 Williamson 13
1.3.3.3 Antonutti and Horsten 13
References 14
PART I ALGORITHM, CONSISTENCY; AND EPISTEMIC
RANDOMNESS
2 Algorithms and the Mathematical Foundations of Computer
Science.19
Walter Dean
2.1 Introduction 19
2.2 Motivating Algorithmic Realism 24
2.3 Algorithms in Theoretical Computer Science 27
2.4 In Search of a Foundational Framework 34
2.5 Procedural Equivalence 41
vi | CONTENTS
2.5.1 Simulation Equivalence 42
2.5.2 The Exigencies of Simulation 44
2.5.2.1 Formalizing the Transitional Condition 45
2.5.2.2 Formalizing the Representational Requirement 47
2.5.2.3 Implementing Recursion 48
2.6 Taking Stock 51
2.6.1 Moschovakis, Gurevich, and the Level-Relativity of Algorithms 51
2.6.2 Algorithms, Identity, and Mathematical Practice 54
Acknowledgement 57
Notes 57
References 63
3 The Second Incompleteness Theorem: Reflections and
Ruminations . . . • .67
Albert Visser
3.1 Introduction 67
3.1.1 Status of the Technical Results in this Chapter 68
3.2 Versions of the Second Incompleteness Theorem 68
3.2.1 A Basic Version of G2 69
3.2.2 G2 as a Statement of Interpretability Power 69
3.2.3 Feferman's Theorem 70
3.2.4 G2 as an Admissible Rule 70
3.3 Meaning as Conceptual Role 70
3.3.1 L-Predicates 71
3.3.2 On HBL-Predicates 74
3.3.3 Feferman on L-Predicates 78
3.3.4 Philosophical Discussion 79
3.4 Solution of the Meaning Problem 80
3.5 Abolishing Arbitrariness 81
3.5.1 Consistency Statements as Unique Solutions of Equations 82
3.5.2 Bounded Interpretations 83
Notes 85
References 86
Appendix A Basic Facts and Definitions 88
A.l Theories 88
A.2 Translations and Interpretations 88
A.3 Sequential Theories 90
A4 Complexity Measures 90
4 Iterated Definability, Lawless Sequences, and Brouwer's
Continuum.92
Joan Rand Moschovakis
4.1 Introduction 92
4.2 Choice Sequences 93
4.2.1 Brouwer s Continuum 93
CONTENTS
vii
4.2.2 The Problem of Defining “Definability” 93
4.2.3 “Lawlike” versus “Lawless” Sequences 94
4.3 The Formal Systems RLS(- ) and FIRM(- ) 94
4.3.1 The Three-Sorted Language £(^) 94
4.3.2 Axioms and Rules for Three-Sorted Intuitionistic
Predicate Logic 95
4.3.3 Axioms for Three-sorted Intuitionistic Number Theory 95
4.3.4 Lawless Sequences, Restricted Quantification,
and Lawlike Comprehension 96
4.3.5 Axioms for Lawless Sequences 96
4.3.6 Well-Ordering the Lawlike Sequences 97
4.3.7 Restricted LEM, the Axiom of Closed Data
and Lawlike Countable Choice 97
4.3.8 Brouwer's Bar Theorem and Troelstra's Generalized
Continuous Choice 98
4.3.9 Classical and Intuitionistic Analysis as Subsystems of FIRM( ~ ) 99
4.3.10 Consistency of FIRM(^) 99
4.4 Construction of the Classical Model and Proof of Theorem 1 100
4.4.1 Definability Over (A, - A) by a Restricted Formula of £(^ ) 100
4.4.2 The Classical Model A4(^n) 100
4.4.3 Outline of the Proof of Theorem 1 102
4.5 The T-Realizability Interpretation 103
4.5.1 Definitions 103
4.5.2 Outline of the Proof of Theorem 2 104
4.6 Epilogue 105
Acknowledgement 106
Notes 106
References 106
5 A Semantics for In-Principle Provability. 108
T. Achourioti
5.1 Introduction 108
5.2 In-Principle Provability and Intensionality 109
5.3 Modelling Epistemic Mathematics: A Theory of Descriptions 111
5.4 Intensional Truth 113
5.5 Towards Axioms for In-Principle Provability 115
5.6 Intensional Semantics for Tt is In-Principle Provable that' 117
5.6.1 Dynamical Proofs 118
5.6.2 Bringing Provability Back into ‘In-Principle Provability' 119
5.6.3 Band Theory T 122
5.7 Conclusion 123
Notes 124
References 125
viii | CONTENTS
PART II MIND AND MACHINES
6 Collapsing Knowledge and Epistemic Church's Thesis.129
Timothy J. Carlson
6.1 Introduction 129
6.2 Knowing Entities and Syntactic Encoding 133
6.3 Hierarchies and Stratification 134
6.4 Collapsing 136
6.5 A Computable Collapsing Relation 137
6.6 A Machine That Knows EA + ECT 138
6.7 Remarks 146
References 147
7 Godel's Disjunction.148
Peter Koellner
7.1 The Disjunction 150
7.1.1 Relative Provability and Truth 150
7.1.2 Absolute Provability 151
7.1.3 Idealized Finite Machines and Idealized Human Minds 153
7.1.4 Summary 154
7.2 Notation 155
7.3 Arithmetic 156
7.4 Incompleteness 156
7.5 Epistemic Arithmetic 157
7.6 Epistemic Arithmetic with Typed Truth 159
7.7 The Disjunction in EAr 160
7.8 The Classic Argument for the First Disjunct 162
7.9 The First Disjunct in EAt 163
7.10 Penrose's New Argument 164
7.11 Type-Free Truth 166
7.12 A Failed Attempt 167
7.13 The System DTK 169
7.14 Basic Results in DTK 170
7.15 The Disjunction in DTK 174
7.16 The Disjuncts in DTK 176
7.17 Conclusion 183
Acknowledgement 185
Notes 185
References 186
8 Idealization, Mechanism, and Knowability.189
Stewart Shapiro
8.1 Lucas and Penrose 189
8.2 Godel 190
CONTENTS
ix
8.3 Idealization 192
8.4 Epistemology 197
8.5 Ordinal Analysis 200
Notes 206
References 206
PART III ABSOLUTE UNDECIDABILITY
9 Provability, Mechanism, and the Diagonal Problem.211
Graham Leach-Krouse
9.1 Two Paths to Incompleteness 212
9.1.1 Post's Path 213
9.1.2 Gödel'sPath 217
9.2 Post's Response to Incompleteness: Absolutely Undecidable Propositions 219
9.2.1 From Unsolvability to Undecidability 221
9.2.2 Encounter, 1938 223
9.3 Gödel's Response to Incompleteness: Anti-Mechanism 223
9.4 Subgroundedness 225
9.5 Studying Absolute Provability 228
9.5.1 Post's Approach 228
9.5.2 Gödel’s Approach 229
9.6 Diagonalization Problem 231
9.7 Conclusions 233
Notes 234
References 240
10 Absolute Provability and Safe Knowledge of Axioms.243
T imothy Williamson
10.1 Absolute Provability 243
10.2 Propositions and Normal Mathematical Processes 244
10.3 The Epistemic Status of Axioms 245
10.4 Gödel on the Human Mind 248
10.5 Mathematical Certainty 250
Acknowledgement 251
Notes 251
References 252
11 Epistemic Church's Thesis and Absolute Undecidability.254
Marianna Antonutti Marfori and Leon Horsten
11.1 Introduction 254
11.2 Absolute Undecidability 255
11.2.1 Absolute Undecidability in Epistemic Arithmetic 255
11.2.2 Other Concepts of Absolute Undecidability 256
11.2.2.1 Fitch's Undecidables 256
11.2.2.2 Formally Undecidable Arithmetical Statements 256
11.2.2.3 Truth-Indeterminate Undecidables 258
x | CONTENTS
11.3 A New Disjunction 258
11.3.1 Epistemic Church's Thesis 259
11.3.2 ECT and Absolute Undecidability 260
11.4 Models for ECT 262
11.4.1 Is ECT True? 262
11.4.2 Simple Machines 263
11.4.3 More Realistic Models? 265
11.5 Conclusion 267
Acknowledgements 268
Notes 268
References 269
Index 273 |
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| spelling | Horsten, Leon 1966- Verfasser (DE-588)1049335686 aut Gödel's disjunction the scope and limits of mathematical knowledge Leon Horsten, Professor of Philosophy, University of Bristol; Philip Welch, Professor of Mathematical Logic, University of Bristol First edition Oxford Oxford University Press 2016 x, 277 Seiten txt rdacontent n rdamedia nc rdacarrier Reichweite (DE-588)4049187-0 gnd rswk-swf Wissen (DE-588)4066559-8 gnd rswk-swf Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Reichweite (DE-588)4049187-0 s Wissen (DE-588)4066559-8 s Mathematik (DE-588)4037944-9 s Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 s DE-604 Welch, Philip Verfasser (DE-588)1112475370 aut Digitalisierung BSB Muenchen - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029094830&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Horsten, Leon 1966- Welch, Philip Gödel's disjunction the scope and limits of mathematical knowledge Reichweite (DE-588)4049187-0 gnd Wissen (DE-588)4066559-8 gnd Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd Mathematik (DE-588)4037944-9 gnd |
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| title | Gödel's disjunction the scope and limits of mathematical knowledge |
| title_auth | Gödel's disjunction the scope and limits of mathematical knowledge |
| title_exact_search | Gödel's disjunction the scope and limits of mathematical knowledge |
| title_full | Gödel's disjunction the scope and limits of mathematical knowledge Leon Horsten, Professor of Philosophy, University of Bristol; Philip Welch, Professor of Mathematical Logic, University of Bristol |
| title_fullStr | Gödel's disjunction the scope and limits of mathematical knowledge Leon Horsten, Professor of Philosophy, University of Bristol; Philip Welch, Professor of Mathematical Logic, University of Bristol |
| title_full_unstemmed | Gödel's disjunction the scope and limits of mathematical knowledge Leon Horsten, Professor of Philosophy, University of Bristol; Philip Welch, Professor of Mathematical Logic, University of Bristol |
| title_short | Gödel's disjunction |
| title_sort | godel s disjunction the scope and limits of mathematical knowledge |
| title_sub | the scope and limits of mathematical knowledge |
| topic | Reichweite (DE-588)4049187-0 gnd Wissen (DE-588)4066559-8 gnd Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Reichweite Wissen Gödelscher Unvollständigkeitssatz Mathematik |
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