Gödel’s Disjunction: The scope and limits of mathematical knowledge

289 Pages·2016·2.141 MB·English

by  Leon Horsten, Philip Welch

GÖDEL’S DISJUNCTION

Gödel’s Disjunction
Thescopeandlimitsofmathematicalknowledge
LEON HORSTEN
ProfessorofPhilosophy,UniversityofBristol
PHILIP WELCH
ProfessorofMathematicalLogic,UniversityofBristol
3
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CONTENTS
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
LeonHorstenandPhilipWelch
1.1 Gödel’sDisjunctionandBeyond 1
1.1.1 TheDisjunctiveThesis 2
1.1.2 TheFirstDisjunct 2
1.1.3 TheSecondDisjunct 3
1.2 FormalFrameworks 4
1.2.1 MathematicalPhilosophytotheRescue? 4
1.2.2 EpistemicMathematics 5
1.2.3 ComputationandtheNatureofAlgorithm 6
1.3 OrganisationoftheContributions 7
1.3.1 Algorithm,Consistency,andEpistemicRandomness 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 MindsandMachines 10
1.3.2.1 Carlson 10
1.3.2.2 Koellner 10
1.3.2.3 Shapiro 12
1.3.3 AbsoluteUndecidability 12
1.3.3.1 Leach-Krouse 12
1.3.3.2 Williamson 13
1.3.3.3 AntonuttiandHorsten 13
References 14
PART I
ALGORITHM, CONSISTENCY, AND EPISTEMIC
RANDOMNESS
2 AlgorithmsandtheMathematicalFoundationsofComputer
Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
WalterDean
2.1 Introduction 19
2.2 MotivatingAlgorithmicRealism 24
2.3 AlgorithmsinTheoreticalComputerScience 27
2.4 InSearchofaFoundationalFramework 34
2.5 ProceduralEquivalence 41
vi | CONTENTS
2.5.1 SimulationEquivalence 42
2.5.2 TheExigenciesofSimulation 44
2.5.2.1 FormalizingtheTransitionalCondition 45
2.5.2.2 FormalizingtheRepresentationalRequirement 47
2.5.2.3 ImplementingRecursion 48
2.6 TakingStock 51
2.6.1 Moschovakis,Gurevich,andtheLevel-RelativityofAlgorithms 51
2.6.2 Algorithms,Identity,andMathematicalPractice 54
Acknowledgement 57
Notes 57
References 63
3 TheSecondIncompletenessTheorem:Reflectionsand
Ruminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
AlbertVisser
3.1 Introduction 67
3.1.1 StatusoftheTechnicalResultsinthisChapter 68
3.2 VersionsoftheSecondIncompletenessTheorem 68
3.2.1 ABasicVersionofG2 69
3.2.2 G2asaStatementofInterpretabilityPower 69
3.2.3 Feferman’sTheorem 70
3.2.4 G2asanAdmissibleRule 70
3.3 MeaningasConceptualRole 70
3.3.1 L-Predicates 71
3.3.2 OnHBL-Predicates 74
3.3.3 FefermanonL-Predicates 78
3.3.4 PhilosophicalDiscussion 79
3.4 SolutionoftheMeaningProblem 80
3.5 AbolishingArbitrariness 81
3.5.1 ConsistencyStatementsasUniqueSolutionsofEquations 82
3.5.2 BoundedInterpretations 83
Notes 85
References 86
AppendixABasicFactsandDefinitions 88
A.1 Theories 88
A.2 TranslationsandInterpretations 88
A.3 SequentialTheories 90
A.4 ComplexityMeasures 90
4 IteratedDefinability,LawlessSequences,andBrouwer’s
Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
JoanRandMoschovakis
4.1 Introduction 92
4.2 ChoiceSequences 93
4.2.1 Brouwer’sContinuum 93
CONTENTS | vii
4.2.2 TheProblemofDefining“Definability” 93
4.2.3 “Lawlike”versus“Lawless”Sequences 94
4.3 TheFormalSystemsRLS(≺)andFIRM(≺) 94
4.3.1 TheThree-SortedLanguageL(≺) 94
4.3.2 AxiomsandRulesforThree-SortedIntuitionistic
PredicateLogic 95
4.3.3 AxiomsforThree-sortedIntuitionisticNumberTheory 95
4.3.4 LawlessSequences,RestrictedQuantification,
andLawlikeComprehension 96
4.3.5 AxiomsforLawlessSequences 96
4.3.6 Well-OrderingtheLawlikeSequences 97
4.3.7 RestrictedLEM,theAxiomofClosedData
andLawlikeCountableChoice 97
4.3.8 Brouwer’sBarTheoremandTroelstra’sGeneralized
ContinuousChoice 98
4.3.9 ClassicalandIntuitionisticAnalysisasSubsystemsofFIRM(≺) 99
4.3.10 ConsistencyofFIRM(≺) 99
4.4 ConstructionoftheClassicalModelandProofofTheorem1 100
4.4.1 DefinabilityOver(A,≺ )byaRestrictedFormulaofL(≺) 100
A
4.4.2 TheClassicalModelM(≺R) 100
4.4.3 OutlineoftheProofofTheorem1 102
4.5 The(cid:2)-RealizabilityInterpretation 103
4.5.1 Definitions 103
4.5.2 OutlineoftheProofofTheorem2 104
4.6 Epilogue 105
Acknowledgement 106
Notes 106
References 106
5 ASemanticsforIn-PrincipleProvability . . . . . . . . . . . . . . . . . . . . . .108
T.Achourioti
5.1 Introduction 108
5.2 In-PrincipleProvabilityandIntensionality 109
5.3 ModellingEpistemicMathematics:ATheoryofDescriptions 111
5.4 IntensionalTruth 113
5.5 TowardsAxiomsforIn-PrincipleProvability 115
5.6 IntensionalSemanticsfor‘ItisIn-PrincipleProvablethat’ 117
5.6.1 DynamicalProofs 118
5.6.2 BringingProvabilityBackinto‘In-PrincipleProvability’ 119
5.6.3 BandTheoryT 122
5.7 Conclusion 123
Notes 124
References 125
viii | CONTENTS
PART II
MIND AND MACHINES
6 CollapsingKnowledgeandEpistemicChurch’sThesis. . . . . . . . . . . .129
TimothyJ.Carlson
6.1 Introduction 129
6.2 KnowingEntitiesandSyntacticEncoding 133
6.3 HierarchiesandStratification 134
6.4 Collapsing 136
6.5 AComputableCollapsingRelation 137
6.6 AMachineThatKnowsEA+ECT 138
6.7 Remarks 146
References 147
7 Gödel’sDisjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148
PeterKoellner
7.1 TheDisjunction 150
7.1.1 RelativeProvabilityandTruth 150
7.1.2 AbsoluteProvability 151
7.1.3 IdealizedFiniteMachinesandIdealizedHumanMinds 153
7.1.4 Summary 154
7.2 Notation 155
7.3 Arithmetic 156
7.4 Incompleteness 156
7.5 EpistemicArithmetic 157
7.6 EpistemicArithmeticwithTypedTruth 159
7.7 TheDisjunctioninEAT 160
7.8 TheClassicArgumentfortheFirstDisjunct 162
7.9 TheFirstDisjunctinEAT 163
7.10 Penrose’sNewArgument 164
7.11 Type-FreeTruth 166
7.12 AFailedAttempt 167
7.13 TheSystemDTK 169
7.14 BasicResultsinDTK 170
7.15 TheDisjunctioninDTK 174
7.16 TheDisjunctsinDTK 176
7.17 Conclusion 183
Acknowledgement 185
Notes 185
References 186
8 Idealization,Mechanism,andKnowability . . . . . . . . . . . . . . . . . . . .189
StewartShapiro
8.1 LucasandPenrose 189
8.2 Gödel 190
CONTENTS | ix
8.3 Idealization 192
8.4 Epistemology 197
8.5 OrdinalAnalysis 200
Notes 206
References 206
PART III
ABSOLUTE UNDECIDABILITY
9 Provability,Mechanism,andtheDiagonalProblem . . . . . . . . . . . . . .211
GrahamLeach-Krouse
9.1 TwoPathstoIncompleteness 212
9.1.1 Post’sPath 213
9.1.2 Gödel’sPath 217
9.2 Post’sResponsetoIncompleteness:AbsolutelyUndecidablePropositions 219
9.2.1 FromUnsolvabilitytoUndecidability 221
9.2.2 Encounter,1938 223
9.3 Gödel’sResponsetoIncompleteness:Anti-Mechanism 223
9.4 Subgroundedness 225
9.5 StudyingAbsoluteProvability 228
9.5.1 Post’sApproach 228
9.5.2 Gödel’sApproach 229
9.6 DiagonalizationProblem 231
9.7 Conclusions 233
Notes 234
References 240
10 AbsoluteProvabilityandSafeKnowledgeofAxioms. . . . . . . . . . . . . .243
TimothyWilliamson
10.1 AbsoluteProvability 243
10.2 PropositionsandNormalMathematicalProcesses 244
10.3 TheEpistemicStatusofAxioms 245
10.4 GödelontheHumanMind 248
10.5 MathematicalCertainty 250
Acknowledgement 251
Notes 251
References 252
11 EpistemicChurch’sThesisandAbsoluteUndecidability. . . . . . . . . . .254
MariannaAntonuttiMarforiandLeonHorsten
11.1 Introduction 254
11.2 AbsoluteUndecidability 255
11.2.1 AbsoluteUndecidabilityinEpistemicArithmetic 255
11.2.2 OtherConceptsofAbsoluteUndecidability 256
11.2.2.1 Fitch’sUndecidables 256
11.2.2.2 FormallyUndecidableArithmeticalStatements 256
11.2.2.3 Truth-IndeterminateUndecidables 258