A Framework for Priority Arguments

By: Manuel Lerman

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Hardcover | 19 April 2010

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194 Pages

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22.86 x 15.24 x 1.27

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"This book presents a unifying framework for using priority arguments to prove theorems in computability. Priority arguments provide the most powerful theorem-proving technique in the field, but most of the applications of this technique are ad hoc, masking the unifying principles used in the proofs. The proposed framework presented isolates many of these unifying combinatorial principles and uses them to give shorter and easier-to-follow proofs of computability-theoretic theorems. Standard theorems of priority levels 1, 2, and 3 are chosen to demonstrate the framework's use, with all proofs following the same pattern. The last section features a new example requiring priority at all finite levels. The book will serve as a resource and reference for researchers in logic and computability, helping them to prove theorems in a shorter and more transparent manner"--Provided by publisher.

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Preface	p. xiii
Introduction	p. 1
Computably Enumerable Sets	p. 1
Degrees	p. 2
Finite Sequences and Trees	p. 3
Systems Of Trees Of Strategies	p. 5
An Overview	p. 7
Systems of Trees	p. 14
Basic Modules	p. 17
The Path Generating Function	p. 18
Links	p. 21
Consistency	p. 23
Blocks	p. 24
The Weight Function	p. 27
Directing and Action Sentences: An Example	p. 29
Directing and Action Sentences: Properties	p. 31
The Framework Theorem	p. 33
A Sampling of Lemmas	p. 36
∑₁ Construction	p. 41
∑₁Constructions	p. 41
The Friedberg-Mucnik Theorem	p. 47
The Basic R_,i-Module	p. 49
T¹-Analysis	p. 49
T⁰-Analysis	p. 49
The Construction	p. 50
The Verification	p. 50
A Low Computably Enumerable Degree	p. 50
The Basic P-Module	p. 51
The Basic N_eModule	p. 52
T¹-Analysis	p. 52
T⁰-Analysis	p. 52
The Construction	p. 52
The Verification	p. 52
A Properly d-c.e. Degree	p. 53
The Basic R_{,, W}-Module	p. 54
T¹-Analysis	p. 55
T⁰-Analysis	p. 55
The Construction	p. 56
The Verification	p. 56
Summary	p. 56
₂ Constructions	p. 57
The ₂ Level	p. 57
Avoiding Upward Cones	p. 58
The Basic P-Module	p. 59
The Basic N-Module	p. 59
T²-Analysis	p. 60
T¹ -Analysis	p. 60
T⁰-Analysis	p. 60
The Construction	p. 61
The Verification	p. 61
The Sacks Splitting Theorem	p. 61
The Basic P_x-Module	p. 62
The Basic N-Module	p. 62
T²-Analysis	p. 63
T¹-Analysis	p. 63
T⁰-Analysis	p. 63
The Construction	p. 64
The Verification	p. 64
Backtracking	p. 64
Permitting	p. 66
The Basic P_,i-Module	p. 68
T²-Analysis	p. 69
T¹ -Analysis	p. 69
T⁰Analysis	p. 70
The Construction	p. 70
The Verification	p. 70
Further Remarks	p. 71
Summary	p. 71
₂ Constructions	p. 73
I₂ Constructions	p. 73
A High Computably Enumerable Degree	p. 75
The Basic P-Module	p. 76
The Basic Q_x-Module	p. 77
T²-Analysis	p. 77
T¹-AnalysJs	p. 77
T⁰-Analysis	p. 78
The Construction	p. 78
The Verification	p. 78
Further Remarks	p. 78
The Jump Inversion Theorem	p. 78
The Basic Q_e-Module	p. 80
The Basic N_e-Module	p. 80
T²-Analysis	p. 80
T¹-Analysis	p. 80
T⁰-Analysis	p. 81
The Construction	p. 81
The Verification	p. 81
The Minimal Pair Theorem	p. 81
The Basic P_,i-Module	p. 84
The Basic N_,-Module	p. 84
T²-Analysis	p. 85
T¹-Analysis	p. 85
T⁰-Analysis	p. 85
The Construction	p. 85
The Verification	p. 86
Further Remarks	p. 86
Embedding the Pentagon	p. 86
The L Requirement	p. 87
The J Requirement	p. 87
The Basic N_,-Module	p. 88
The Basic P_A,-Module and the Basic P_C,-Module	p. 88
The Basic Q-Module	p. 88
T²-Analysis	p. 89
T¹-Analysis	p. 89
T⁰-Analysis	p. 89
The Construction	p. 90
The Verification	p. 90
Further Remarks	p. 90
Summary	p. 91
₃ Constructions	p. 93
₃ Constructions	p. 93
Properties of ₃ Constructions	p. 94
The Restraint Requirements	p. 95
The Correction Requirements	p. 96
Bounding A above by V	p. 97
The Density Theorem	p. 98
The R Requirement	p. 98
The Basic N-Module	p. 98
The Basic PModule	p. 99
T³-Analysis	p. 100
T²-Analysis, T^-Analysis	p. 101
T⁰-Analysis	p. 101
The Construction	p. 101
The Verification	p. 101
Summary	p. 102
₃ Consructions	p. 103
Module-Respecting Derived Assignments	p. 103
Strong Minimal Pairs	p. 104
The Strong Minimal Pair Theorem	p. 106
The Basic P-Module	p. 106
The Basic Q-Module	p. 106
The Basic M_,W-Module	p. 107
T³-Analysis..'	p. 108
T²-Analysis	p. 109
T¹-Analysis	p. 109
T⁰-Analysis	p. 110
The Construction	p. 110
The Verification	p. 110
Some Comments	p. 111
Paths and Links	p. 113
Paths	p. 113
Switching and Nonswitching Extensions	p. 117
Links	p. 119
Free Nodes	p. 124
Relationships Between Links	p. 126
Framework Lemmas	p. 132
Monotonic Sentence Decomposition	p. 132
Thrived Assignments	p. 133
Level Analysis Lemmas	p. 136
The ₁ Level	p. 136
The ₂ Level	p. 137
The ₃ Level	p. 138
Restraint Lemmas	p. 140
Miscellaneous Lemmas	p. 144
Backtracking	p. 147
Backtracking	p. 147
Properties of PL Sets	p. 149
Extenders	p. 152
Backtracking and Links	p. 154
Set Approximation	p. 155
Constraint	p. 158
REA Sets	p. 159
Level-Specific Backtracking	p. 161
Higher-Level Constructions	p. 163
Jump Posets with Least Element	p. 163
The Requirements	p. 164
T^l-Analysis	p. 166
Tⁱ-Analysis, I < l	p. 166
The Construction	p. 166
The Construction	p. 167
Proof	p. 168
Infinite Systems of Trees	p. 173
Infinite Systems: A Brief Analysis	p. 173
References	p. 175
Table of Contents provided by Ingram. All Rights Reserved.

A Framework for Priority Arguments

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Hardcover

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