A Framework for Priority ArgumentsThis 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. |
Contents
Systems of Trees of Strategies | 5 |
Σ1 Constructions | 41 |
2 Constructions | 57 |
Π2 Constructions | 73 |
3 Constructions | 93 |
Σ3 Constructions | 103 |
Paths and Links | 113 |
Backtracking | 147 |
HigherLevel Constructions | 163 |
Infinite Systems of Trees | 173 |
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Common terms and phrases
action sentences activated action allow Analysis antiderivative apply assume axiom backtracking basic module begin block bounds Chapter clause Clauses i)–(iii completed computably enumerable degrees computably enumerable set conclude consistent constrained construction contradiction correct corresponding current path defined Definition describe determined directing sentence easily element ensure existence extender finite first follows from Lemma framework functional Furthermore given Hence holds immediate successor implemented incompatible induction infinite initial assignment initial derivative Limit Path monotonic nodes obtain oracle outcome pair parameters present Preservation primary principal derivative priority proof properties prove quantifiers Remark requirements resp restraint satisfaction satisfied satisfy sentence decomposition shared specify stage standard derived assignment standard initial supported Suppose switched Theorem trees true path validated action Verification weight witnesses