A limit on relative genericity in the recursively enumerable sets

Author:

Lempp Steffen,Slaman Theodore A.

Abstract

AbstractWork in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X′, its Turing jump, is recursive in ∅′ and high if X′ computes ∅″. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of AW is recursive in the jump of A. We prove that there are no deep degrees other than the recursive one.Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turing functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that (WA)′ is forced to disagree with Φ(−;A′). The conversion has some ambiguity; in particular, A cannot be found uniformly from W.We also show that there is a “moderately” deep degree: There is a low nonzero degree whose join with any other low degree is not high.

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy

Cited by 9 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. A join theorem for the computably enumerable degrees;Transactions of the American Mathematical Society;2004-02-27

2. Definable Relations on the Computably Enumerable Degrees;Computability and Models;2003

3. An almost deep degree;Journal of Symbolic Logic;2001-06

4. Implicit measurements of dynamic complexity properties and splittings of speedable sets;Journal of Symbolic Logic;1999-09

5. Σ5-completeness of index sets arising from the recursively enumerable Turing degrees;Annals of Pure and Applied Logic;1996-06

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