Domination, forcing, array nonrecursiveness and relative recursive enumerability

Journal of Symbolic Logic 77 (1):33-48 (2012)
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Abstract

We present some abstract theorems showing how domination properties equivalent to being $\overline{GL}_{2}$ or array nonrecursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of some known results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang, and Yu [2009] that every degree above any in $\overline{GL}_{2}$ is recursively enumerable in a 1-generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below ANR degrees also reveal a new level of uniformity in these types of results

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Citations of this work

A Hyperimmune Minimal Degree and an ANR 2-Minimal Degree.Mingzhong Cai - 2010 - Notre Dame Journal of Formal Logic 51 (4):443-455.

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References found in this work

Lowness for genericity.Liang Yu - 2006 - Archive for Mathematical Logic 45 (2):233-238.
Direct and local definitions of the Turing jump.Richard A. Shore - 2007 - Journal of Mathematical Logic 7 (2):229-262.
Turing degrees of certain isomorphic images of computable relations.Valentina S. Harizanov - 1998 - Annals of Pure and Applied Logic 93 (1-3):103-113.
A 2-minimal non-gl2 degree.Mingzhong Cai - 2010 - Journal of Mathematical Logic 10 (1):1-30.

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