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Domination, forcing, array nonrecursiveness and relative recursive enumerability

Published online by Cambridge University Press:  12 March 2014

Mingzhong Cai  and
Richard A. Shore
Mingzhong Cai
Affiliation:
Department of Mathematics, Cornell University, Ithaca NY 14853, USA, E-mail: yiyang@math.cornell.edu
Richard A. Shore
Affiliation:
Department of Mathematics, Cornell University, Ithaca NY 14853, USA, E-mail: shore@math.cornell.edu
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Abstract

We present some abstract theorems showing how domination properties equivalent to being 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 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 33 - 48
Copyright
Copyright © Association for Symbolic Logic 2012

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References

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