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A limit on relative genericity in the recursively enumerable sets

Published online by Cambridge University Press:  12 March 2014

Steffen Lempp*
Affiliation:
Department of Mathematics, Yale University, New Haven, Connecticut 06520
Theodore A. Slaman
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
*
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Abstract

Work 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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1989

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References

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