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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  and
Theodore A. Slaman
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
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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 A ⊕ W 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 (W ⊕ A)′ 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
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 376 - 395
Copyright
Copyright © Association for Symbolic Logic 1989

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References

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