An almost deep degree

Abstract

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A join theorem for the computably enumerable degrees

“…By the definability of jump classes other than low 1 , for each n ≥ 1, both PL n+1 and PH n are definable ideals of R. Cholak, Groszek and Slaman [2] stated without proof (as remarked above) that PH 1 = {0}. They also stated that it is conceivable that PL 2 = {0}, and the same may hold for PL n , n ≥ 3.…”

“…S. Lempp and T. Slaman [5] showed that the only deep degree is 0. Extending the notion of deep degree, P. Cholak, M. Groszek and T. Slaman [2], page 900, called a c.e. degree a n-deep if for every c.e.…”

“…In the same paper [2], Cholak, Groszek and Slaman introduced the notion of almost deep degrees, the c.e. degrees a such that for any low c.e.…”

“…They obtained the following result. Theorem 1.2 (Cholak, Groszek and Slaman [2]). There is a nonzero almost deep degree.…”

Five Lectures on Algorithmic Randomness

“…That construction was a careful analog of the construction of an almost deep c.e. degree by Cholak, Groszek and Slaman [5], which was a c.e. degree a such that for all low c.e.…”