A weakly 2-generic which Bounds a minimal degree

Journal of Symbolic Logic 84 (4):1326-1347 (2019)

Abstract

Jockusch showed that 2-generic degrees are downward dense below a 2-generic degree. That is, if a is 2-generic, and $0 < {\bf{b}} < {\bf{a}}$, then there is a 2-generic g with $0 < {\bf{g}} < {\bf{b}}.$ In the case of 1-generic degrees Kumabe, and independently Chong and Downey, constructed a minimal degree computable from a 1-generic degree. We explore the tightness of these results.We solve a question of Barmpalias and Lewis-Pye by constructing a minimal degree computable from a weakly 2-generic one. While there have been full approximation constructions of ${\rm{\Delta }}_3^0$ minimal degrees before, our proof is rather novel since it is a computable full approximation construction where both the generic and the minimal degrees are ${\rm{\Delta }}_3^0 - {\rm{\Delta }}_2^0$.

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References found in this work

Minimal Degrees Recursive in 1-Generic Degrees.C. T. Chong & R. G. Downey - 1990 - Annals of Pure and Applied Logic 48 (3):215-225.
On $\pi^0_1$ Classes and Their Ranked Points.Rod Downey - 1991 - Notre Dame Journal of Formal Logic 32 (4):499-512.
Degrees of Unsolvability.Leonard P. Sasso - 1975 - Journal of Symbolic Logic 40 (3):452-453.
Arithmetical Sacks Forcing.Rod Downey & Liang Yu - 2006 - Archive for Mathematical Logic 45 (6):715-720.
Degrees of Unsolvability.Joseph R. Shoenfield - 1975 - Studia Logica 34 (3):284-288.

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