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A WEAKLY 2-GENERIC WHICH BOUNDS A MINIMAL DEGREE
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- Journal:
- The Journal of Symbolic Logic / Volume 84 / Issue 4 / December 2019
- Published online by Cambridge University Press:
- 04 November 2019, pp. 1326-1347
- Print publication:
- December 2019
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- Article
- Export citation
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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 
${\rm{\Delta }}_3^0 - {\rm{\Delta }}_2^0$.
