Terminology:


Cohesive sets: $A\subset \omega$, for each recursively enumerable set $W\_e$, either $A\cap W\_e$ is finite or $A\cap(\omega\setminus W\_e)$ is finite.


Non-high degrees: Degree $a$ such that $a'\not\geq 0''$.


I'm wondering if it is possible to construct a cohesive set using some non-high 1-generic degree as an oracle? i.e. are there $A$ cohesive, and $B$ non-high 1-generic such that $A\leq\_T B$? Thanks in advance!



Cohesive sets with degree below some non-high 1-generic degrees?
The answer to this question is in Jockusch and Stephan's 1993 paper 'A cohesive set which is not high.'
<http://www.comp.nus.edu.sg/~fstephan/coh.ps>


Take $A$ and $B$ such that $A \le\_T B$, $A$ is Cohesive and $B' \not\ge\_T 0''$. This implies that $A' \not \ge\_T 0''$ and so by the paper mentioned above $A$ computes a diagonally-not-computable function. But no 1-generic can compute such a function hence $B$ cannot be 1-generic.


