Abstract:
We demonstrate that for every pair of computably enumerable degrees $\mathbf a<_\mathbf Q\mathbf b$ there exists a properly 2-computably enumerable degree $\mathbf d$, $\mathbf a<_\mathbf Q\mathbf d<_\mathbf Q\mathbf b$, such that $\mathbf a$ isolates $\mathbf d$ from below and $\mathbf b$ isolates $\mathbf d$ from above. As a corollary we prove that there exists a 2-computably enumerable degree which is $Q$-incomparable with any nontrivial (i.e., different from $\boldsymbol0$ and $\boldsymbol0'$) computably enumerable degree, and that every nontrivial computably enumerable degree isolates some 2-computably enumerable degree from below and some 2-computably enumerable degree from above.