Abstract

In this paper, we first prove that there exist computably enumerable (c.e.) degrees a and b such that ${\bf a\not\leq b}$ , and for any c.e. degree u, if ${\bf u\leq a}$ and u is cappable, then ${\bf u\leq b}$ , so refuting a conjecture of Lempp (in Slaman [1996]); secondly, we prove that: (A. Li and D. Wang) there is no uniform construction to build nonzero cappable degree below a nonzero c.e. degree, that is, there is no computable function $f$ such that for all $e\in\omega,$ (i) $W_{f(e)}\leq_{\rm T}W_e$ , (ii) $W_{f(e)}$ has a cappable degree, and (iii) $W_{f(e)}\not\leq_{\rm T}\emptyset$ unless $W_e\leq_{\rm T}\emptyset.$