The d.r.e. degrees are not dense (Q1182487)

The main result of this paper is to prove that \(\mathbf{0}'\) is a minimal cover in the d.r.e. Turing degrees. The proof is a complex \(\mathbf{0}'''\) priority argument with a central strategy akin to the Lachlan join strategy [the third author, Ann. Math. Logic 9, 307-365 (1975; Zbl 0357.02040)]. A number of further technical devices are needed to make the proof go through. The paper is exceedingly well written and is quite accessible. The result should be contrasted not only with Sacks' density theorem, but Cooper's result that the \(\text{low}_ 2\) d.r.e. degrees are dense [the first author, to appear] and the Cooper-Lempp- Watson result [the first and the fourth author and \textit{P. Watson}, Isr. J. Math. 67, 137-152 (1989; Zbl 0691.03023)] that if \({\mathbf a}<{\mathbf b}\) are \(n\)-r.e. there exists an \(n+1\) r.e. degree \({\mathbf c}\) with \({\mathbf a}<{\mathbf c}<{\mathbf b}\).