Isolation and the high/low hierarchy
A subset \(D\) of \(\omega\) is called a d.c.e. (difference of c.e. sets) if there are c.e. (computably enumerable) sets \(A_0\) and \(A_1\) such that \(D=A_0 -A_1\). A d.c.e. degree \(\mathbf d\) is called isolated by a c.e. degree \(\mathbf b\) if \({\mathbf b}<{\mathbf d}\) and any c.e. degree \(\mathbf c\) below \(\mathbf d\) is also below \(\mathbf b\). The second author, \textit{Guohua Wu}, proved that there is a high d.c.e. degree \(\mathbf d\) isolated by a low\(_2\) c.e. degree \(\mathbf b\) [Math. Log. Q. 47, 525-534 (2001; Zbl 1003.03039)]. In this paper, the authors improve Wu's result and show that there are a high d.c.e. degree \(\mathbf d\) and a low c.e. degree \(\mathbf a\) such that \(\mathbf d\) is isolated by \(\mathbf a\).
scientific article; zbMATH DE number 1982474