Noncappable Enumeration Degrees Below 0'e

We prove that there exists a noncappable enumeration degree strictly below 0 0 e. Two notions of relative computability, Turing and enumeration re-ducibility, are basic to any natural ne-structure theory for the classes of computable and incomputable objects. Of the theories for the corresponding degree structures (D D D and D D D e), that for the Turing degrees is the better developed, mainly due to the depth of knowledge of spe-ciic local structure (see for example Lerman Le83], Odifreddi Od89] and Soare So87]). Despite its importance (see Co90]) in applications to nondeterministic computations, to relative computability involving partial information, in providing models of-calculus, and in setting 1991 Mathematics Subject Classiication. 03D30.