1-Generic Enumeration Degrees Below O e ’

Enumeration reducibility is the formalisation of the natural concept of relative enumerability between sets of natural numbers. A set A is said to be enumeration reducible to a set B iff there is some effective procedure which gives an enumeration of A from any enumeration of B. This can be shown to be equivalent to the following definition:Definition 1.1 A set of natural numbers A is enumeration reducible (e-reducible,≦_e) to a set of natural numbers B iff there is an i such that for all x$$x \in A \Leftrightarrow \exists z\left[ {\langle x,z\rangle \in {W_i}\& {D_z} \subset B} \right]$$ where W_i and D_z are, respectively, the ith recursively enumerable set and the zth finite set in appropriate standard listing of such sets.