1-Generic Enumeration Degrees Below O ^’_e

Abstract

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 i ^th recursively enumerable set and the z ^th finite set in appropriate standard listing of such sets.