An Alternative Proof of 1-Generic Splittings (Proof theory and proving)

Abstract

Wu (2006) showed that every nonzero computably enumerable degree splits into two 1-generic degrees, and therefore, no two computably enumerable degrees bound the same class of 1-generic degrees. By relativizing this result with respect to the Lachlan set, it can be shown that (*) every nonzero d.c.e. degree splits into four 1-generic degrees. Here, a set A is d.c.e. (or, 2-c.e.) if there are two computably enumerable sets B and C such that A = B-C (set difference). Turing degree of a d.c.e. set is called a d.c.e. degree. By (*), no two d.c.e. degrees bound the same class of 1-generic degrees. Chong and Yu (2016) improved the result (*). In fact, it is split into two 1-generic degrees. In this note, we propose a construction with rollbacks of stages. By means of this construction, we give an alternative proof of (*).