Members of thin Π₁⁰ classes and generic degrees

Abstract

A Π 1 0 \Pi ^{0}_{1} class P P is thin if every Π 1 0 \Pi ^{0}_{1} subclass Q Q of P P is the intersection of P P with some clopen set. In 1993, Cenzer, Downey, Jockusch and Shore initiated the study of Turing degrees of members of thin Π 1 0 \Pi ^{0}_{1} classes, and proved that degrees containing no members of thin Π 1 0 \Pi ^{0}_{1} classes can be recursively enumerable, and can be minimal degree below 0 ′ \mathbf {0}’ . In this paper, we work on this topic in terms of genericity, and prove that all 2-generic degrees contain no members of thin Π 1 0 \Pi ^{0}_{1} classes. In contrast to this, we show that all 1-generic degrees below 0 ′ \mathbf {0}’ contain members of thin Π 1 0 \Pi ^{0}_{1} classes.