Members of thin Π₁⁰ classes and generic degrees

Abstract

A Pi(0)(1) class P is thin if every Pi(0)(1) subclass Q of P is the intersection of P with some clopen set. In 1993, Cenzer, Downey, Jockusch and Shore initiated the study of Turing degrees of members of thin Pi(0)(1) classes, and proved that degrees containing no members of thin Pi(0)(1) classes can be recursively enumerable, and can be minimal degree below 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 Pi(0)(1) classes. In contrast to this, we show that all 1-generic degrees below 0' contain members of thin Pi(0)(1) classes.