members of thin Π₁⁰ classes and generic degrees

2020/08/11 by Stephan, Frank, Wu, Guohua, Yuan, Bowen
#03D28 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2008.04539

Abstract

A Π⁰₁ class P is thin if every Π⁰₁ 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 Π⁰₁ classes, and proved that degrees containing no members of thin Π⁰₁ classes can be recursively enumerable, and can be minimal degree below \bf 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 Π⁰₁ classes. In contrast to this, we show that all 1-generic degrees below \bf 0' contain members of thin Π⁰₁ classes.

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