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### Members of thin Π₁⁰ classes and generic degrees

Frank Stephan, Guohua Wu, Bowen Yuan
Published: 29 March 2022

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 $\mathbf {0}’$. In this paper, we work on this topic in terms of genericity, and prove that all 2-generic degrees contain no members of thin $\Pi ^{0}_{1}$ classes. In contrast to this, we show that all 1-generic degrees below $\mathbf {0}’$ contain members of thin $\Pi ^{0}_{1}$ classes.
Keywords: Members / math/mathml / inline formula / thin Π / content type / generic degrees / formula content

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