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Members of thin Π₁⁰ classes and generic degrees

Frank Stephan, Guohua Wu, Bowen Yuan

Abstract: A Π10\Pi ^{0}_{1} class PP is thin if every Π10\Pi ^{0}_{1} subclass QQ of PP is the intersection of PP with some clopen set. In 1993, Cenzer, Downey, Jockusch and Shore initiated the study of Turing degrees of members of thin Π10\Pi ^{0}_{1} classes, and proved that degrees containing no members of thin Π10\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 Π10\Pi ^{0}_{1} classes. In contrast to this, we show that all 1-generic degrees below 0\mathbf {0}’ contain members of thin Π10\Pi ^{0}_{1} classes.
Keywords: Members / math/mathml / inline formula / thin Π / content type / generic degrees / formula content
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