Members of thin π⁰₁ classes and generic degrees

• Main Authors: Stephan, Frank, Wu, Guohua, Yuan, Bowen
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2022
• Subjects: Science::Mathematics; Turing Degrees; Genericity
• Online Access: https://hdl.handle.net/10356/162480
• Institution: Nanyang Technological University

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 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 0' contain members of thin π⁰₁ classes.