Members of thin π⁰₁ classes and generic degrees
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 enu...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/162480 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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. |
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