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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sg-ntu-dr.10356-1624802022-10-25T01:03:13Z Members of thin π⁰₁ classes and generic degrees Stephan, Frank Wu, Guohua Yuan, Bowen School of Physical and Mathematical Sciences Science::Mathematics Turing Degrees Genericity 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. Ministry of Education (MOE) Nanyang Technological University The first author was supported by Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2016-T2-1-019 / R146-000-234-112 and MOE2019-T2-2-121 / R146-000-304-112. The second author was supported by Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2016-T2-1-083 (M4020333); NTU Tier 1 grants RG32/16 (M4011672) and RG111/19 (M4012245). 2022-10-25T01:03:13Z 2022-10-25T01:03:13Z 2022 Journal Article Stephan, F., Wu, G. & Yuan, B. (2022). Members of thin π⁰₁ classes and generic degrees. Proceedings of the American Mathematical Society, 150(7), 3125-3131. https://dx.doi.org/10.1090/proc/15325 0002-9939 https://hdl.handle.net/10356/162480 10.1090/proc/15325 2-s2.0-85130992333 7 150 3125 3131 en MOE2016-T2-1-019 / R146-000-234-112 MOE2019-T2-2-121 / R146-000-304-112 MOE2016-T2-1-083 (M4020333) RG32/16 (M4011672) RG111/19 (M4012245) Proceedings of the American Mathematical Society © 2022 American Mathematical Society. All rights reserved. |
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Science::Mathematics Turing Degrees Genericity Stephan, Frank Wu, Guohua Yuan, Bowen Members of thin π⁰₁ classes and generic degrees |
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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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School of Physical and Mathematical Sciences |
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School of Physical and Mathematical Sciences Stephan, Frank Wu, Guohua Yuan, Bowen |
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Stephan, Frank Wu, Guohua Yuan, Bowen |
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Stephan, Frank |
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Members of thin π⁰₁ classes and generic degrees |
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Members of thin π⁰₁ classes and generic degrees |
title_full |
Members of thin π⁰₁ classes and generic degrees |
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Members of thin π⁰₁ classes and generic degrees |
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Members of thin π⁰₁ classes and generic degrees |
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members of thin π⁰₁ classes and generic degrees |
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2022 |
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https://hdl.handle.net/10356/162480 |
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