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A recursion theoretic property of $ \mathbf{\Sigma^1_1}$ equivalence relations


Author: Howard Becker
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 03E15
DOI: https://doi.org/10.1090/proc/14118
Published electronically: July 23, 2018
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Abstract: Let $ E$ be a $ \mathbf {\Sigma ^1_1}$ equivalence relation on $ 2^\omega $ which does not have perfectly many equivalence classes. For $ a \in 2^\omega $, define $ L^a_E$ to be the set $ \{[x]_E: (\exists y) (y \in [x]_E$ and $ \omega _1^{\langle a,y \rangle } = \omega _1^a)\}$. For a Turing cone of $ a$'s, $ L^a_E$ is countable. This is proved assuming $ \mathbf {\Pi }^1_2$-determinacy.


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Additional Information

Howard Becker
Affiliation: Suite 3, 4711 Forest Drive, Columbia, South Carolina 29206
Email: hsbecker@hotmail.com

DOI: https://doi.org/10.1090/proc/14118
Received by editor(s): October 27, 2017
Received by editor(s) in revised form: January 28, 2018
Published electronically: July 23, 2018
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2018 American Mathematical Society

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