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Degree-invariant, analytic equivalence relations without perfectly many classes


Author: Antonio Montalbán
Journal: Proc. Amer. Math. Soc. 145 (2017), 395-398
MSC (2010): Primary 03D99, 03E99
DOI: https://doi.org/10.1090/proc/13218
Published electronically: July 12, 2016
MathSciNet review: 3565390
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Abstract: We show that there is only one natural Turing-degree invariant, analytic equivalence relation with $ \aleph _1$ many equivalence classes: the equivalence $ X \equiv _{\omega _1} Y\iff \omega _1^X=\omega _1^Y$. More precisely, under $ PD+\neg CH$, we show that every Turing-degree invariant, analytic equivalence relation with $ \aleph _1$ many equivalence classes is equal to $ \equiv _{\omega _1}$ on a Turing cone.


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Additional Information

Antonio Montalbán
Affiliation: Department of Mathematics, University of California, Berkely, California 94720
Email: antonio@math.berkeley.edu

DOI: https://doi.org/10.1090/proc/13218
Received by editor(s): January 23, 2016
Received by editor(s) in revised form: March 13, 2016
Published electronically: July 12, 2016
Additional Notes: The author was partially supported by NSF grant DMS-0901169 and the Packard Fellowship
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2016 American Mathematical Society