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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Borel reducibility and symmetric models


Author: Assaf Shani
Journal: Trans. Amer. Math. Soc. 374 (2021), 453-485
MSC (2010): Primary 03E15, 03E25, 03E75
DOI: https://doi.org/10.1090/tran/8250
Published electronically: November 3, 2020
MathSciNet review: 4188189
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Abstract:

We develop a correspondence between Borel equivalence relations induced by closed subgroups of $S_\infty$ and weak choice principles, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998).

For example, we show that the equivalence relation $\cong ^\ast _{\omega +1,0}$ is strictly below $\cong ^\ast _{\omega +1,<\omega }$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong ^\ast _{\omega +1,<\omega }$ provides invariants for $\Sigma ^0_{\omega +1}$ equivalence relations induced by actions of $S_\infty$, while $\cong ^\ast _{\omega +1,0}$ provides invariants for $\Sigma ^0_{\omega +1}$ equivalence relations induced by actions of abelian closed subgroups of $S_\infty$.

We further apply these techniques to study the Friedman-Stanley jumps. For example, we find an equivalence relation $F$, Borel bireducible with $=^{++}$, so that $F\restriction C$ is not Borel reducible to $=^{+}$ for any non-meager set $C$. This answers a question of Zapletal, arising from the results of Kanovei-Sabok-Zapletal (2013).

For these proofs we analyze the symmetric models $M_n$, $n<\omega$, developed by Monro (1973), and extend the construction past $\omega$, through all countable ordinals. This answers a question of Karagila (2019).


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Additional Information

Assaf Shani
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
MR Author ID: 1169593
ORCID: 0000-0002-8357-8910
Email: shani@math.harvard.edu

Received by editor(s): November 14, 2018
Received by editor(s) in revised form: May 6, 2020
Published electronically: November 3, 2020
Article copyright: © Copyright 2020 American Mathematical Society