# Assaf Shani: Borel equivalence relations and symmetric models

HUJI Logic Seminar

We will have a meeting of the Logic Seminar this Wednesday 11/7, 11-13 Ross 70A.

Title:  Borel equivalence relations and symmetric models
Speaker: Assaf Shani
Abstract. We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models of set theory without choice, 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 $\cong^\ast_{\omega+1}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{\omega+1}$ corresponds to $\Sigma^0_{\omega+1}$ actions of $S_\infty$, while $\cong^\ast_{\omega+1,0}$ corresponds to $\Sigma^0_{\omega+1}$ actions of “well behaved” closed subgroups of $S_\infty$, e.g., abelian groups.
We further apply these techniques to study the Friedman-Stanley jumps. For example, we find a topology on the domain of $=^{++}$ so that $=^{++}\restriction C$ is Borel bireducible with $=^{++}$, for any comeager set $C$. This answers a question of Zapletal, based on the results of Kanovei-Sabok-Zapletal (2013).
For these proofs we analyze the models $M_n$, $n<\omega$, developed by Monro (1973), and extend his construction past $\omega$, through all countable ordinals. This answers a question of Karagila (2016), e.g., establishing separation between the $\omega$ and $\omega+1$’th Kinna-Wagner principle.

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