Borel reducibility and symmetric models

We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models 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).