We develop a correspondence between Borel equivalence relations induced by closed subgroups of $S_\infty$ and 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 $\cong^\ast_{\omega+1,< \omega}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{\omega+1,< \omega}$ bounds the complexity of $\Sigma^0_{\omega+1}$ actions of $S_\infty$, while $\cong^\ast_{\omega+1,0}$ bounds the complexity of $\Sigma^0_{\omega+1}$ actions of ``well behaved'' closed subgroups of $S_\infty$, such as abelian groups. The notions mentioned above will be defined in the talk, and I will also survey the results of Hjorth-Kechris-Louveau.