We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of S_∞, 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 ≅^*_ω+1,0 is strictly below ≅^*_ω+1 in Borel reducibility. By results of Hjorth-Kechris-Louveau, ≅^*_ω+1 corresponds to Σ⁰_ω+1 actions of S_∞, while ≅^*_ω+1,0 corresponds to Σ⁰_ω+1 actions of "well behaved" closed subgroups of S_∞, for example abelian groups. For these proofs we analyze the models M_n, n<ω, developed by Monro (1973), and extend his construction past ω, through all countable ordinals. This answers a question of Karagila (2016).