Borel reducibility and symmetric models

Abstract

We develop a correspondence between Borel equivalence relations induced by closed subgroups of S ∞ 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 ≅ ω + 1 , 0 ∗ \cong ^\ast _{\omega +1,0} is strictly below ≅ ω + 1 , > ω ∗ \cong ^\ast _{\omega +1,>\omega } in Borel reducibility. By results of Hjorth-Kechris-Louveau, ≅ ω + 1 , > ω ∗ \cong ^\ast _{\omega +1,>\omega } provides invariants for Σ ω + 1 0 \Sigma ^0_{\omega +1} equivalence relations induced by actions of S ∞ S_\infty , while ≅ ω + 1 , 0 ∗ \cong ^\ast _{\omega +1,0} provides invariants for Σ ω + 1 0 \Sigma ^0_{\omega +1} equivalence relations induced by actions of abelian closed subgroups of S ∞ S_\infty . We further apply these techniques to study the Friedman-Stanley jumps. For example, we find an equivalence relation F F , Borel bireducible with = + + =^{++} , so that F ↾ C F\restriction C is not Borel reducible to = + =^{+} for any non-meager set C 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 M_n , n > ω n>\omega , developed by Monro (1973), and extend the construction past ω \omega , through all countable ordinals. This answers a question of Karagila (2019).