## Borel reducibility and symmetric models. (arXiv:1810.06722v1 [math.LO])

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 $\cong^\ast_{\omega+1,&lt;\omega}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{\omega+1,&lt;\omega}$ 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 qu查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 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 qu
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