Borel equivalence relations and symmetric models

We develop a relationship between Borel equivalence relations and weak choice principles. Specifically, we show that questions about Borel reducibility and strong ergodicity between equivalence relations which are classifiable by countable structures can be translated to questions about fragments of choice holding in certain symmetric models. We then use tools developed in the '60s and '70s to analyze such symmetric models and solve several problems about Borel equivalence relations.This relationship is explained in Chapter~\ref{chapter:symmetric-models-borel-reducibility}.These techniques are applied to the study of equivalence relations high in the Borel reducibility hierarchy in Chapter~\ref{chapter:jumps-HKL}.In \cite{HKL98} Hjorth, Kechris and Louveau refined the Friedman-Stanley jump hierarchy by defining equivalence relations $\cong^\ast_{\alpha+1,\beta}$, $\beta<\alpha$.For example, they show that $\cong^\ast_{\omega+1,<\omega}$ 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 \textit{abelian} closed subgroups of $S_\infty$.Whether $\cong^\ast_{\omega+1,0}$ is strictly below $\cong^\ast_{\omega+1,<\omega}$ in Borel reducibility was left open. We show that they are distinct and prove generally that the entire hierarchy defined by Hjorth-Kechris-Louveau is strict, establishing their conjecture. We further study the Friedman-Stanley jumps. For example, we find an equivalence relation $F$ on a Polish space $Y$ such that $F$ is Borel bireducible with $=^{++}$ and $F\rest C$ is Borel bireducible with $=^{++}$ for any non-meager set $C\subset Y$. This answers a question of Zapletal, arising from the results of Kanovei-Sabok-Zapletal \cite{ksz}. In the terminology of \cite{ksz}, the result states ``$=^{++}$ is in the spectrum of the meager ideal''.For these proofs we analyze the symmetric models $M_n$, $n<\omega$, developed by Monro \cite{Mon73} to separate Kinna-Wagner principles. We extend Monro's construction past $\omega$, through all countable ordinals, answering a question of Karagila \cite{Kar17}.Chapter \ref{chapter:products-ctbl-relations} is devoted to study equivalence relations lower in the hierarchy, ``just above'' the countable products of countable Borel equivalence relations.We show that for a countable Borel equivalence relation $E$, if $E$ is strongly ergodic with respect to a measure $\mu$ then $E^\N$ is strongly ergodic with respect to $\mu^\N$.Similarly we establish strong ergodicity results between the recently defined $\Gamma$-jumps of Clemens and Coskey \cite{CC19}. In particular, we show that the $\mathbb{F}_2$-jump of $E_\infty$ is strictly above the $\mathbb{Z}$-jump of $E_\infty$, answering a question of Clemens and Coskey.We study a notion of equivalence relations which can be classified by infinite sequences of ``definably countable sets''. In particular, we define an interesting example of such equivalence relation which is strictly above $E_\infty^\N$, strictly below $=^+$, and is incomparable with the $\Gamma$-jumps of countable equivalence relations.The proofs rely on a fine analysis of the very weak choice principles ``every sequence of $E$-classes admits a choice sequence'', for various countable Borel equivalence relations $E$