Classifying invariants for $E_1$: A tail of a generic real

Abstract

Let $E$ be an analytic equivalence relation on a Polish space. We introduce a framework for studying the possible "reasonable" complete classifications and the complexity of possible classifying invariants for $E$, such that: (1) the standard results and intuitions regarding classifications by countable structures are preserved in this framework; (2) this framework respects Borel reducibility; (3) this framework allows for a precise study of the possible invariants of certain equivalence relations which are not classifiable by countable structures, such as $E_1$. In this framework we show that $E_1$ can be classified, with classifying invariants which are $κ$-sequences of $E_0$-classes where $κ=\mathfrak{b}$, and it cannot be classified in such a manner if $κ<\mathbf{add}(\mathcal{B})$. These results depend on analyzing the following sub-model of a Cohen real extension, introduced by Kanovei-Sabok-Zapletal (2013) and Larson-Zapletal (2020). Let $\left $ be a generic sequence of Cohen reals, and define the tail intersection model $$M=\bigcap_{n<ω}V[\left ].$$ An analysis of reals in $M$ will provide lower bounds for the possible invariants for $E_1$. We also extend the characterization of turbulence from Larson-Zapletal (2020) in terms of intersection models.