Classifying Invariants for E1: 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; and (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 $\mathrm{\kappa }=\mathfrak{b}$, and it cannot be classified in such a manner if $\mathrm{\kappa }<\mathbf{add}(\mathcal{B})$. These results depend on analyzing the following submodel of a Cohen-real extension. Let $⟨c_n:n<\mathit{\omega }⟩$ be a generic sequence of Cohen reals, and define the tail intersection model

$$M=\bigcap _{n<\mathit{\omega }}V[⟨c_m:m\ge n⟩].$$

An analysis of reals in M will provide lower bounds for the possible invariants for $E_1$. We also extend a characterization of turbulence, due to Larson and Zapletal, in terms of intersection models.