Equivalence of generics

Abstract

Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, with particular focus on Cohen and random forcing. We prove, amongst other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, while the latter is neither amenable nor treeable.

18 pages. We have made minor stylistic changes and corrected an error in the statement of Lemma 3.5, now appearing as Lemmas 3.5 and 3.6