Equivalence of generics

TL;DR

This paper studies the complexity of an equivalence relation on generic filters over a countable transitive model of set theory, focusing on Cohen and random forcing, revealing their distinct structural properties.

Contribution

It introduces a natural equivalence relation on generic filters and analyzes its complexity for Cohen and random forcing, showing their different structural behaviors.

Findings

Cohen forcing equivalence relation is an increasing union of hyperfinite Borel relations.

Random forcing equivalence relation is neither amenable nor treeable.

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.