Equivalence of generics
Logic
2024-07-22 v2
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.
Cite
@article{arxiv.1810.04704,
title = {Equivalence of generics},
author = {Iian B. Smythe},
journal= {arXiv preprint arXiv:1810.04704},
year = {2024}
}
Comments
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