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, focusing on Cohen and random forcing. We prove, among other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, and hence is amenable, while the latter is neither amenable nor treeable.
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Notes
We do not really need M to be countable; for a given partial order \({\mathbb {P}}\), it suffices that \({\mathcal {P}}({\mathbb {P}})\cap M\) is countable in \({\mathbf {V}}\). In particular, we could allow M to be a transitive class in \({\mathbf {V}}\), such as when \({\mathbf {V}}\) is a generic extension of M after sufficient collapsing, or \(M={\mathbf {L}}\) under large cardinal hypotheses. In such cases, only the proofs of Lemmas 2.6 and 2.9 need alteration, instead relying on Theorem 2.16.
Being hyperfinite is a \(\Sigma ^1_2\) property in the codes, but whether it is \(\Sigma ^1_2\)-complete, and thus not absolute for countable models, appears to be unknown.
We would like to thank the anonymous referee for clarification on this point.
As per footnote 1, we may allow \(\omega _1\subseteq M\), provided \(|{\mathcal {P}}({\mathbb {B}})\cap M|=|{\mathcal {P}}({\mathbb {R}})\cap M|\) is still countable in \({\mathbf {V}}\). In this case, M will be \(\mathbf {\Sigma }^1_2\)-correct by Shoenfield absoluteness (Theorem 25.20 in [13]).
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Acknowledgements
I would like to thank Samuel Coskey, Joel David Hamkins, Andrew Marks, and Simon Thomas for many helpful conversations and correspondences that contributed to this work.
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Smythe, I.B. Equivalence of generics. Arch. Math. Logic 61, 795–812 (2022). https://doi.org/10.1007/s00153-021-00813-3
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DOI: https://doi.org/10.1007/s00153-021-00813-3