Forcing, genericity and CBERS
Filippo Calderoni, Dima Sinapova
TL;DR
The paper addresses the Borel complexity of the generic-equivalence relation $\equiv_{M}^{\mathbb{P}}$ across forcing notions, focusing on smoothness and hyperfiniteness. It introduces a condensation-based criterion to characterize smoothness, and proves hyperfiniteness for Prikry forcing by encoding equality of generic sequences through tail behavior. The authors also explore non-homogeneous forcing where the equivalence is not smooth and develop a mutual-genericity framework linking invariant uniformization to hyperhyperfiniteness, with partial results for Cohen forcing and negative outcomes for Random forcing. Together, these results advance understanding of the union problem and the descriptive-set-theoretic complexity of generic-extensions, with implications for how generics determine when different filters yield the same model.
Abstract
In this paper we continue the study of equivalence of generics filters started by Smythe in [Smy22]. We fully characterize those forcing posets for which the corresponding equivalence of generics is smooth using the purely topological property of condensation. Next we leverage our characterization to show that there are non-homogeneous forcing for which equivalence of generics is not smooth. Then we prove hyperfiniteness in the case of Prikry forcing and some additional results addressing the problem whether generic equivalence for Cohen forcing is hyperfinite.