Perfect set dichotomy theorem in generalized Solovay model
Hiroshi Sakai, Toshimasa Tanno
TL;DR
This work proves that the perfect set dichotomy holds for all equivalence relations on the reals in the Solovay model and extends the result to the generalized Solovay model for μ^μ, using a niceness-based forcing framework to overcome quotient-forcing obstacles. It develops a comprehensive approach combining Coll-based forcing, definability arguments, and case-splitting to derive either well-orderability of quotient spaces or existence of E-inequivalent perfect sets, with analogous results at higher cardinals. The paper further establishes a three-element basis theorem for uncountable linear orders in Solovay-type models, deducing the absence of Aronszajn and Suslin lines under certain large-cardinal assumptions and AD-related settings. Overall, it provides a robust, general methodology for transfer of regularity properties and dichotomies from the classical Solovay model to its higher-cardinal generalizations and derives strong structural consequences for uncountable linear orders.
Abstract
We prove that the perfect set dichotomy theorem holds in the Solovay model $V ((ω^ω)^{V[G]})$. Namely, for every equivalence relation $E$ on $\mathbb{R}$, either $\mathbb{R}/E$ is well-orderable or there exists a perfect set consisting of $E$-inequivalent reals. Furthermore we consider a generalization of the Solovay model for an uncountable regular cardinal $μ$ and show the perfect set dichotomy theorem for $μ^μ$ also holds in that model. We establish the three element basis theorem for uncountable linear orders in the Solovay model for a weakly compact cardinal, in a general form covering the uncountable case.