Perfect set dichotomy theorem in generalized Solovay model

TL;DR

This paper proves the perfect set dichotomy theorem in the Solovay model and its generalizations for uncountable cardinals, extending classical descriptive set theory results to these models.

Contribution

It establishes the perfect set dichotomy in the Solovay model and its uncountable generalizations, and proves a three-element basis theorem for uncountable linear orders.

Findings

Perfect set dichotomy holds in the Solovay model.

The theorem extends to models for uncountable regular cardinals.

A three-element basis theorem for uncountable linear orders is established.

Abstract

We prove that the perfect set dichotomy theorem holds in the Solovay model $V ((\omega^\omega)^{V[G]})$. 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 $\mu$ and show the perfect set dichotomy theorem for $\mu^\mu$ 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.