Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations

Abstract

Assume $\mathsf{ZF}+\mathsf{A}{\mathsf{D}}^{+}+\mathsf{V}=\mathsf{L}(\mathcal{P}(\mathbb{R}))$. Let $\approx$ denote the relation of being in bijection. Let $\kappa \in \mathrm{ON}$ and $⟨E_{\alpha }:\alpha <\kappa ⟩$ be a sequence of equivalence relations on $\mathbb{R}$ with all classes countable and for all $\alpha <\kappa$, $\mathbb{R}/E_{\alpha }\approx \mathbb{R}$. Then the disjoint union ${⨆}_{\alpha <\kappa }\mathbb{R}/E_{\alpha }$ is in bijection with $\mathbb{R}\times \kappa$ and ${⨆}_{\alpha <\kappa }\mathbb{R}/E_{\alpha }$ has the Jónsson property. Assume $\mathsf{ZF}+\mathsf{A}{\mathsf{D}}^{+}+\mathsf{V}=\mathsf{L}(\mathcal{P}(\mathbb{R}))$. A set $X\subseteq [{\omega }_1{]}^{<{\omega }_1}$ has a sequence $⟨E_{\alpha }:\alpha <{\omega }_1⟩$ of equivalence relations on $\mathbb{R}$ such that $\mathbb{R}/E_{\alpha }\approx \mathbb{R}$ and $X\approx {⨆}_{\alpha <{\omega }_1}\mathbb{R}/E_{\alpha }$ if and only if $\mathbb{R}\bigsqcup {\omega }_1$ injects into $X$. Assume $\mathsf{AD}$. Suppose $R\subseteq [{\omega }_1{]}^{\omega }\times \mathbb{R}$ is a relation such that for all $f\in [{\omega }_1{]}^{\omega }$, $R_f=\{x\in \mathbb{R}:R(f,x)\}$ is nonempty and countable. Then there is an uncountable $X\subseteq {\omega }_1$ and function $\Phi :[X{]}^{\omega }\to \mathbb{R}$ which uniformizes $R$ on $[X{]}^{\omega }$: that is, for all $f\in [X{]}^{\omega }$, $R(f,\Phi (f))$. Under $\mathsf{AD}$, if $\kappa$ is an ordinal and $⟨E_{\alpha }:\alpha <\kappa ⟩$ is a sequence of equivalence relations on $\mathbb{R}$ with all classes countable, then $[{\omega }_1{]}^{\omega }$ does not inject into ${⨆}_{\alpha <\kappa }\mathbb{R}/E_{\alpha }$.