Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations

Abstract

Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. Let $\approx$ denote the relation of being in bijection. Let $\kappa \in \mathrm{ON}$ and $\langle E_\alpha : \alpha < \kappa\rangle$ 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 $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$ is in bijection with $\mathbb{R} \times \kappa$ and $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$ has the J\'onsson property. Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. A set $X \subseteq [\omega_1]^{<\omega_1}$ has a sequence $\langle E_\alpha : \alpha < \omega_1\rangle$ of equivalence relations on $\mathbb{R}$ such that $\mathbb{R} / E_\alpha \approx \mathbb{R}$ and $X \approx \bigsqcup_{\alpha < \omega_1} \mathbb{R} / E_\alpha$ if and only if $\mathbb{R} \sqcup \omega_1$ 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 $\Phi : [X]^\omega \rightarrow \mathbb{R}$ 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 $\langle E_\alpha : \alpha < \kappa\rangle$ is a sequence of equivalence relations on $\mathbb{R}$ with all classes countable, then $[\omega_1]^\omega$ does not inject into $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$.