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 $κ\in \mathrm{ON}$ and $\langle E_α: α< κ\rangle$ be a sequence of equivalence relations on $\mathbb{R}$ with all classes countable and for all $α< κ$, $\mathbb{R} / E_α\approx \mathbb{R}$. Then the disjoint union $\bigsqcup_{α< κ} \mathbb{R} / E_α$ is in bijection with $\mathbb{R} \times κ$ and $\bigsqcup_{α< κ} \mathbb{R} / E_α$ has the Jónsson property. Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. A set $X \subseteq [ω_1]^{<ω_1}$ has a sequence $\langle E_α: α< ω_1\rangle$ of equivalence relations on $\mathbb{R}$ such that $\mathbb{R} / E_α\approx \mathbb{R}$ and $X \approx \bigsqcup_{α< ω_1} \mathbb{R} / E_α$ if and only if $\mathbb{R} \sqcup ω_1$ injects into $X$. Assume $\mathsf{AD}$. Suppose $R \subseteq [ω_1]^ω\times \mathbb{R}$ is a relation such that for all $f \in [ω_1]^ω$, $R_f = \{x \in \mathbb{R} : R(f,x)\}$ is nonempty and countable. Then there is an uncountable $X \subseteq ω_1$ and function $Φ: [X]^ω\rightarrow \mathbb{R}$ which uniformizes $R$ on $[X]^ω$: that is, for all $f \in [X]^ω$, $R(f,Φ(f))$. Under $\mathsf{AD}$, if $κ$ is an ordinal and $\langle E_α: α< κ\rangle$ is a sequence of equivalence relations on $\mathbb{R}$ with all classes countable, then $[ω_1]^ω$ does not inject into $\bigsqcup_{α< κ} \mathbb{R} / E_α$.