By "small cardinality" I mean a Scott cardinality onto which $\mathbb{R}$ surjects ($0$ isn't interesting here). $\mathcal{R}=(\mathbb{R};+,\times,\mathbb{Z})$ is the field of real numbers expanded by a predicate naming the integers, and $\Sigma$ is the language of $\mathcal{R}$ plus a fresh unary relation symbol $D$ and binary relation symbol $E$. Below we work in $\mathsf{ZF+DC+AD_\mathbb{R}}$.
For a (not-necessarily-well-orderable) small cardinality $\kappa$, we say $\kappa$ is detectable iff there is some $\Sigma$-formula $\varphi$ such that the following are equivalent for each set $X\subseteq\mathbb{R}$ and each equivalence relation $R$ on $X$:
$\vert X/R\vert=\kappa$.
$\varphi$ holds in the expansion of $\mathcal{R}$ gotten by interpreting $D$ as $X$ and $E$ as $R$.
The variety of small cardinalities is enormous; see, for example, Woodin's The cardinals below $\vert[\omega_1]^{<\omega_1}\vert$. On the other hand, the theory $\mathsf{ZF+DC+AD_\mathbb{R}}$ is in my experience "relatively complete" for statements about not-too-big combinatorial objects. Broadly I'm interested in what this theory can, or cannot, prove about detectability. To keep things reasonably concrete, the following seems like a good test question:
Which of the four distinct cardinalities of $$\mathbb{R},\mathbb{R}\cup\omega_1,\mathbb{R}\times\omega_1,[\omega_1]^\omega$$ are detectable?
(That these are distinct is Woodin's Lemma 17, for which $\mathsf{AD_\mathbb{R}}$ is unnecessary. Woodin also includes $\omega_1$, but the detectability of $\omega_1$ is provable in $\mathsf{ZF}$ alone by an easy argument.) An earlier question of mine focused on the situation in $\mathsf{ZFC}$, and I don't have any good sense about how the relevant arguments transfer (or don't) from one setting to the other.
