For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is **$\mathcal{X}$-detectable** iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary predicate symbol $U$ such that for all $A\subseteq X$, the expansion of $\mathcal{X}$ gotten by interpreting $U$ as $A\subseteq X$ satisfies $\varphi$ iff $\vert A\vert\ge\kappa$.

For example, $\omega_1$ is $(\omega_1;<)$-detectable since the uncountable subsets of $\omega_1$ are exactly the unbounded ones. By contrast, Alex Kruckman observed that by a result of Robinson no uncountable cardinal is $\mathcal{R}=(\mathbb{R};+,\times)$-detectable.

I'm interested in the expansion $\mathcal{R}_\mathbb{N}:=(\mathbb{R};+,\times,\mathbb{N})$ of $\mathcal{R}$ gotten by adding a predicate naming the natural numbers (equivalently, adding all projective functions and relations). Since we can talk about one real enumerating a list of other reals, $\omega_1$ is $\mathcal{R}_\mathbb{N}$-detectable ("there is no real enumerating all elements of $U$"). More pathologically, if $\mathfrak{c}=2^\omega$ is regular and there is a projective well-ordering of the continuum of length $\mathfrak{c}$ then $\mathfrak{c}$ is $\mathcal{R}_\mathbb{N}$-detectable. So for example it is consistent with $\mathsf{ZFC}$ that $\omega_2$ is $\mathcal{R}_\mathbb{N}$-detectable.

I'm curious whether this type of situation is the only way to get $\mathcal{R}_\mathbb{N}$-detectability past $\omega_1$. There are multiple ways to make this precise, of course. At present the following two seem most natural to me:

Is it consistent with $\mathsf{ZFC}$ that there are at least two distinct regular cardinals $>\omega_1$ which are $\mathcal{R}_\mathbb{N}$-detectable?

Is it consistent with $\mathsf{ZFC}$ that there is a singular cardinal which is $\mathcal{R}_\mathbb{N}$-detectable?

Note that an affirmative answer to either question requires a large continuum, namely $\ge\omega_3$ and $\ge\omega_{\omega+1}$ respectively. Although my primary interest is in *first-order* definability, I'd also be interested in answers for other logics which aren't too powerful (e.g. $\mathcal{L}_{\omega_1,\omega}$).

6more comments