Throughout, work in $\mathsf{ZF+DC+AD_\mathbb{R}}$.
Given a theory $T$, let $[T]$ be the set of isomorphism types of models of $T$ with domain $\subseteq\omega$. This question is an outgrowth of this old MSE question asking about $\vert[\mathsf{Lin}]\vert$, where $\mathsf{Lin}$ is the theory of linear orders, which was answered by Andres Caicedo by $(1)$ showing that there is an injection $[\mathsf{Lin}]\rightarrow[\omega_1]^{<\omega_1}$ and $(2)$ quoting a result of Woodin showing that $\vert[\omega_1]^{<\omega_1}\vert$ is extremely large (see Woodin's paper The cardinals below $\vert[\omega_1]^{<\omega_1}\vert$).
I'm more generally interested in the question of which cardinalities (not necessarily well-orderable!) are of the form $\vert[T]\vert$ for a first-order theory $T$ in a countable language. Call such cardinalities model-theoretic. There are a number of questions which could be asked here; I've picked one in particular and I hope it's neither trivial nor impossible.
Assume (the absolute version of) Vaught's Conjecture. Is $\omega_1+2^{\aleph_0}$ a model-theoretic cardinal?
The need for VC here is that, since $T$ is not required to be complete, the model-theoretic cardinalities are trivially closed under addition. Since $2^{\aleph_0}$ is obviously a model-theoretic cardinality, a counterexample to VC would give a positive answer to the question for silly reasons. On the other hand, it is easy to show that $\omega_1$ is a model-theoretic cardinal iff VC fails; this is because if $\vert [T]\vert=\omega_1$ then $T$ is WLOG complete, as any incomplete theory has either countably many or continuum-many distinct completions. So, put another way, I'm asking whether the model-theoretic-ness of $\omega_1+2^{\aleph_0}$ is equivalent to VC.
