Originally asked and bountied at MSE without success:
Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ring of continuous functions from $\mathcal{X}$ to $\mathbb{R}$.
My question is:
Is spatiality definable in $\mathsf{SOL}_{\infty,\infty}$?
Here $\mathsf{SOL}_{\infty,\infty}$ is the "large" infinitary version of second-order logic (with the standard semantics!), in which arbitrary set-sized Boolean combinations and homogeneous blocks of quantifiers are allowed. This is an incredibly powerful logic - already in $\mathsf{FOL}_{\infty,\infty}$ (defined here), each structure is characterized by a single sentence - but it still has limitations. The characterizations of spatiality I'm aware of (see Eric Wofsey's answer to the above-linked question) seem to require more than $\mathsf{SOL}_{\infty,\infty}$, but that logic is strong enough that I have no idea how to begin attacking a negative answer.
I've focused on $\mathsf{SOL}_{\infty,\infty}$ since - for me at least - it seems a plausible upper bound on what could count as an "algebraic" characterization of something.
