TITLE: Are there structures in a finite signature that are recursively categorically axiomatizable in SOL but not finitely categorically axiomatizable?
QUESTION [6 upvotes]: Recall that a structure $\mathcal{M} = \langle M, I^\sigma_M \rangle$ in a signature $\sigma$ is categorically axiomatized by a second-order theory $T$ when, for any $\sigma$-structure $\mathcal{N} = \langle N, I^\sigma_N \rangle$, $\langle N, \mathcal{P}(N), I^\sigma_N \rangle \vDash T$ just in case $\mathcal{N}$ is isomorphic to $\mathcal{M}$.
It is fairly easy to find a structure in a finite signature that is categorically second-order axiomatizable but not finitely categorically second-order axiomatizable. Add a single function symbol $f$ to the language of second-order arithmetic, and choose a non-second-order-definable $\zeta: \mathbb{N} \rightarrow \mathbb{N}$. Then consider the theory $T$ that adds to the axioms of second-order arithmetic ($\mathsf{Z}^2$) the sentence $f(\bar{n}) = \overline{\zeta(n)}$ for each natural number $n$, where $\bar{m}$ is the canonical numeral for $m$. (I owe the idea for this example to Andrew Bacon.)
This theory $T$, however, is not recursively axiomatizable. Is there a structure in a finite signature that has a recursive categorical second-order axiomatization but no finite categorical second-order axiomatization?
I believe that it is possible to find a recursively axiomatizable second-order theory $T$ whose spectrum (i.e., the set $\{\kappa \in \mathsf{Card}: \exists \mathcal{M} (\mathcal{M} \vDash T$ and $\vert \mathscr{M} \vert = \kappa)\}$) is shared by no finitely axiomatizable second-order theory, using partial truth predicates. (Consider the theory with $\mathsf{Z}^2$ relativized to some predicate $N$ and $\{$"The cardinality of the non-$N$s is not $\Sigma^1_n$-characterizable"$: n \in \omega\}$.) But I cannot see how to turn this into a categorical theory.

REPLY [2 votes]: There is even an example of a cardinal  $\kappa$ and an r.e. categorical second-order theory $T$ such that for no finitely axiomatized second-order theory $U$, the spectrum of $U$ has $\kappa$ as its least element.
Let $T_0$ be the theory $\mathsf{ZC}_2+\forall x\exists \alpha(x\in V_\alpha)$, here $\mathsf{ZC}_2$ is the second-order version of Zermelo set theory where in the separation scheme we allow formulas without class quantifiers, but with class variables. It is easy to see that the second-order models of this theory are precisely $(V_\alpha,\in)$, for limit $\alpha>\omega$. Further I denote by $\mathcal{L}$ the set of sentences of the language of this theory.
Let $T$ be the extension of $T_0$ by

$\forall \alpha (\exists \varphi\in \mathcal{L}) (((V_\alpha,\in)\models_2\varphi)\land \forall \beta<\alpha ((V_\beta,\in)\not\models_2\varphi))$.
$\varphi\to\exists \alpha ((V_\alpha,\in)\models_2 \varphi)$, where $\varphi$ ranges over $\mathcal{L}$.

Obviously the only second-order model of $T$ is $(V_\alpha,\in)$, where $\alpha$ is the least ordinal such that for any $\varphi\in\mathcal{L}$ if $(V_\alpha,\in)\models \varphi$, then for  $(V_\beta,\in)\models \varphi$, for some $\beta<\alpha$. Clearly, $\alpha>\omega^2$ and hence the cardinality of this model is $\beth_\alpha$, which will be our $\kappa$.
Suppose for a contradiction that there is a second-order sentence $\varphi$ such that the smallest model of $\varphi$ is in the cardinality $\beth_\alpha$. Let $\varphi'$ be the naturally constructed sentence of second-order set theory expressing that there exists a class-model of $\varphi$. By construction $(V_\alpha,\in)\models_2\varphi'$, but for no $\beta<\alpha$ we have $(V_\beta,\in)\models_2\varphi'$, contradiction.