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.
