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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.

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Sure. Let $\mathcal{N}=(\mathbb{N};+,\times)$ be the standard model of (first-order) arithmetic and let $R\subseteq\mathbb{N}^2$ be the binary relation given by $(a,b)\in R$ iff the $a$th $\Sigma^1_b$ sentence (in arithmetic, according to some standard enumeration) is true in $\mathcal{N}$. That is, $R$ codes the second-order theory of $\mathcal{N}$.

The structure $(\mathbb{N};+,\times, R)$ is second-order recursively axiomatizable: we use a single second-order sentence to fix the $\{+,\times\}$-reduct, and then recursively say that each column of $R$ codes the appropriate set. By the non-collapsing of the second-order hierarchy, this can't be done by a single second-order sentence.

Note that this trick can be generalized to a wide variety of logics. For logics like $\mathcal{L}_{\kappa,\lambda}$ which don't "fit" into $\mathbb{N}$ in a reasonable way, we need to $(i)$ shift to a different "base" and $(ii)$ possibly adopt a different notion of "recursive," but an analogous result should hold.

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