In this post, Joel David Hamkins (Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2, URL (version: 2015-05-10): <https://mathoverflow.net/q/206178>) answers a questions about categorical theories extending second-order ZFC. 


The way he obtains the categoricity of a theory T is by adding to ZFC2 axioms that make reference to models of T, i.e., ZFC2 +"there is a unique model of T, and no inaccessible cardinals above the size of that model", thus relying on Zermelo's theorem that the cardinality of any model of ZFC2 is an inaccessible cardinal. 


My question is whether there is a way of making a theory categorical that doesn't rely on Zermelo's theorem, and thus might work for theories that were *not* extensions of ZFC2. 


More specifically, I want to know whether the union of $\omega$-many $V\_{\kappa}$ for $\kappa$ inaccessible (i.e. the countably-infinite union of domains of models of ZFC2) could be categorically characterized. Clearly, the union is not itself a model of ZFC2 since it has countable cofinality. Hence, we cannot use the "extension of ZFC2" method mentioned above. 



Categorical set theories that are not extensions of second-order ZFC
Answering your more specific question: yes, it$^\*$ can be so characterized, in the following way:


* First, start with the axioms of second-order Zermelo set theory with choice (without Replacement). Note that $V\_\mu$ is a model of this if $\mu$ is the first limit of inaccessibles - indeed, as long as $\lambda$ is a limit ordinal greater than $\omega$ we have that $V\_\lambda$ satisfies second-order Zermelo set theory with choice.
* Next, add "For every ordinal $\alpha$ there is an inaccessible cardinal $>\alpha$."
* EDIT: Now, add "Every set is contained in a set model of ZFC$\_2$." Note that since ZFC$\_2$ has only finitely many axioms, this *is* in fact expressible by a single second-order sentence. This axiom gives "local replacement" - in particular, it implies that $V\_\alpha$ exists for each ordinal $\alpha$ in the model.
* Finally, add "There is no ordinal which is a limit of inaccessibles."


This characterizes $V\_\mu$ up to isomorphism. 




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$^\*$Or rather, the union of the **first** $\omega$-many $V\_\kappa$s with $\kappa$ inaccessible is so characterizable; in general, if $(\kappa\_i)\_{i\in\omega}$ is an increasing sequence of inaccessibles there is of course no reason to believe that $\bigcup\_{i\in\omega} V\_{\kappa\_i}=V\_{\sup\{\kappa\_i:i\in\omega\}}$ is so characterizable.


