Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

Question

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an axiom that there are no inaccessable cardinals, or that there is exactly one inaccessible cardinal). In particular, the examples from the referenced question/answer place an arbitrary limit on the height of the cumulative hierarchy, where any one of the suggested 'new' axioms could just be replaced by another axiom specifying an even larger inaccessable cardinal.

So I'm wondering if it can be shown that there is a 'largest' axiomatization, in the sense that no categorical extension of ZFC2 has a larger model (or if the opposite can be shown i.e. that there is no largest)?

Of course, if we place arbitrary restrictions on large cardinals in our meta-theory, we could trivially make this question true (e.g. by assuming there is exactly one inaccessible in the meta-theory). A reasonable set of metatheory assumptions for the purpose of this question would be ZFC + large cardinal axioms i.e. assuming that relevant large cardinals do exist (if necessary). I'm assuming standard (e.g. non-Henkin) semantics for 2nd order logic.

Also, my understanding is that the 2nd order Lowenheim number does at least provide an upper bound on the size of the models associated with categorical extensions of ZFC2 [2], and per this question [3], it sounds like the Lowenheim bound would not be any larger for nth order logic compared to 2nd order logic.

[1] https://math.stackexchange.com/questions/317729/what-axioms-need-to-be-added-to-second-order-zfc-before-it-has-a-unique-model-u

[2] http://www.logic.math.helsinki.fi/people/jouko.vaananen/JV96.pdf

[3] Is the lowenheim-skolem number of nth order logic larger than the corresponding number for 2nd order logic

Answer 1

$\newcommand\ZFC{\text{ZFC}}$Assume that $T$ is a categorical extension of second-order set theory $\ZFC_2$. So it has a unique model, which by Zermelo's theorem must be of the form $V_\kappa$ for some inaccessible cardinal $\kappa$.

Let $T^+$ be the theory asserting $\ZFC_2+$ there is a unique model of $T$, and no inaccessible cardinals above the size of that model. It follows (in the theory $\ZFC+$ there is a model of $T$ and a larger inaccessible cardinal) that the unique model of $T^+$ is $V_\delta$, where $\delta$ is the smallest inaccessible cardinal above $\kappa$. So $T$ was not largest in the sense you described. So there can be no such largest theory.

For example, if $T$ is the theory asserting $\ZFC_2+$ there is no inaccessible cardinal, then $T^+$ asserts that there is exactly one inaccessible cardinal, and $T^{++}$ asserts that there are exactly two inaccessible cardinals, and so on. So these theories are the theories that you seem already to have known about.

Notice that I am not placing any anti-large cardinal impositions in the meta-theory, only in the theory $T^+$, which I am considering but not asserting. Any much stronger large cardinal theory (asserting the existence of much larger large cardinals than a $T$-cardinal and one more inaccessible) will agree that $T^+$ is categorical, with a larger model than the unique model of $T$. So this argument seems to agree with your intentions for the large cardinal meta-theory.