While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see ), these axioms seem 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 there is a 'largest' axiomatization, in the sense that no categorical extension of ZFC2 has a larger model?
In particular, my current understanding is that the largest model for a categorical 2nd-order theory corresponds to the 2nd-order lowenheim number (which is apparently larger than the first measurable cardinal, if such a cardinal exists ). So it would seem that a natural candidate for an additional axiom would be one that fixes the size of the unique model to be equal to the lowenheim number, e.g. perhaps specifying that the largest inaccessible cardinal corresponds to the one immediately prior to the second-order lowenheim number (possibly with the lowenheim number specified by an encoded description, since I believe the actual cardinal is not currently known?).
Unlike the axiomatizations from the link, the intention is that this would have the advantage that no 'larger' categorical axiomatization could be constructed to replace it. I suppose one potential issue is the possibility that an even higher order logic than 2nd-order could be used to axiomatize an even larger categorical ZFC2 extension; however, my understanding is that given the equi-satisfiability results for 2nd-order logic versus nth-order logic, the lowenheim number shouldn't be larger than the 2nd order value, in which case even the higher order ZFC extensions would not be larger than the second order one (though even if they were it would be interesting to know the maximal second order axiomatization)