In Asaf's answer to this question:
he confirmed that it is possible to categorically characterize a model of second-order ZFC by adding an axiom stating that ``There are exactly $\alpha$ inaccessible cardinals'', where $\alpha$ is an ordinal parameter.
Since $\alpha$ is an ordinal not a cardinal, presumably the axiom would really be asserting existence of a bijection from $\alpha$ to the inaccessibles in the model.
I have been trying to prove that second order ZFC + ``There's a bijection from $\alpha$ to the inaccessibles'' is categorical for all $\alpha$. Intuitively, it seems like it should be easy; we know from Zermelo that for any two models of ZFC2, either they are isomorphic, or one is isomorphic to a proper initial segment of the other. Moreover, any model of ZFC has height equal to an inaccessible.
I have been trying to prove this by showing that you get a contradiction by supposing that the heights of the models differ; if I can show that they have the same height, isomorphism follows by another theorem of Zermelo.
But I just can't get the right contradiction - I've tried violations of injectivity and surjectivity of the functions in each model, but since the bijections don't guarantee any sort of order on the range, I can't use the standard methods to show that the "shorter" model wouldn't have enough inaccessibles to satisfy the bijection.
Am I missing something obvious here?