TITLE: Are there any complete, first-order and unstable theories which have non-categorical second-order formulations?
QUESTION [9 upvotes]: Since it's not stable, $PA$ fails at being categorical in a power in the worst possible way, having $2^{\lambda}$ models in any uncountable $\lambda$. But $PA$ regains its categoricity in the move to second order logic, by which I mean, once we replace the induction schema with a single, second-order axiom. Are there natural examples of unstable first-order theories for which categoricity is not recoverable in this way? 
If the restriction to unstable $T$ is too strong, are there natural examples of complete, non-categorical first-order $T$ which are also non-categorical as second-order theories?
I'm struggling to think of such  examples because I don't know of any axiomatizable, semantically complete but non-categorical second-order theory. Also, of course, unlike $PA$, not every theory has a natural second-order extension. 
Thanks.

REPLY [10 votes]: Another spin on this question, that gets around the fact that there are only countably many second order sentences, is to look at categoricity in $L^2_{\kappa,\omega}$ where we allow conjunctions of size less that $\kappa$ and second order quantifiers. Hyttinen, Kangas and Vaananen (see http://www.math.helsinki.fi/logic/people/jouko.vaananen/HytKanVaa.pdf) show that, for suitable cardinals $\kappa$, all models of a complete first order theory $T$ of size $\kappa$ are characterized by and $L^2_{\kappa,\omega}$-sentence if and only if $T$ is
``classifiable" in the sense of Shelah's Main-Gap i.e., $T$ is stable, NDOP, shallow, and NOTOP.