TITLE: Categoricity of second order theories - precisely what does it mean?
QUESTION [9 upvotes]: As far as I am aware, it's been proven that the second order theories of $\mathbb{N}, \mathbb{Z},\mathbb{Q},\mathbb{R}, \mathbb{C}$ are categorical. I am sure that this is the case at least for $\mathbb{N},\mathbb{R}$.
But it's unclear in second order logic what actually is a "set". IE. axioms need to be added for sets. So, given that, what does the categoricity imply? Does it mean that in specific if we consider the sets of first-order ZFC, then the categoricity holds and all models are equivalent?

REPLY [9 votes]: To supplement Asaf's answer, let me observe that there are natural second-order theories whose categoricity is undecidable in ZFC. Specifically, let's take second-order ZFC ("ZFC$_2$") itself! 
It's easy to show in ZFC that any set-sized model of ZFC$_2$ (yes, models are required to be sets already, so this is redundant; but it's worth mentioning explicitly since we do sometimes care about class-sized "models," and when working with second-order logic is often such a time) is (up to isomorphism) of the form $(V_\alpha,\in)$ for $\alpha$ a strongly inaccessible cardinal. This tells us:

It is consistent with ZFC that ZFC$_2$ has no (set-sized) models. This nonexistence statement is just "There is no strongly inaccessible cardinal."
Now make the mild (hehe) assumption that ZFC + "There is exactly one inaccessible" and ZFC + "There are two inaccessibles" are each consistent (obviously the second implies the first, so this is a bit redundant, but what the hey). Then the former proves that ZFC$_2$ has a set-sized model and is in fact categorical, while the latter proves that ZFC$_2$ is not categorical.

The moral I'm trying to get at here is that second-order categoricity (and satisfiability) claims can't be made on their own; we really need a background universe of sets. Of course, every logic has some sort of set-ish commitment, but with second-order logic the overhead is just ridiculous.