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This builds off of (and reuses some of the examples from) this question about failures of categoricity not related to size.

Second-order $\mathsf{ZFC}$ isn't categorical, but it's almost-categorical, meaning $\mathcal{M} = \langle D, \in \rangle$ is a model of second-order $\mathsf{ZFC}$ iff there is an (strongly) inaccessible cardinal $\kappa$ such that $\mathcal{M}$ is isomorphic to $V_k$. In my understanding, since the Full Semantics for second-order logic relies on set theory (and assuming that set theory is $\mathsf{ZFC}$), both set theory and (consequently) second-order logic lack the ability to discriminate between inaccessible cardinals.

First-order Peano arithmetic fails to be categorical at least because of the Upward Löwenheim-Skolem theorem. The theory's models should have a cardinality of $\aleph_0$, but if it is has a model of $\aleph_0$ it has a model of every infinite cardinality, and so it has larger models as well (the non-standard ones). In a sense, because of the Löwenheim-Skolem theorems, first-order theories can't distinguish infinite cardinalities.

So it seems we have two cases of failure of categoricity related to the background logic's inability to distinguish certain cardinalities.

In the first-order setting we can blame this failure on the Löwenheim-Skolem theorems; what is the analogous result that we blame the inability to distinguish certain cardinalities on in a second-order setting?

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Not every second-order theory has to be categorical. It doesn't work like that, and it has nothing to do with the Lowenheim-Skolem theorems.

The failure of categoricity of $\sf ZFC_2$ is simply because there are no additional axioms of infinity. What do I mean by that? The axiom of infinity is a special axiom in $\sf ZFC$ because when replacing this axiom with its negation we have a theory bi-interpretable with $\sf PA$, meaning it is quite weak compared to set theory with the axiom of infinity.

The axiom of infinity asserts the existence of a set which is infinite, which is larger than what we could have achieved "by hand". Strong axioms of infinity will tell you more information about what sort of large cardinals are there. If you add the axiom that there are no inaccessible cardinals, then the theory is categorical; if you add an axiom saying there are exactly five, then the theory is categorical. And so on.

Where does it break? If you add the statement that there is a proper class of inaccessible cardinals, then this can be true in many points along the way, it means nothing. You can say all sort of things about how "big" is the class of inaccessible cardinals, or inaccessible limits of inaccessible cardinals and so on; but at some point our ability to express these things stops and from there on end it's an open gate.

So the issue here is two-fold:

  1. The lack of strong infinity axioms.

  2. The inability to describe exactly the structure of the large cardinals in the universe. In the case of $\omega$, everything below it is finite, and therefore fully describable with a single axiom. Here, once we allow infinitely many inaccessibles things start to get messy because it depends on what sort of infinite structures we can describe.

But neither of these have much to do with the Lowenheim-Skolem theorems.

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  • $\begingroup$ This is all very helpful. I just have the follow up question of what explains the second issue (our inability to describe exactly the structure of the large cardinals)? Is it that these cardinals aren't second-order characterizable? Something else? What determines the sorts of infinite structures we can describe? $\endgroup$ – Dennis Mar 30 '15 at 20:08
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    $\begingroup$ Well, just assuming you only have inaccessible cardinals already gives you a significant gap. There are only so-many order types you can describe, even as a second-order statement, and your axioms can ostensibly be of the type "the set of inaccessible with this property has order type like this" and conjunctions of that, or similar statements. But at some point you run out of statements, and since we don't know for sure that all large cardinals are inconsistent, you can just consider $\Pi^n_m$-indescribable cardinals of sufficient type. $\endgroup$ – Asaf Karagila Mar 30 '15 at 20:17
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    $\begingroup$ I mean, it's not that we don't have enough power to characterize large cardinals; in most cases we do. It's just that categoricity depends on which axioms you add to the theory, which will mean they describe a unique large cardinal structure (e.g. "there is a single measurable which is the limit of inaccessible cardinals, each of which is a limit of weakly compact cardinals which is a limit of Woodin cardinals, and there are no inaccessible cardinals above this measurable"). $\endgroup$ – Asaf Karagila Mar 30 '15 at 20:20

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