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The paradigmatic cases I know of theories failing to be categorical are cases that seem tied to the language's inability to distinguish between different infinite cardinalities of the theories domain. For example, 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 second-order $\mathsf{PA}$, however, it has access to resources of set theory in the semantics and so can distinguish between models of different infinite cardinalities (i.e., the Löwenheim-Skolem theorems fail in a second-order setting). So second-order $\mathsf{PA}$ gets to be categorical---its models all have cardinality $\aleph_0$.

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 lose the ability to discriminate between inaccessible cardinals. Again, we have a failure of categoricity related to the background logic's inability to distinguish certain cardinalities [Side question: 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?].

Are all failures of categoricity related to size like the examples above? Are there examples of theories that fail to be categorical in ways wholly unrelated to the size of their models?

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Consider a group axiom, coupled with the axiom "There are exactly six objects". This is not categorical, because there are two non-isomorphic six-element groups (the symmetric group and the cyclic group).

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