it's known that there are non-isomorphic structures that satisfy the same first-order sentences. Likewise it's known (by cardinality arguments) that there are non-isomorphic structures that satisfy the same second-order sentences, and more generally, that safisfy the same nth-order logic sentences, for all n.
TWo questions:
Is it known a concrete example of a complete but non-categorical second-order theory, or n-th order theory (i.e. two structures that satisfy the same second-order sentences, yet not isomorphic)?
Is there a logic that can distinguish between any two non-isomorphic structures (i.e. complete theories of this logic are always categorical)?
Thanks in advance
EDIT: I've found the answer to my second question. By the same cardinality arguments for the second-order case, the set of sentences of the logic must be a proper class. But if we use the "limit" logic of the infinitary logics $\mathcal{L}_{\kappa \kappa}$, we can distinguish between two non-isomorphic structures by using the same argument we use to show that in first-order logic, finite structures can be characterized up to isomorphism by a single first-order sentence (i.e. the sentence says what is the cardinality of the structure, and the behaviour of the non-logical symbols under the structure interpretation.
My first question still remains.