 Second-order theories can typically have just one model up to isomorphism. Thus if we investigate the class of models of a second-order sentence, the class may very well consist of just one model, up to isomorphism. With just one model one cannot really develop a model theory. One ends up investigating the one and only model and there are no compelling reasons to use logic in the study of it. If the one and only model is an algebraic structure, one should use algebra to study it. If it is an ordered structure, one should use the general theory of ordered structures to study it, and so on. It is unlikely that being second-order characterizable (see  §7.1)  tells much about the structure. It reveals more about second-order logic in general. For example, being able to characterize the structure of natural numbers means that the Compactness Theorem and the Completeness Theorem fail. Being able to characterize the ordered field of real numbers means that both the Downward and the Upward Löwenheim-Skolem Theorems fail. It is from this perspective—to understand second-order logic itself—that it is interesting to know exactly which structures are second-order characterizable up to isomorphism.