 The situation changed somewhat when Henkin proved the Completeness Theorem for second-order logic with respect to so-called general models  (§9).  Now it became possible to use second-order logic in the same way as first order logic, if only one keeps in mind that the semantics is based on general models. The ability of second-order logic to characterize mathematical structures up to isomorphism does not extend to general models (except in the form of internal categoricity—see  §10).  Any second-order theory with an infinite general model has general models of all infinite cardinalities, and therefore cannot be categorical in the context of general models. We shall discuss general models and the related Henkin models in  §9.