 Which model theory is more appropriate for second-order logic? There is a trade-off between the attractive features of the metatheory of second-order logic with Henkin models, on the one hand, and the expressive power of second-order logic with standard models, on the other. Frege’s own interest in second-order quantification had much to do with its ability to express the concept of the ancestral of a dyadic relation \(R\), which allowed him in turn to define the concept natural number in terms of \(0\) and successor. For details, the reader may consult the entry on  Frege’s Theorem and Foundations of Arithmetic.  There are, in addition, categorical axiomatizations of arithmetic, where a set of sentences is categorical if, and only if, all of its models are isomorphic, which means that there is only one model up to isomorphism. Dedekind (1893) famously proved, for example, that the second-order formulation of the Peano Axioms characterized the structure of the natural number system. There is a similar result for real analysis. For second-order set theory, Zermelo (1930) observed that given two models of second-order Zermelo-Fraenkel set theory, one is isomorphic to a strongly inaccessible initial segment of the other. The entry on  set theory  describes the axioms of set theory. Other examples of the expressive resources afforded by second-order quantification with standard models are discussed in Shapiro (1991). The adoption of a model theory based on Henkin models requires one to surrender all these applications.