 In so-called standard semantics, sometimes called full semantics, monadic second-order variables range over the entire powerset of the domain; two-place second-order variables range over the entire class of ordered pairs of members of the domain, etc. It can be shown that second-order languages, with standard semantics, can characterize many mathematical notions and structures, up to isomorphism. Examples include the notions of finitude, countability, well-foundedness, minimal closure, and structures like the natural numbers, the real numbers, and the complex numbers. As a result, none of the limitative theorems of classical, first-order logic hold: there is no effective deductive system is both sound and complete, the logic is not compact, and both Löwenheim-Skolem theorems fail. Some, such as Quine [1986], argue that second-order logic, with standard semantics is not really logic, but is a form of mathematics, set theory in particular. For more on this, see Shapiro [1991] and the entry on  higher-order logic,  along with the many references cited there.