On the type structure of standard ML

Standard ML is a useful programming language with a polymorphic type system and a flexible module facility. One notable feature of the core expression language of ML is that it is implicdy typed: no explicit type information need be supplied by the programmer. In contrast, the module language of ML is explicitly typed; in particular, the types of parameters in parametric modules must be supplied by the programmer. We study the type structure of Standard ML by giving an explicitly-typed, polymorphic function calculus that captures many of the essential aspects of both the core and module language. In this setting, implicitly-typed core language expressions are regarded as a convenient short-hand for an explicitly-typed counterpart in our function calculus. In contrast to the Girard-Reynolds polymorphic calculus, our function calculus is predzcatiw: the type system may be built up by induction on type levels. We show that, in a precise sense, the language becomes inconsistent if restrictions imposed by type levels are relaxed. More specifically, we prove that the important programming features of ML cannot be added to any impredicative language, such as the Girard-Reynolds calculus, without implicitly assuming a type of all types.