System U

In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies between the sorts). They were both proved inconsistent by Jean-Yves Girard in 1972. This result led to the realization that Martin-Löf's original 1971 type theory was inconsistent as it allowed the same "Type in Type" behaviour that Girard's paradox exploits. System U is defined as a pure type system with three sorts ; two axioms ; and five rules . System U− is defined the same with the exception of the rule. The sorts and are conventionally called “Type” and “Kind”, respectively; the sort doesn't have a specific name. The two axioms describe the containment of types in kinds () and kinds in (). Intuitively, the sorts describe a hierarchy in the nature of the terms. All values have a type, such as a base type (e.g. is read as “b is a boolean”) or a (dependent) function type (e.g. is read as “f is a function from natural numbers to booleans”). is the sort of all such types ( is read as “t is a type”). From we can build more terms, such as which is the kind of unary type-level operators (e.g. is read as “List is a function from types to types”, that is, a polymorphic type). The rules restrict how we can form new kinds. is the sort of all such kinds ( is read as “k is a kind”). Similarly we can build related terms, according to what the rules allow. is the sort of all such terms. The rules govern the dependencies between the sorts: says that values may depend on values (functions), allows values to depend on types (polymorphism), allows types to depend on types (type operators), and so on. The definitions of System U and U− allow the assignment of polymorphic kinds to generic constructors in analogy to polymorphic types of terms in classical polymorphic lambda calculi, such as System F. An example of such a generic constructor might be (where k denotes a kind variable) This mechanism is sufficient to construct a term with the type (equivalent to the type ), which implies that every type is inhabited.