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:352 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:353 (where k denotes a kind variable)
This mechanism is sufficient to construct a term with the type , which implies that every type is inhabited. By the Curry–Howard correspondence, this is equivalent to all logical propositions being provable, which makes the system inconsistent.
- Girard, Jean-Yves (1972). "Interprétation fonctionnelle et Élimination des coupure de l'arithmétique d'ordre supérieur" (PDF).
- Sørensen, Morten Heine; Urzyczyn, Paweł (2006). "Pure type systems and the lambda cube". Lectures on the Curry–Howard isomorphism. Elsevier. ISBN 0-444-52077-5.
- Barendregt, Henk (1992). "Lambda calculi with types". In S. Abramsky, D. Gabbay and T. Maibaum. Handbook of Logic in Computer Science. Oxford Science Publications. pp. 117–309.
- Coquand, Thierry (1986). "An analysis of Girard's paradox". Logic in Computer Science. IEEE Computer Society Press. pp. 227–236.