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 JeanYves Girard in 1972.^{[1]} This result led to the realization that MartinLöf's original 1971 type theory was inconsistent as it allowed the same "Type in Type" behaviour that Girard's paradox exploits.
YouTube Encyclopedic

1/3Views:2 96921 5691 488

System U aujourd'hui

FreedomRail Closet Organization System Install  Glenbrook U

E36 BMW 328is UUC System U Exhaust
Transcription
Formal definition
System U is defined^{[2]}^{: 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 typelevel 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.
Girard's paradox
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^{[2]}^{: 353 } (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. By the Curry–Howard correspondence, this is equivalent to all logical propositions being provable, which makes the system inconsistent.
Girard's paradox is the typetheoretic analogue of Russell's paradox in set theory.
References
 ^ Girard, JeanYves (1972). "Interprétation fonctionnelle et Élimination des coupures de l'arithmétique d'ordre supérieur" (PDF).
 ^ ^{a} ^{b} Sørensen, Morten Heine; Urzyczyn, Paweł (2006). "Pure type systems and the lambda cube". Lectures on the Curry–Howard isomorphism. Elsevier. doi:10.1016/S0049237X(06)800157. ISBN 0444520775.
Further reading
 Barendregt, Henk (1992). "Lambda calculi with types". In S. Abramsky; D. Gabbay; T. Maibaum (eds.). 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.