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.
Formal definition.
System U is defined as a pure type system with
System U− is defined the same with the exception of the formula_4 rule.
The sorts formula_5 and formula_6 are conventionally called “Type” and “Kind”, respectively; the sort formula_7 doesn't have a specific name. The two axioms describe the containment of types in kinds (formula_8) and kinds in formula_7 (formula_10). Intuitively, the sorts describe a hierarchy in the "nature" of the terms.
The rules govern the dependencies between the sorts: formula_21 says that values may depend on values (functions), formula_22 allows values to depend on types (polymorphism), formula_23 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 (where "k" denotes a kind variable)
This mechanism is sufficient to construct a term with the type formula_25, 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 type-theoretic analogue of Russell's paradox in set theory.