System U

Formal definition

System U is defined as a pure type system with

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.

1. All values have a type, such as a base type (e.g. b : B o o l is read as “b is a boolean”) or a (dependent) function type (e.g. f : N a t → B o o l is read as “f is a function from natural numbers to booleans”).
2. ∗ is the sort of all such types ( t : ∗ 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. L i s t : ∗ → ∗ 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.
3. ◻ is the sort of all such kinds ( k : ◻ is read as “k is a kind”). Similarly we can build related terms, according to what the rules allow.
4. △ 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^:353 (where k denotes a kind variable)

This mechanism is sufficient to construct a term with the type ( ∀ p : ∗ , p ) = ⊥ , 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.