System U is defined^[2]: 352 as a pure type system with

• three sorts ${\displaystyle \{\ast ,\square ,\triangle \}}$ ;
• two axioms ${\displaystyle \{\ast :\square ,\square :\triangle \}}$ ; and
• five rules ${\displaystyle \{(\ast ,\ast ),(\square ,\ast ),(\square ,\square ),(\triangle ,\ast ),(\triangle ,\square )\}}$ .

System U⁻ is defined the same with the exception of the ${\displaystyle (\triangle ,\ast )}$  rule.

The sorts ${\displaystyle \ast }$  and ${\displaystyle \square }$  are conventionally called “Type” and “Kind”, respectively; the sort ${\displaystyle \triangle }$  doesn't have a specific name. The two axioms describe the containment of types in kinds (${\displaystyle \ast :\square }$ ) and kinds in ${\displaystyle \triangle }$  (${\displaystyle \square :\triangle }$ ). 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. ${\displaystyle b:\mathrm {Bool} }$  is read as “b is a boolean”) or a (dependent) function type (e.g. ${\displaystyle f:\mathrm {Nat} \to \mathrm {Bool} }$  is read as “f is a function from natural numbers to booleans”).
2. ${\displaystyle \ast }$  is the sort of all such types (${\displaystyle t:\ast }$  is read as “t is a type”). From ${\displaystyle \ast }$  we can build more terms, such as ${\displaystyle \ast \to \ast }$  which is the kind of unary type-level operators (e.g. ${\displaystyle \mathrm {List} :\ast \to \ast }$  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. ${\displaystyle \square }$  is the sort of all such kinds (${\displaystyle k:\square }$  is read as “k is a kind”). Similarly we can build related terms, according to what the rules allow.
4. ${\displaystyle \triangle }$  is the sort of all such terms.

The rules govern the dependencies between the sorts: ${\displaystyle (\ast ,\ast )}$  says that values may depend on values (functions), ${\displaystyle (\square ,\ast )}$  allows values to depend on types (polymorphism), ${\displaystyle (\square ,\square )}$  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^[2]: 353 (where k denotes a kind variable)

${\displaystyle \lambda k^{\square }\lambda \alpha ^{k\to k}\lambda \beta ^{k}\!.\alpha (\alpha \beta )\;:\;\Pi k:\square ((k\to k)\to k\to k)}$ .

This mechanism is sufficient to construct a term with the type ${\displaystyle (\forall p:\ast ,p)}$  (equivalent to the type ${\displaystyle \bot }$ ), 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.

• 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.
• Hurkens, Antonius J. C. (1995). Dezani-Ciancaglini, Mariangiola; Plotkin, Gordon (eds.). A simplification of Girard's paradox. Second International Conference on Typed Lambda Calculi and Applications (TLCA '95). Edinburgh. pp. 266–278. doi:10.1007/BFb0014058.