System U

Short description: Special forms of a typed lambda calculus

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

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

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

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

The rules govern the dependencies between the sorts: [math]\displaystyle{ (\ast,\ast) }[/math] says that values may depend on values (functions), [math]\displaystyle{ (\square,\ast) }[/math] allows values to depend on types (polymorphism), [math]\displaystyle{ (\square,\square) }[/math] 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)

[math]\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) }[/math].

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

Further reading

• Barendregt, Henk (1992). "Lambda calculi with types". Handbook of Logic in Computer Science. Oxford Science Publications. pp. 117–309. ftp://ftp.cs.ru.nl/pub/CompMath.Found/HBK.ps. 
• 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). "A simplification of Girard's paradox". in Dezani-Ciancaglini, Mariangiola; Plotkin, Gordon. Second International Conference on Typed Lambda Calculi and Applications (TLCA '95). Edinburgh. pp. 266–278. doi:10.1007/BFb0014058. https://link.springer.com/chapter/10.1007/BFb0014058.