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 $\{\ast ,◻,\mathrm{△}\}$;
• two axioms $\{\ast :◻,◻:\mathrm{△}\}$; and
• five rules $\{(\ast ,\ast ),(◻,\ast ),(◻,◻),(\mathrm{△},\ast ),(\mathrm{△},◻)\}$.

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

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

The rules govern the dependencies between the sorts: $(\ast ,\ast )$ says that values may depend on values (functions), $(◻,\ast )$ 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 (where k denotes a kind variable)

$\lambda k^{◻}\lambda {\alpha }^{k\to k}\lambda {\beta }^k.\alpha (\alpha \beta ):\mathrm{\Pi }k:◻((k\to k)\to k\to k)$.

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