A type-theoretical Curry paradox and its solution

Abstract The Curry–Howard correspondence, according to which propositions are types, suggests that every paradox formulable in natural deduction has a type-theoretical counterpart. I will give a purely type-theoretical formulation of Curry’s paradox. On the basis of the definition of a type $\Gamma (A)$, Curry’s reasoning can be adapted to show the existence of an object of the arbitrary type A. This is paradoxical for several reasons, among others that A might be an empty type. The solution to the paradox consists in seeing that $\Gamma (A)$ is not a well-defined type.