I am confused. I think I've read two usages of the word impredicative in type theory:When people talk about the impredicative version of Martin-Lf's type theory, which they say it is inconsistent. Apparently they call it impredicative because of Type : Type.When people talk about the impredicative Prop in CoC. Apparently they call it impredicative because you can quantify over Prop and get Prop stuff.Is it the case? One notion of impredicativity implies the other? Is there no problem (i.e. inconsistency) by assuming Prop impredicative?
Impredicative in type theory
type theory;calculus of constructions