  Given (1) and (2) we should have a type of propositions (as in simple type theory), and given (3) this should also be the type of all types. Girard’s paradox shows that one cannot have (1),(2) and (3) simultaneously. Martin-Löf’s choice was to take away (2), restricting type theory to be predicative (and, indeed, the notion of universe appeared first in type theory as a predicative version of the type of all types). The alternative choice of taking away (3) is discussed in Coquand 1986.   