In Per Martin-Löf (1998) "An Intuitionistic Theory of Types" in G Sambin and JM Smith (eds) Twenty-five years of constructive type theory Clarendon Press (original work written 1972 but unpublished), at p. 133, Martin-Löf says that a universe $V$ which is the type of all types, including itself, so $V : V$ (or in his notation, $V \in V$) is inconsistent because of Girard's paradox.
Martin-Löf goes on to give a proof that where a type $U$ is defined as the type of all orderings without infinite descending chains, there is an ordering on $U$ itself which both has and does not have infinite descending chains, which is paradoxical. I understand the proof itself, but the problem is that I do not understand why the proof depends on $V$ being the type of itself. Any help would be greatly appreciated.