Intro is dubious 

The intro claims that the problem with a universal set stems from the axiom of comprehension.

But shortly afterwards it points out that the axiom of comprehension causes a problem even for sets which aren't the universal set. So it isn't the universal set which is the problem at all; it's the axiom of comprehension.

With the axiom of comprehension removed (and the axiom of separation substituted for instance), it is not clear why the universal set causes problems.