I want to introduce you to a formal set theory. (The theory is my own, but it builds on work by many other people.) According to this theory, the sets are arranged into stages, but every set has an absolute complement. In fact, the theory proves that the sets are arranged in well-ordered levels, but constitute a Boolean algebra. The theory puts the empty set and the universal set on a par; just how 'on a par' they are will emerge during the talk. But in the end, this theory suggests that the claim ‘there is no universal set’ is not just an unjustified dogma; it is a dogma without determinate content.
See the seminar webpage http://users.ox.ac.uk/~philmath/pomseminar.html for titles and abstracts of other speakers as available.
Philosophy of Mathematics Seminar Convenors: Daniel Isaacson and Volker Halbach