In several set theories, among which Quine's NF is one of the best known and most extensively
investigated, the existence of a Universal set V can be proved. V is the set of all sets-indeed
all "proper classes", including V itself, are elements of V. Although it might be considered
"meaningless" to ask what the "cardinal number" of V is, there are some questions of this type
which one could ask. As far as I know, the consistency of NF relative to ZF is still an open
question. But have any of the well known "large cardinal axioms" (assuming that they can be 
expressed in NF) been proved to be inconsistent with NF? If so, this might throw some light
on the question of how "large" V really is.



A question about Universal sets.
I believe that not much is known about NF itself and large cardinals - keep in mind that NF disproves the axiom of choice (Specker, 1953), and many large cardinal notions are not robustly defined in the absence of choice, in the sense that definitions equivalent over ZFC become non-equivalent without AC.


The system NFU (=NF + urelements), however, is much better behaved. It is known to be consistent - in fact, consistent relative to PA - and NFU does not disprove choice. (Although, having urelements, a universal set, and choice leads to the interesting situation in which the powerset of the universe is strictly smaller than the universe itself.) Over NFU+AC, we can attempt to define at least the smaller large cardinals in much the usual way.


That said, I don't' know anything about large cardinals in NFU, but I'd look at work of Ali Enayat (for example, <http://academic2.american.edu/~enayat/Slides%20of%20Talks/Cambridge%20Slides.pdf>) for starters.


