Having a universal set [in naive set theory] is a sufficient condition for Russel's paradox. See Naive Set Theory[1] bottom of page 6. This is why we can have no Universe in any consistent set theory.
Edit: @mafribe makes the point that there are some set theories that can still have universal sets by culling other features that ZF-style set theories have. I was mostly referring to ZF-style set theory (hence my citation). Indeed, one could even make a ZF-style set theory paraconsistent and still have Universal sets.
Having a universal set is a sufficient condition for Russel's paradox.
That's not true. There are set-theories, e.g. Quine's NF [1] which allow universal sets, and other things like the set of all ordinals, that are forbidden in ZF-style set-theories. The problem in ZF is caused by unlimited comprehension. NF circumvents this by restricting comprehension. Tom Forster [2] has written a great deal about set theories with universal sets, including the wonderful [3]. He makes the historical point that set theory was born with universal sets.
True, I was mostly referring to ZF-style set theories (which is what the thread is mainly about). Your point could even be extended by saying that there are proofs for a paraconsistent ZF with a universal set.
Your [3] link doesn't work by the way, I'm interested in reading Forster!
Edit: @mafribe makes the point that there are some set theories that can still have universal sets by culling other features that ZF-style set theories have. I was mostly referring to ZF-style set theory (hence my citation). Indeed, one could even make a ZF-style set theory paraconsistent and still have Universal sets.
[1] http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Al...