There are various set theories that have a universal set (which would be absolutely infinite)...one involves using 3-valued logic.
The thing to research regarding this is the axiom of comprehension, as it is called.
If set theory A has a completely unrestricted axiom of comprehension, set theory A will automatically then have a universal set, namely the A-set consisting of all A-sets that equal themselves.
For the axiom of comprehension is a powerful way to "generate" new X-sets from the other X-sets known to exist. The general idea is that a set can be defined in an unambiguous way as being the ensembles all of whose members share a common property. For example, the description "integers having remainder 2 when divided by 3" "generates" a set as described.
If we take an unrestricted axiom of comprehension, then it states something along these lines:
For every description there is a set whose elements fit that description.
(Hmm, I wonder if "grammatical systems" would help here.)
Seems innocent enough, right? Well Russell came along and showed that this unrestricted axiom of comprehension, devised by Cantor, leads to a set theory in which every statement is both true and false; a set theory in which every statement is both true and false is probably not going to be considered interesting at all.
Russell came up with a particularly clever and brutal disproof of the axiom of unrestricted comprehension that Cantor believed was true. This, some speculate, is why Cantor finished his days in an asylum because, deep down, he had lost his faith and lost his way by even acknowledging the possibility that God does not exist. He thought set theory was a mathematical description of God as the ultimate set so big it contains all sets.
If anthropomorphizing God (God to man) is tricky and/or wrong, then going in reverse (man to God) is probably equally as uncertain. Cantor was trying to do this with his set theory, imho.
If only Cantor had lived a bit longer. His successors eventually came up with different set theories in which there is an ultimate set so big it contains all sets.
To Cantor it might have been his goal to prove that God exists but that argument which still happens a lot to this day often boil down to one's definition of the word God, or say undefined if not.
In NFU, the universal set is "generated" by an axiom and there is an axiom of restricted comprehension. In a paper by Skolem, three-valued logic was used to reveal a set theory with an unrestricted axiom of comprehension was, in fact, a theorem!
I am not sure which of these set theories are stronger or weaker than the others (except in the obvious cases).
Imho, Cantor is one of the giants on whose shoulders his successors do rest. I'd have to say he's one of my favorite mathematicians.
