I have that power. And yes, I do know about the barber pair of docks. I also know the resolutions to it.
The whole realization that you could describe things that were not sets, and the whole need to verify that what looked like a set was a set was big news to mathematicians / logicians, as well as the general "don't define a set by negative parameters". Basically, unless you know the whole universe of objects, a "what's not in" is not well defined.
Modern set theory and axioms are ... confusing, at best. I have read the sections on Wikipedia several times, and I still don't get it. But I also know that it's only one of several equivalent systems, and not all of them require an infinite set of axioms to define sets.
Finite universes are trivial. But those can't describe math. It's the whole Godel issue.