> > I'm vaguely aware of some of the several paradoxes of set theory.
Informal accounts of sets (as distinct from set theory) are paradoxical. It was partly in response to that, that set theories were created. ("Partly" because one of the founders--Zermelo--created his set theory to prove a particular theorem.)
> Another question I have is whether the structure of the diagonal > argument doesn't include something like that. People seem happy to > live with these paradoxes,
"Investigate" rather than "live with", I'd say. Unless by "paradox" you just mean something counter-intuitive, say like Banach-Tarski.
> if there's one more I don't suppose that > upsets the apple cart, but I might find it - interesting.
One more what? That infinities come in different sizes may be paradoxical in the sense of being counter-intuitive, but it is not paradoxical in the sense of being contradictory.
-- I have seen elephants paint very competently... but not in Cumbria.