I notice that Russell's paradox, Burali-Forti's paradox, and even Cantor's paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking if it wouldn't be a good way to stop the paradoxes, to just prohibit sets containing themselves, via a modification in the axiom scheme of comprehension, probably.
But is there some branch of mathematics, maybe something close to recursion theory, that depends on sets that contain themselves at some level of depth?
Also, is there any other paradox of naive set theory that doesn't depend on sets that contain themselves?
Thanks in advance.