I know pretty much nothing about set theory beyond first year undergraduate maths, so apologies if this is a stupid question.
The axiom of regularity in ZFC as I have understood it would forbid the existence of the following sets:
Why should such sets not exist? Can it be shown that their existence leads to a contradiction? I know about Russell's paradox but isn't forbidding sets like the above a bit overkill, as they seem not to lead to paradoxes like Russell's set does.
Also is it possible to create a set of axioms that allow the maximum number of sets to exist such that they do not lead to a contradiction?