TITLE: What is wrong with sets like $\{a,\{a\},\{\{a\}\},\ldots\}$
QUESTION [7 upvotes]: 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:
$\{a,\{a\},\{\{a\}\},\{\{\{a\}\}\},\ldots\}$
$\{a,\{a\},\{a,\{a\}\},\{a,\{a\},\{\{a\}\}\},\ldots\}$
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?

REPLY [5 votes]: You have it backwards, those sets are perfectly fine. In fact, the ordinal numbers are a hugely important class of sets of the form of the second example.
The axiom of regularity says that there are no infinite descending chains under inclusion, i.e. that there do not exist infinitely many $S_i$ such that $\cdots \in S_{i+2}\in S_{i+1}\in S_i\in\cdots\in S_1$. It is a consequence of the axiom of regularity that no set is an element of itself.
The axiom of regularity effective prevents infinitely nested sets like $\cdots\{\{a\}\}\cdots$. If we look at the example in the title, the issue at stake here is the existence of the limit of the sequence defined by the elements of the sets. That limit does not correspond to a set