An example for a set $\mathcal S$ in which the whole set is an element

Question

Can you suggest an example for a set $\mathcal S$ in which the whole set is an element. That is $\mathcal S \in \mathcal S$ (other than "the collection of all sets in the universe")

Answer 1

No set is an element of itself in ZFC, the standard system of set theory. See here.

If you just want a set which is an element of a different set you could consider $\emptyset$, which is a set and also an element of $1 = \{\emptyset\}$.

Answer 2

While (as nbritten said) the usual axioms of set theory do not allow self-containing sets (in particular, they do not allow a universal set), there are alternative set theories which do permit such things. These include ZFC-variants gotten by replacing the axiom of foundation/regularity with an "antifoundation" axiom, as well as very non-ZFC flavored theories like Quine's NF and its variants and positive set theories like GPK$_\infty^+$.

The former (the ZFC-like theories) do not permit a universal set. Instead, self-containing sets in such theories are much "smaller," intuitively speaking. The standard example is a Quine atom - this is a set satisfying $a=\{a\}$. If we replace foundation with Aczel's antifoundation axiom, we get a theory proving the existence of exactly one Quine atom; if we use Boffa's antifoundation axiom instead, we get many (indeed, a proper class of) distinct Quine atoms.

Many of the latter (the non-ZFC-like theories) - indeed, most that I'm aware of - allow "big" sets like the universal set; they also allow more mysterious things like the "co-Russell set" of all sets which do contain themselves (whose self-containing status seems unclear).