Does there exist a set $A$ such that $A=\{A\}$ ?
Edit(Peter LL): Such sets are called Quine atoms.
Naive set theory By Paul Richard Halmos On page three, the same question is asked.
Using the usual set notation, I tried to construct such a set: First
with finite number of brackets and it turns out that after deleting
those finite number of pairs of brackets, we circle back to the original
question.
For instance, assuming $A=\{B\}$, we proceed as follows:
$\{B\}=\{A\}\Leftrightarrow
B=A \Leftrightarrow B=\{B\}$ which is equivalent to the original
equation.
So the only remaining possibility is to have infinitely many pairs of
brackets, but I can't make sense of such set. (Literally, such a set is
both a subset and an element of itself. Further more, It can be shown
that it is singleton.)
For some time, I thought this set is unique and corresponded to $\infty$
in some set-theoretic construction of naturals.
To recap, my question is whether this set exists and if so what
"concrete" examples there are. (Maybe this set is axiomatically
prevented from existing.)
Best Answer
In standard set theory (ZF) this kind of set is forbidden because of the axiom of foundation.
There are alternative axiomatisations of set theory, some of which do not have an equivalent of the axiom of foundation. This is called non-well-founded set theory. See e.g. Aczel's anti-foundation_axiom, where there is a unique set such that $x = \{x\}$.