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][1] 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.) [1]: http://dc108.4shared.com/download/YNLXDssM/Halmos_-_Naive_set_theory.djvu?tsid=20100725-105252-871bd162