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