TITLE: $x\in \{\{\{x\}\}\}$ or not?
QUESTION [5 upvotes]: I wonder if we can we say $x\in \{\{\{x\}\}\}$?  
In one viewpoint the only element of $\{\{\{x\}\}\}$ is $\{\{x\}\}$. In the other viewpoint $x$ is in $\{\{\{x\}\}\}$, for example all people in Madrid are in Spain.

REPLY [3 votes]: As many here explained, this certainly don't have to be true, and assuming the axiom of regularity this can be disproved quite easily.
I will remark that it is consistent that the axiom of regularity fails, and that this situation does happen. Of course it would mean that $x=\{\{x\}\}$, but it is a plausible scenario (which, for example, follows from certain "anti-foundation" axioms).
But perhaps one thing missing from the answers here is the mention of a transitive closure. Given a set $X$ we define the transitive closure of $X$ as the set of all elements which can be "reached" by taking elements of elements and so on. The transitive closure of $X=\{\{\varnothing\}\}$ would be to take $X\cup\{\varnothing\}\cup\varnothing=\{\varnothing,\{\varnothing\}\}$, for example.
So while $x$ need not be an element of $\{\{\{x\}\}\}$, it is an element of the transitive closure of $\{\{\{x\}\}\}$.