I'm not sure if said set exist or whether it is unique, but what name could I use to find more about it and what kind of interesting properties does it have?
Clarification edit: I meant a set $V$ such that $V = V^V$. I am curious about the concept and would like to read more about it; I don't mind which formalism this is in. (From the answers, I understand that this isn't possible in most.)