Notation for the set of all sets I know that it doesn't make sense mathematically (Russell's paradox), but is there any nice way or notation to express the set of all sets?  
Edit: I want the set to contain itself, even though it might break some definition.
 A: In set theory the class of all sets is often denoted $V$.
A: This concept is usually referred to as a "class". This concept is formalized in Von Neumann–Bernays–Gödel set theory which is essentially the usual Zermelo–Fraenkel set theory (ZF) + classes. The basic rule is a class is just some predicate. A set is a predicate restricted to a set. We also allow ourselves the axioms of ZF to define sets to avoid having a self-referential definition.
Every set is a class because if we have a set $\{x \in A : \phi(x)\}$ (i.e. the predicate $\phi$ restricted to the set $A$) then we have a class $\{x : x \in A \wedge \phi(x)\}$. What distinguishes a set from a "proper class" (a class that is not a set) is that sets are allowed to be members of other classes. That is, for a set $A$ we are allowed to talk about $A \in B$ where $B$ is a class. The class of all sets may be defined as
$$ C = \{A : A = A\}. $$
Russel's paradox tells us that the statement $C \in C$ leads to a contradiction. Since it doesn't make sense to talk about whether or not $C$ is a member of something, that makes $C$ a proper class.
A: In category theory, you can refer to the category of all sets as “the category Set,” and its objects are precisely the “class of all sets”. I can’t remember what is popular for denoting the objects of a category, but I think obj(Set) is one option.
