An element not in a set Having a set I need to take an arbitrary element which is not in this set.
I know that existence of such elements for every set can be proved in ZF.
My question: Are there any established notation/terminology for such elements?
 A: So the result, I am assuming you are using is that $V = \{x : x = x\}$ is not a "set" in $ZF$, i.e. the entire universe is not a set. Therefore given any set $x$, $"x \neq V"$ since $V$ is a proper class and $x$ is a set. Hence there exists some set $y$ such that $y \notin x$. 
I am not sure there exists a notation for this element since given any set $x$ there is certainly more than one element with this property. 
More interestingly, in ZFC (adding the axiom of choice), you get a function that gives you the element you want. To see this: let $F$ be a set. In ZF, $\bigcup F$ is a set. By the above, there exists a set $y \notin \bigcup F$. Let $Z = \bigcup F \cup \{y\}$. Define $E = \{Z - x : x \in F\}$. $E$ is a set of nonempty sets. By the axiom of choice, there is a choice function $f : E \rightarrow \bigcup E$ such that $f(z) \in z$. Let $g$ be the function that takes $x \mapsto Z - x$. Let $h = f \circ g$. Then $h(x) \notin x$ for all $x$. So this shows that in $ZFC$, for any set of sets, there is actually a functions that gives you the element not in the set. 
A: One often uses $\infty$ to represent some element not in the set--this is particularly common in topology, when dealing with an Alexandrov one-point compactification of a given topological space. Of course, in some contexts, $\infty$ may already have another pre-specified meaning, and so one must use something else to denote said element. There isn't really a standard terminology, either. "Some specific element not in the set" is pretty much it, in my experience.
A: You may be thinking the complement of set x with respect to (some "universal") set U. I put universal in quotes because, of course, in ZF, there exists no truly universal set and no complement of any set wrt to such a "universal" set. 
Here, U is an arbitrary set, a universe or domain of discourse, e.g. the set of real numbers. We define the complement of x wrt U as x' such that:
$\forall a (a\in x' \leftrightarrow (a\in U\wedge a\notin x))$
Also called the absolute complement. Other commonly used notation:
$x'=U\setminus x=x^c$
See Wiki article: http://en.wikipedia.org/wiki/Complement_(set_theory)
