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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?

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I don't know if that really answers your question but given the set $E$, you could just use $E \notin E$. –  Joel Cohen Jun 13 '12 at 16:28
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Why the downvotes? –  Olivier Bégassat Jun 13 '12 at 16:35
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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.

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In ZF, since the universal set cannot exist, every set x must exclude something. But the set of all such exclusions (the complement of x wrt the universal set) also cannot exist. So, there cannot be a well defined terminology for all those things excluded from x. –  Dan Christensen Jun 14 '12 at 14:13
    
@Dan: Are you saying that we cannot denote classes definable by a formula with parameters by a letter? What about $L(x), HOD[A]$ and so on? –  Asaf Karagila Jun 15 '12 at 6:24
    
@Asaf Karagila: Sorry, I don't understand your notation, but I suppose I should have said, a well defined terminology within the context of ZF (or similar) set theory, i.e. one without a formal notion of classes. –  Dan Christensen Jun 15 '12 at 14:13
    
@Dan: In every model of ZF there is a definable class called $L$ which is Godel's constructible universe. We can define $L[A]$ which is of relative constructible sets, there is another kind of relative constructibility denoted by $L(A)$. All these are usual notations and terminology for proper classes defined using a parameter $A$. There are more classes as these two which can be defined using parameters. There is nothing wrong denoting by $V)x($, for example, the class $\{y\mid y\notin x\}. The question is usability, and how many people will use that notation. My guess? Not many. –  Asaf Karagila Jun 15 '12 at 14:16
    
@Asaf Karagila: Somehow, I don't think this is what the OP had in mind. See my answer. –  Dan Christensen Jun 15 '12 at 14:24
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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.

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"Infinity" isn't a math term (unlike "infinite cardinal" or "the neighborhood of positive infinity" (in real analysis)), but all infinities have in common that every thing called "infinity" is greater than every natural number. Your proposed $\infty$ does not posses this property, so I'd not like to use $\infty$ to represent it. Now I use "None" as the notation of an element which is not an element of the set in consideration (because it is none of its elements). –  porton Jun 13 '12 at 18:24
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That isn't quite true, actually. In the realm of complex variables, for example, one often considers "the point at infinity" for various reasons--in fact, this is one instance of one-point compactification--and in complex numbers, there isn't really a notion of order that accords with the field arithmetic of $\Bbb C$. It simply denotes (in this context) something that is beyond the scope of a given set. That said, your choice of notation also works just fine. –  Cameron Buie Jun 13 '12 at 19:01
    
It is greater than every natural number in the partial order $x\leq y \Leftrightarrow |x|\leq |y|$ of complex numbers. –  porton Jun 13 '12 at 19:29
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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)

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