Define $A = \{ 1,2,A \}$, $A$ can not be a set (Axiom of regularity). Can $A$ be a "class" or a "collection" of elements. This question is probably very naive but it does bother me and I am not sure where to look for an answer.
Define $A = \{ 1,2,A \}$, $A$ can not be a set (Axiom of regularity). Can $A$ be a "class" or a "collection" of elements. Suppose it is. Don't we get the Russel's paradox again? Take the collection of all collections which does not contain themselves as an element. Does it contain itself as an element?
Is there a short answer or is there a lot of knowledge involved?
Thank you  
Remark: I know that $A$ cannot be a set because of axiom of regilarity. But, what prevents it from being a class or a collection?
 A: A class in standard (ZF) set theory generally coincides with a property that any set may or may not have.  For instance, the property might be "$\phi(x) \equiv x \not\in x$".  Then $\{x \;:\; x\not\in x\}$, while not necessarily a set, is a well-defined class.  But note that the elements of a class are sets, not classes.  In your case, you want to know whether there is a class corresponding to $A$.  Such a class would contain exactly $1$, $2$, and a third set equal to the entire class.  This could only occur if $A$ were itself a set, which you've already noted is ruled out by the axiom of regularity.
A: Luiz Cordeiro already mentioned that you can't define $A$ like this,
but as you are interested in the details of axiomatic set theory let
me try to explain why exactly you can't do it.
You will know that in ZFC $\{a,b\}$ is just an abbreviation.  You
usually introduce the pairing axiom which says that for any sets $a$
and $b$ there's a (unique) set which contains exactly the elements $a$
and $b$.  This unique set is then written as $\{a,b\}$ and it is
obvious how this can be extended to one or (using union) three
elements.
So, when we later write something like "let $A$ be the set
$\{1,2,c\}$" we are actually applying a couple of axioms under the
assumption that $1$, $2$, and $c$ are sets.  What we should
have said is: "We know that $1$, $2$, and $c$ are sets and by the
axioms of pairing, union, and extensionality we know there's a unique
set which has exactly these three elements.  We'll call it
$A$."
Now it should be clear why this fails for the attempted "definition"
of your $A$ above.  The step "we know that $1$, $2$, and $A$ are sets"
doesn't work because at this point $A$ hasn't been defined yet.
A: Classes are really just "too big of a set" (at least in the context of separation axioms for every formula). So sets are really just "small classes". In particular, a class with three elements is a set.
The issue with defining $A=\{1,2,A\}$ is that the definition is circular. Just because you wrote something which looks like it has meaning, doesn't mean that it has meaning (e.g. "The table ate lunch, while the dogs policed the dust" is even syntactical, but has no real meaning).
Sets with certain properties cannot be defined explicitly. We can at best prove their existence is consistent with some axioms, then deduce that such set exists. For example you can't define $x=\{x\}$, but you can prove that it is possible that the axiom of regularity fails and such $x$ does in fact exists, then you can just pick one.
As for "collection", this is an informal notion which means either set, or a class, or something else, depending on the context. Since you haven't specified the context, it seems that you mean "set or class". So it's not a collection. 
