Definitions of $2$ in set theory? I have seen different definitions of the number $2$ in set theory. The simplest I have seen is the sequence $\mathbf{1}=\{\emptyset\}$, $\mathbf{2}=\{\emptyset, \{\emptyset\}\}$, $\mathbf{3}=\{\emptyset, \{\emptyset\},\{\emptyset, \{\emptyset\}\}\}$ and so on. 
But I have also read Russell's definition in Principia Mathematica which defines "twoness" as something along the lines of "The set of all sets such that there exist elements $x$ and $y$ beloning to the set where $x$ is not $y$ and if also $z$ belongs to the set then $z=x$ or $z=y$." i.e. the set of all sets with two unique elements.
So in modern set notation I would try to express this as:
$$\mathbf{2}= \{S: \exists xy:(x\subset S \land y\subset S \land x \neq y \land \forall z:(z\subset S \implies z=x \lor z=y))  \}$$
(Not sure if I have that quite right). Now, the second definition seems more intuitive albeit more complicated. Whereas the first definition seems to be like a code with no logical meaning. I can see why a statement like $\mathbf{1+1=2}$ would take a few pages to prove in the later case.
So which is the "correct" definition of 2?
 A: The first definition is a good practical example of a set that has two elements. See it as the "standard" example of what a set with two elements looks like, a representative of the concept of $2$.
The second class you describe is the collection of all things that have two elements, so it completely describes the concept of having the quantity $2$. The problem is that the second class is not a set: it has too many elements, and is therefore a proper class. This makes doing mathematics with it a little troublesome.
What the second class describes, is the concept of cardinality: it describes all the sets that have cardinality $2$. Another way to define this class, would be to take all the sets that have a bijective function to the representative set $\{\varnothing,\{\varnothing\}\}$.
A: Your first example (Von Neumann's definition) is the most typical set-theoretical definition of $2.$ There are many other choices that could be made, but this one has advantages, most prominently that it generalizes to the usual nice definition of ordinals.
Your second example (I think there are typos but what you intend is clear) is not a set at all in the standard framework (ZF), but rather a proper class. This is the class of all sets with cardinality two (or equivalently with the "twoness" property). Your first example could be said to be a canonical representative of this class. 
However, it is perfectly reasonable to have a definition of two that does not have the twoness property. For instance, the Zermelo definition has $0=\emptyset,$ $1= \{\emptyset\},$ $2=\{\{\emptyset\}\},$ etc. What's important is that there is some way of mapping the set back to the intuitive idea of the number two in the context of the definitions of the rest of the natural numbers, not that it literally has two elements. Though one might view the fact that the cardinality of a natural number is the natural number itself as another advantage of the Von-Neumann definition.
A: To answer your question directly, don't try to think that one of these definitions is "the correct one".  Each of these definitions is a different means of establishing 'twoness', each serves a different mathematical and/or philosophical purpose, and each is "correct" in its proper context.  I like to think of them as different 'standard rulers' for measuring 'twoness'.  [personally, I think the second definition is closer to 'correct' for general 'twoness', but unfortunately, as Russell pointed out, the underlying mathematics doesn't actually work right]
A: Well, there is no correct definition of two. All these things are set-theoretical constructs that no other mathematicians really need. (This is not to diminish the set theory. We do the same thing with other concepts, e.g. Euclidean geometry, real numbers, etc.)
There are only various ways how to define 2 using axioms. One takes a set denoted $0$ whose existence is guaranteed by the axioms, and then defines $1=\{0\}$ and $2=\{1,0\}$. The other one basically says "any natural number $n$ is represented by any set with exactly $n$ elements. The only question is: which one is easier to use to construct whatever you need to construct? This is difficult, and most importantly, again, not really important for practical use.
I personally like the simple one as there $n<m$ (as numbers) if and only $n\in m$ (as sets), so you have the ordering of natural numbers very easy, also, $n+1 = S(n) = n\cup\{n\}$, giving the successor function. In the Russel's way, you have to say that $n<m$ if there exists a set representing $m$ such that $n\subset m$, so it's a bit more complicated.
