Understanding successor sets Is there an intuitive way to understand them? Let's use $ \mathbb N$ as an example.  $2$ is defined as $1 \,{\cup}\,\{1\}$, but I fail to understand why is it so and how it connects with the intuitive meaning of $2$.
If you're trying to understand why the definition has to be that way, don't - there's no reason it has to be that way. But the definition you're talking about works out very nicely - it's one of the reasons Von Neumann got the big money:

Don't think of it in terms of $S(n)=n\cup\{n\}$, think of it as $n=\{0,1,\dots,n-1\}$.

We want to give definitions of $0,1,2\dots$ in terms of sets.

It seems reasonable that whatever set we choose to represent the natural number $n$, it should be a set with exactly $n$ elements.

So what should $0$ be? The only choice is $\emptyset$, since that's the only set with no elements. 

Now what should $1$ be? It should be a set with one element. What should that element be? We could say $1=\{\{\{\emptyset\}\}\}$,  since that set has exactly one element. But that seems needlessly complicated. Why not take that one element to be the simplest set possible? So we define $1=\{\emptyset\}$.

And now we notice that hmm, $1=\{0\}$ because of our previous definition.

Now $2$ should be a set with two elements. We already have the sets $0$ and $1$ lying around, and it might be cool or convenient to have $1$ be a subset of $2$, because after all $2$ is supposed to be "larger" than $1$. So why not say $2=\{0,1\}$?

And now we notice that $1=0\cup\{0\}$ and $2=1\cup\{1\}$...







It really does work out very nicely. Consider another possibility. We could have said $0=\emptyset$, $1=\{\emptyset\}$, $2=\{\{\emptyset\}\}$, etc. Let's say those are the ugly naturals. First, they're just not as pretty. For example, $0$ has no elements, while $1,2,\dots$ all have exactly one element.

More important, there's no ntaural way to extrapolate from the ugly naturals to infinite ordinals (saying $\omega=\{\{\dots\emptyset\dots\}\}$  or in other words $\omega=\{\omega\}$  messes up our set theory). But with the Von Neumann construction, as Akiva Weinberger pointed out, it's perfectly natural to say that the next thing after all the natural numbers is $\omega=\{0,1,2,\dots\}$. Looks just like a natural number except bigger. And then $\omega+1=\omega\cup\{\omega\}$ gives the next one.
As the $n$th successor of $0$: $1 = s(0), 2 = s^2(0), 3 = s^3(0)$, where $s^n(x) = \overbrace{s(\cdots s}^n(x))$.