Is there any notation for referring to a general set of $N$ elements? Currently I'm using $\{1, \dots, N\}$, but the fact that the set consists of natural numbers is irrelevant. I'd prefer to just write something like ${\mathfrak{S}}(N)$. :-)
For instance, I know natural numbers are some times encoded as $0 = \{\}$, $1 = \{0\}$, $2 = \{0, 1\}$. The $N$:th element in such encoding could for instance do.
If we decode $0=\varnothing, 1=\{0\}, 2=\{0,1\}$ and generally $n+1=n\cup\{n\}$, then we have that $n$ itself is a set of $n$ elements.
If you don't want to confuse with the natural numbers you can use $[n]$ for example.
Whatever notation you chooses, though, just write it down. For example:
We will use ${\frak S}(n)$ to denote a fixed set of $n$ elements.
Or if you prefer
Let $A_n$ be a set of size $n$.
Then you can continue with $A_n$ instead.
As for a concrete notation for the set of size $n$, well.. there is none which I can recall.
