Question on von Neumann integers and power set If $2^X$ is the set of functions from $X$ to $2$ (or the set of two elements). And we use the von Neumann notation to denote integers by sets $0=\emptyset$ and $n+1=n\cup\{n\}$ then is it correct to make the identification between the set $2^X$ when $X$ is an integer (since integers are sets) and the von Neumann integer that $2^X$ computes?  
For example, $2^3=8$ so is it appropriate to identify $8=\{0,\ldots,7\}$ with the set of functions from $3$ to $2$?  My intuition is that the $2^X$ notation obeys von Neumann's notation, but I'm not sure if this is generally true or if I'm way off.  This question was motivated when I came accross $2^{2^m}$ and I wanted to identify it with $2^4$ (for $m=2$), but wasn't sure if the top $2^m$ was a number of the set of functions from $m$ to $2$, or if that distinction matters.
 A: Identify in what sense?
Identifying means that you forgo some structure to take on another. If you want to identify $\mathcal P(\{0,1,2\})$ with $2^{\{0,1,2\}}$ or $2^3$, then you exchange subsets by indicator functions. And that's fine and well known.
But if you exchange $2^3$ by $8$, you need to be more specific how you make the identification. Of course, you might say that you do it by mapping each number to its binary representation in three digits. That's fine, and probably the canonical way to think of such identification.
But then comes the question, what does the identification gives you? When identifying $\mathcal P(X)$ and $2^X$ we have a canonical and very nice way to transform $\mathcal P(X)$ into a module (and in fact an algebra) over $\Bbb F_2$. That's an excellent excuse for doing this.
What do you expect to gain from identifying $2^3$ with $8$? If you expect to gain some cardinal arithmetical things, you should probably note that cardinal arithmetic is already defined as identifying $2^3$ with $8$. We define $2^3$ as the cardinal number of $\mathcal P(\{0,1,2\})$, which happens to be $8$.
If you expect some other structural benefits, you are probably going to need to argue for them explicitly. But can you do it? Sure.
