I thought about it and just figured out a way to count $A$. Each element of $A$ is a finite set constructed of left brackets, $\{$, and right brackets, $\}$, so in replacing each left bracket with a $0$ and each right bracket with a $1$, we can convert each element of $A$ to a unique finite binary string:

Let $f(\{) = 0$ and let $f(\}) = 1$.

Arbitrarily, let $a = \{ \{ \{ \} \}, \{ \{ \}, \{ \{ \} \} \} \}$.

Then $f[a] = f(\{) f(\{) f(\{) f(\}) f(\}), f(\{) f(\{) f(\}), f(\{) f(\{) f(\}) f(\}) f(\}) f(\}) = 00011001001111$.

Define $A = \{ x : \text{ there exists some model of ZF set theory such that } x \text{ is an element of the model and } f[x] = \text{ a finite binary string} \}$.

Now that my definition is clear, I am curious, what is the difference between $L_w$ and $A$?