Meanwhile, did you find the link on the hereditarily finite sets of interest?

Is your idea different in some way?

Also can you please say in English what #5 means?

What is the domain of f?

Yes, I was interested in the hereditarily finite sets that you linked to in your initial post, however, the first time I read your initial post, I don't recall seeing that link. You may have edited the post after I read it as I am just now seeing that for the first time.

No, my idea is not different. I was more interested in knowing what the difference was between the $A$ I was describing and $L_{\omega}$ (or $V_{\omega}$, as $L_{\omega} = V_{\omega}$). But, as clarified in the wiki article you cited, $A = V_{\omega} = \bigcup_{k=0}^{\infty} V_k$. I understand now why $L_{\omega}$ is the set of all hereditarily finite sets, but my first reading of the definition of the Constructible Universe had me thinking that $L_{\omega}$ could contain infinite sets so long as every element of each infinite set in $L_{\omega}$ was an element of some $L_{k}$ where $k \in \mathbb{N}$. Further, I understand why $L_{\alpha + 1}$ is a subset of $V_{\alpha + 1}$ for all $\alpha \geq \omega$ (because there are more elements of $\mathcal{P}(L_{\omega})$ than there are in $\text{DEF}(L_{\omega})$ on account of there being non-computable elements of $\mathcal{P}(L_{\omega})$ while every element of $\text{DEF}(L_{\omega})$ must be the result of some first order formula using only variables contained in $L_{\omega}$.

#5 said nothing different than #'s 1 through 4. I was just continuing the pattern in an initial effort to define in 'MyMathForum' style the hereditarily finite sets. With a little more thought I could have given a better definition, but you understood it anyways as you wrote:

Oh I see. You're just saying they're finite. Ok.

Where $f$ was only defined for the inputs '$\{$' and '$\}$', that was its domain. I used $f$ within an alternative definition of $A$ in which I applied $f$ to every $x$ such that $x$ was an element of some model of ZF set theory so as to say, if $f[x]$ was a finite binary string, then $x$ was in $A$. This all is trivial anyways. Using $f^{-1}$ across the finite binary strings is simply an alternative way of defining $A$.