I want to find a function $\phi : P (\Sigma_{\text{bool}}^{*}) \to [0,1]$.
I know that this can be done by defining
$$ \phi(\{N(w_1), N(w_2), N(w_3), ..., N(w_n)\}) = N(w_1) + \frac{1}{N(w_2) + \frac{1}{...+\frac{1}{N(w_n)}}} $$
where $N(w_k)$ appends a $1$ to $w_k$ and interprets it as a number, and using the fact that simple continuous fractions are unique.
I also tried to do this using the function
$$ \begin{equation} \psi(\{w_1, ..., w_n\}) = 0.w_13w_23w_33...3w_n \end{equation} $$
that is, mapping each set to the numer whose base-3 decimal representation is the concatentation of all the the words. I think this works, but I am not sure.
I searched on Math Stackexchange but couldn't find this exact question, please acccept my apologies if it is a duplicate.