Is there any known explicit bijection between these two sets?
I know it can be proved that such bijection exists using two injections and Schröder–Bernstein theorem, but I wanted to know whether some explicit bijection is known. I failed to find any except ones constructed awkwardly from the Schröder–Bernstein theorem.
Best Answer
First note that the set $\mathcal P_{\text{fin}}(\Bbb N)$ of finite subsets of $\Bbb N_0$ is in bijection with $\Bbb N_0$: $$ \begin{align}\alpha\colon \mathcal P_{\text{fin}}(\Bbb N)&\to \Bbb N\\A&\mapsto \sum_{k\in A}2^k\end{align}$$
Every real number $a\in[0,1)$ as a binary expansion $a=\sum_{k=0}^\infty a_k2^{-k-1} $ with $a_k\in\{0,1\}$. For those cases with two expansions we pick the one ending in zeroes. Now we map $$ \begin{align}\beta\colon [0,1)&\to \mathcal P(\Bbb N)\\a&\mapsto \{\,k\in\Bbb N\mid a_k=0\,\}\end{align}$$ This one fails to be bijective: We leave out precisely the finite subsets of $\Bbb N$. In other words, we have a bijection $$ \beta\colon [0,1)\to\mathcal P(\Bbb N)\setminus \mathcal P_{\text{fin}}(\Bbb N)$$ The rest is glueing and playing Hilbert's Hotel: We have a bijection $$ \begin{align}\gamma\colon \Bbb R&\to (0,\infty)\\x&\mapsto e^x\end{align}$$ and a bijection $$ \begin{align}\delta\colon (0,\infty)&\to [0,\infty)\\x&\mapsto \begin{cases}x-1,&x\in\Bbb N\\x,&\text{otherwise}\end{cases}\end{align}$$ and a bijection $$ \begin{align}\epsilon \colon [0,\infty)&\to [0,1)\\x&\mapsto\begin{cases} \frac1{1+x},&x>0\\0,&x=0\end{cases}\end{align}$$ All in all this gives us a bijection $$ \zeta\colon \Bbb R\stackrel{\beta\circ\epsilon\circ\delta\circ\gamma}\longrightarrow \mathcal P(\Bbb N)\setminus\mathcal P_{\text{fin}}(\Bbb N)$$ To complete the construction we have to hide countably many finite sets by defining for example $$ \begin{align}\eta \colon \Bbb R&\to \mathcal P(\Bbb N)\\x&\mapsto\begin{cases}\alpha^{-1}(x),&x\in\Bbb N\\ \zeta(x-\sqrt 2),&x=n+m\sqrt 2\text{ with }n\in\Bbb N, m\in\Bbb N_{>0}\\ \zeta(x),&\text{otherwise}\end{cases}\end{align}$$