Does Cantor’s theorem rely on the Empty Set being in the power set of a set?

Question

As I understand, Cantor’s diagonal set can be empty, that is, there could be a mapping from the the Natural Numbers to the Power Set of the Natural Numbers in which the empty set is not mapped. The function is still not a bijection but there is only one element not mapped (one set that is there only because it is vacuously true). So, does Cantor theorem depend on the empty set being part of the Power set? And how could there be a cardinality in between if in this mapping only one element is not mapped?

Answer 1

I believe the discussion is transparent if we work with the set-theoretical version of Cantor's theorem. The theorem says that given any function $f:X\to P(X)$, the set $H(f,X)=\{x\in X\,:\, x\notin f(x)\}$ is not in $\operatorname{im}(f)$. Therefore, a proof that there is no surjection $X\to P(X)$ would be like this:

1. consider a function $f:X\to P(X)$;
2. by Cantor's theorem $\forall y\in X,[f(y)\ne H(f,X)]$;
3. since $H(f,X)\in P(X)$, $f$ is not surjective.

Now, the same argument doesn't work if we substitute $P(X)$ with $P(X)\setminus\{\emptyset\}$, because the claim $H(f,X)\in P(X)\setminus\{\emptyset\}$ in (3) may fail. In fact, $f(x)=\{x\}$ is an easy example of a function $f:X\to P(X)$ such that $H(f,X)=\emptyset$ and, moreover, there are bijections $X\to P(X)\setminus\{\emptyset\}$ if $\lvert X\rvert\le1$.

For the special case where $X=\Bbb N$ (or more generally $X$ Dedekind-infinite) a proof that there isn't a surjection $X\to P(X)\setminus\{\emptyset\}$ could be like this:

• consider an appropriate pair $(g,x)$ such that $x\in X$ and $g:X\to X\setminus\{x\}$ is a bijection (for $X=\Bbb N$ that could be $x=0$ and $g(t)=t+1$);
• consider a hypothetical surjection $f:X\to P(X)\setminus \{\emptyset\}$;
• the function $h(t)=\begin{cases}\emptyset &\text{if }t=x\\ f(g^{-1}(t))&\text{if }t\ne x\end{cases}$ is a surjection $X\to P(X)$;
• this in contrast with Cantor's theorem.

The second question about the "cardinality in-between" seems nonsense to me.