I was wondering if the empty set is at most countable. Specifically, if it has to be defined in some sense like finite or countable or if it can be proved from the definition of countability what it is. Then I thought if I could be able to show that there is at least a surjective function between the natural numbers and the empty set. So I started thinking.
Let $\alpha$ be a function: $\alpha: \mathbb N \rightarrow \emptyset$.
Then we have that $\alpha(n) \notin Im(\alpha)$ for all $\ n \in \mathbb N$, where $\alpha(n)$ is the value of the function $\alpha$ at $n$, i. e., $\alpha(1)$, $\alpha(2)$ ... , $\alpha(n),$ ...
In this manner, the image of $\alpha$ would be empty. But so it is the empty set, an there are no distinct empty sets. Therefore $\alpha$ is surjective (because the codomain equals the image of $\alpha$). As the empty set has no elements, it is at most countable, or better, it is finite.
What you guys think of this proof?
Thank you everybody, now I believe I know where I went wrong, in thinking that there are such functions!