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!
Your argument shows that if $\alpha:\mathbb{N}\to\emptyset$ is a function, then $\alpha$ is surjective. However, this does not prove that there is a surjection from $\mathbb{N}$ to $\emptyset$: what if there are no such functions at all? In fact, there aren't: what would $\alpha(1)$ be, for instance?
In any case, if your goal is to prove the empty set is at most countable "by the definition", then you need to use the definition. The definition of "at most countable" is not that there exists a surjection from $\mathbb{N}$ (though it is a theorem that if such a surjection exists, then your set is at most countable). You said your definition is just that a set is at most countable if it is either countable or finite. So you need to prove either that $\emptyset$ is countable or that it is finite. It's not countable (assuming that by "countable" you mean countably infinite), so you'll have to prove that it's finite.
Again, if you want to prove this, you'll have to use the definition of "finite". There are several different possible definitions, but the most common is that a set $S$ is finite if there is a bijection between $S$ and the set $\{m\in\mathbb{N}:m<n\}$ for some $n\in\mathbb{N}$. In the case $S=\emptyset$, it is equal to the set $\{m\in\mathbb{N}:m<0\}$, so you can take $n=0$ and use the identity map as your bijection.