I understand how we can show the existence of a choice function for any (finite or infinite) collection of (finite or infinite) subsets of, say, $\mathbb{N}$ or $\mathbb{Z}$ without using the axiom of choice, by showing that a well-ordering over the union of sets in the collection exists.
What I don't see though is how to do the same for what appears to be a much simpler case: a (possibly infinite) collection of finite sets, where nothing else is said about the sets other than that each of them is finite.
I would need to show a well-ordering exists over the union of those sets, but since they were not explicitly defined to be subsets of $\mathbb{N}$ or $\mathbb{Z}$, I first need to show the following, for example:
(i) there exists a surjection from $\mathbb{N}$ to $\bigcup X_i$ for an arbitrary collection of finite sets $X_i$
Correct so far?
Alternatively, I'm on the wrong track, and this is precisely a case that (unintuitively for me then) requires application of the axiom.
If so, I suppose I don't see why (i) wouldn't hold, when it seems that any definition within ZF of a finite number of elements is equivalent (in some sense to be made precise) to a finite subset of $\mathbb{N}$, so, given that the union of a collection of finite subsets of $\mathbb{N}$ is a subset of $\mathbb{N}$, the required well-ordering exists.