[carstimon]

> I thought the axiom of choice says that it is possible to obtain an [infinite] set of votes from an [infinite] set of voters each with a non-empty set of choices

The axiom of choice does say this but,

1) This is not where we are applying the axiom of choice

2) You don't need the axiom of choice here: you can always just pick the "voting set" where everyone votes for choice # 1.

The axiom of choice says that the product of infinitely many non-empty sets is empty. This is needed only when you might have a different and non-computable set of choices to make at each step.

Where we are applying the axiom of choice here is: the voting scheme is just a rephrasing of an "Ultrafilter on the power set of a set": https://en.wikipedia.org/wiki/Ultrafilter#Ultrafilter_on_the... This is implied by the axiom of choice in a way that requires some working out (and is sketched in wikipedia). But you can roughly see how the axiom of choice comes about if you start trying to construct a voting scheme (the following is just intuition):

1) We know the result if the number of "Nays" or "Yays" is finite.

2) Pick any set of "Yays" so the number of "Nays" and "Yays" is both infinite. Declare this set (arbitrarily) as a victory for either "Nay" or "Yay"

3) The set in (2) implies that a bunch of other sets are either Nay or Yay by the axioms in the article (e.g. flipping all votes flips the outcome, etc)

4) While there are sets we haven't decided on, repeat steps (2) and (3).

Of course (4) is the whole tricky part, since you have to describe with this "looping forever" means rigorously... but you can see how the idea of "making infinitely distinct choices" comes up.

[bmacho]

> The axiom of choice says that the product of infinitely many non-empty sets is empty.

You mean non-empty.