Okay, I got this, the axiom of choice is not even needed for finite sets of choices, it comes into play elsewhere. But there are quite a few more points where I

Okay, I got this, the axiom of choice is not even needed for finite sets of choices, it comes into play elsewhere. But there are quite a few more points where I can not understand the reasoning behind. Not saying that they are incorrect, only that they are not obvious to me. Given any set of voters, either it or its complement is a majority, but not both Why no draws, like the odd numbered voters versus the even numbered voters? The result of removing a single voter from a majority is still a majority. This is not true for finite sets of voters, why would we require it for infinite sets of voters? This seems more like a possible consequence of dealing with an infinite set of voters than something one would require in advance. The intersection of two majorities is a majority Also not true for finite sets of voters, so the same question as above.