Probability theory doesn't automatically work on infinite sets. If you approach this problem as the well-defined limit of a finite problem, the answer is simple.

1. You roll a fair die and it rolls out of sight. You assign a probability of 1/6 to the proposition that you rolled a heads, since you have no reason to suspect any particular face came up.
2. An angel appears to you and informs you that you are one of N almost identical twins who independently rolled a fair die that rolled out of sight, and that similar angels are appearing to them all and telling them all the same thing. The twins all reason by the same principles and their past lives have been practically indistinguishable. Still 1/6, no new relevant information here.
3. The angel adds that X of the twins rolled six and N-X didn't. Woah, something is messed up, unless X is approximately N/6...
4. The angel then tells you that the angels have worked out a list of pairs of identifiers of you and your twins (you're not exactly alike), such that each twin who rolled six is paired with a twin who didn't roll six. Yep, something is messed up. We have X=N/2, which is nowhere near N/6. A few things could happen now, depending on our priors. In real life, we would probably stop trusting the angel, stop trusting that the dice are fair, or both; it depends on how large N is (and hence how unlikely the "even split with fair dice" is), and how strong our faith was in the angel and the dice. Not in real life, following the conceit of the problem, an incredibly unlikely event just occurred. But no matter; we confidently assign a probability of 1/2 to having rolled a six, based on what the angel told us and no other information to identify our roll.
5. The angel then informs you that each pair of paired twins will be transported into a room for themselves. And, poof!, it is so. You are sitting across from someone who looks very much like you, and you each know that you rolled six if and only if the other did not. Of course, still 1/2; no new relevant information here.

ETA: Say we take the limit as N goes to infinity, with everything else kept constant. We can end up with two countably infinite sets of twins, and apparently the same probabilities at each step, so we have a probability of 1/2 in the infinite case. Now, pretend that instead, we had X=N/3 (and the angel only tells us that, and nothing else). Then our probability of having a six is 1/3. As N approaches infinity, apparently our probability is still 1/3. But in this infinity land, we can still do the room pairing thing! There is a bijection between any two countably infinite sets. In fact, we could approach the infinite case with any ratio of sixes to non-sixes, as long as it's positive and less than one, and still end up with a bijection between the sixes and the non-sixes. Without a well-defined limit approaching the infinite case, we can produce any probability we want; limits need to be well-defined.

This is explained in Jaynes (2003). I don't have it with me, but if I recall it is in Chapter 15 or thereabouts on marginalization and other paradoxes.