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It's only if the sets are countable that we can probabilistically predict ahead of time that there is a pairing. To get the existence of a pairing, we need to know that the cardinality of those who rolled six is equal to the cardinality of those who didn't. It is a consequence of the Law of Large Numbers (or can be easily proved directly) that there are infinitely many sixes and infinitely many non-sixes. And any two infinite subsets of a countable set have the same cardinality. But in the uncountable case, while we can still conclude that there are there are infinitely many sixes and infinitely many non-sixes, I don't see how to get that the cardinality is the same. (In fact, events of the form "there are aleph_1 sixes" aren't going to be measurable in the usual product measure used to model independent events, I suspect.)
But of course if there are uncountably many rollers, then, assuming the Axiom of Countable Choice, we can choose a countably infinite subset and work with that.
Thanks for all the comments. For those who are drawn to not changing the probability all the way through the story--which, I agree, is the intuitively right answer--I recommend looking at this variant where sticking to your guns leads to incoherent probabilities.