= 27d567bc2d48d9f40e9ccc20ea370b15)
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.
= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
The sets don't have to be countable; if there are continuum-many of you indexed by the reals from 0 to 1, the angels could match the interval from 0 to 1/6 with the interval from 1/6 to 1. However, doing this does not preserve measure (as jimrandomh pointed out above), which is the real sleight-of-hand that makes this thought experiment akin to the one where everyone who rolled a six gets unwittingly duplicated up to five copies.
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.