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.
The real interval [0,1) with Lebesgue measure is commonly used as a probability space; in this case, the measure of cases where one rolls a 6 has Lebesgue measure 1⁄6, we can without loss of generality say it’s the interval [0,1/6), and we can linearly pair every point in this set with a point in the set [1/6,1) to get a nice measurable correspondence. It’s just that this fails to preserve measure.
Also, welcome to Less Wrong! You might like to introduce yourself and your interests on a welcome thread (huh, looks like we need a new one).
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.
Fair. The issue I identified as secondary is in fact primary.
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.
The real interval [0,1) with Lebesgue measure is commonly used as a probability space; in this case, the measure of cases where one rolls a 6 has Lebesgue measure 1⁄6, we can without loss of generality say it’s the interval [0,1/6), and we can linearly pair every point in this set with a point in the set [1/6,1) to get a nice measurable correspondence. It’s just that this fails to preserve measure.
Also, welcome to Less Wrong! You might like to introduce yourself and your interests on a welcome thread (huh, looks like we need a new one).