This puzzle is inspired by a reflection on (a) a talk [PDF] by John Norton, and (b) the problem of finding probability measures on multiverses. It is very, very similar—quite likely equivalent—to an example [PDF] discussed by John Norton. Suppose you are one of infinitely many blindfolded people. Suppose that the natural numbers are written on the hats of the people, a different number for each person, with every natural number being on some person's hat. How likely is it that the number on your hat is divisible by three?
The obvious answer is: 1/3. But Norton's discussion of neutral evidence suggests that this obvious answer is mistaken. And here is one way to motivate the idea that the answer is mistaken. Suppose I further tell you this. Each person also have a number on her scarf, a different number for each person, with every natural number being on some person's scarf. Moreover, the following is true: the number on x's scarf is divisible by three if and only if the number on x's hat is not divisible by three. (Thus, you can have 3 on your scarf and 17 on your hat, but not 16 on your scarf and 22 on your hat.) This can be done, since the cardinality of numbers divisible by three equals the cardinality of numbers not divisible by three.
If you apply the earlier hat reasoning to the scarf numbers, it seems you conclude that the likelihood that the number on your scarf is divisible by three is 1/3. But this is incompatible with the conclusion from the hat reasoning, since if the likelihood that the scarf number is divisible by three is 1/3, the likelihood that the hat number is divisible by three must be 2/3.
If there are numbers on hats and scarves as above, symmetry, it seems, dictates that the probability of your hat number being divisible by three is the same as the probability of your scarf number being divisible by three, and hence is equal to 1/2. But this conclusion seems wrong. For the numbers on scarves, even if anti-correlated with those on the hats, should not affect the probability of the hat number being divisible by three. Nor should it matter in what order the hat and scarf numbers were written—hats first, and then scarves done so as to ensure the right anti-correlation between divisibilities, or scarves first, and then hats. But if the hat numbers are written first, then surely the probability of divisibility by three is 1/3, and this should not change from the mere fact that scarf numbers are then written.
One of several conclusions might be drawn:
- Actual infinities are impossible.
- Uniform priors on infinite discrete sets make no sense.
- Probabilities on infinite sets are very subtle, and do not follow the standard probability calculus, but there is a very intricate account of dependence such that whether the hat numbers are assigned first or the scarf numbers are assigned first actually affects the probabilities. I don't know if this can be done—but when I think about it, it seems to me that it might be possible. I seem to be seeing glimpses of this, though the fact that as of writing this (a couple of hours after my return from Oxford) I've been up for 21 hours may be affecting the reliability of my intuitions.