The rearrangement property you’re rejecting is basically what Paul is calling the “rules of probability” that he is considering rejecting.
If you have a probability distribution over infinitely (but countably) many probability distributions, each of which is of finite support, then it is in fact legal to “expand out” the probabilities to get one distribution over the underlying (countably infinite) domain. This is standard in probability theory, and it implies the rearrangement property that bothers you.
The rearrangement property you’re rejecting is basically what Paul is calling the “rules of probability” that he is considering rejecting.
If you have a probability distribution over infinitely (but countably) many probability distributions, each of which is of finite support, then it is in fact legal to “expand out” the probabilities to get one distribution over the underlying (countably infinite) domain. This is standard in probability theory, and it implies the rearrangement property that bothers you.
Oh, thanks, I did not think about that! Now everything makes much more sense.