Those probabilities are multiplied by Xis, which makes it more complicated. If I try running it with Xs being the real numbers (which is probably the most popular choice for utility measurement), the proof breaks down. If I, for example, allow negative utilities, I can rearrange the series from a divergent one into a convergent one and vice versa, trivially leading to a contradiction just from the fact that I am allowed to do weird things with infinite series, and not because of proposed axioms being contradictory. EDIT: concisely, your axioms do not imply that the rearrangement should result in the same utility.
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.
Those probabilities are multiplied by Xis, which makes it more complicated.
If I try running it with Xs being the real numbers (which is probably the most popular choice for utility measurement), the proof breaks down. If I, for example, allow negative utilities, I can rearrange the series from a divergent one into a convergent one and vice versa, trivially leading to a contradiction just from the fact that I am allowed to do weird things with infinite series, and not because of proposed axioms being contradictory.
EDIT: concisely, your axioms do not imply that the rearrangement should result in the same utility.
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.