Well, technically you can still restrict to finite-support probability distributions even if your outcome space is infinite. So even if you allow all real numbers as distances (and have utilities for each), you can restrict your set of beliefs to have finite support at any given time (i.e. at one point in time you might believe the distance to be one of {3,3.1,pi} or any other finite set of reals, and you may pick any distribution over that finite set). This setup still avoids Paul’s paradoxes.
Having said that, I have trouble seeing why you’d need to do this for the specific case of distances. Computers already use float-point arithmetic (of finite precision) to estimate real numbers, and not much goes wrong there. So computers are already restricting the set of possible distances to a finite set.
Any gradualation is likely to not hit the exact distance on the spot. Then If I was faced to be in a situation where I could become sensitive to that distinction I would need to go from not having included it in the support to having inluded it in the support ie from zero probablity to non-zero probablity. This seems like a smell that things are not genuine comparable to the safety of avoiding unbounded utilities, so i am not sure whether it is an improvement.
Even computers can do symbolic manipulation where they can get exact results. They are not forced to numerically simulate everything. Determining the intersection of two lines can be done exactly in finite computation despite doing it by “brute force” point-for-point whether they are in the same location would call for more than numerable steps.
I have intuitiion/introspective impression that there are objects like “distance is {3,3.1, between 3.2 and 3.3}” where the three categories are equiprobable and distances within “3.2 to 3.3″ are equiprobable to each other but that “3.2 to 3.3” is not made up of listable separate beliefs. (More realisticially they tend to not feel exactly equiprobable within the whole range).
Well, technically you can still restrict to finite-support probability distributions even if your outcome space is infinite. So even if you allow all real numbers as distances (and have utilities for each), you can restrict your set of beliefs to have finite support at any given time (i.e. at one point in time you might believe the distance to be one of {3,3.1,pi} or any other finite set of reals, and you may pick any distribution over that finite set). This setup still avoids Paul’s paradoxes.
Having said that, I have trouble seeing why you’d need to do this for the specific case of distances. Computers already use float-point arithmetic (of finite precision) to estimate real numbers, and not much goes wrong there. So computers are already restricting the set of possible distances to a finite set.
Any gradualation is likely to not hit the exact distance on the spot. Then If I was faced to be in a situation where I could become sensitive to that distinction I would need to go from not having included it in the support to having inluded it in the support ie from zero probablity to non-zero probablity. This seems like a smell that things are not genuine comparable to the safety of avoiding unbounded utilities, so i am not sure whether it is an improvement.
Even computers can do symbolic manipulation where they can get exact results. They are not forced to numerically simulate everything. Determining the intersection of two lines can be done exactly in finite computation despite doing it by “brute force” point-for-point whether they are in the same location would call for more than numerable steps.
I have intuitiion/introspective impression that there are objects like “distance is {3,3.1, between 3.2 and 3.3}” where the three categories are equiprobable and distances within “3.2 to 3.3″ are equiprobable to each other but that “3.2 to 3.3” is not made up of listable separate beliefs. (More realisticially they tend to not feel exactly equiprobable within the whole range).