with surreal math ω+10000000 > ω. I am doubtful on whther the two possible options would actually end in the same situation. And in a way I don’t care about how people are label, I care abuot peoples lifes. It could just be that the relationship between labels for peoples and people amounts diverge. For example in the infinite realm cardinality and ordinality diverge while in the finite realm they coincicde. It could be that your proof shows that there is no ordinal improvement but what if the thing to be cared about behaves like a cardinality?
And we can as well scale down the numbers. Given that there are infinite bad lives and infinite good lives taking one person from bad to good seems like it ought to improve mattes but technically it seems it results in the same state.
I do wonder in that if I have the reals colored white for x>0 and black for 0<=x 0 chould be colored black. Now if I change the color of 0 to white I should be getting the result that now there is more white than previously and less black (even if relatively only infinidesimally so). Now you if you technically take the measure of the point 0 and the white portion of the line one might end up saying things like “measure of a single point is 0”. What I would say is that often measures are defined to be reals but here the measure would need to be surreal for it to accomodate questions about infinities. For example in normal form we could write a surreal with “a + b ω^1 + c ω^2 … ” with a,b,c being real factors. Then it woud dmake sense in that a finite countable collection would have b=0 but it doesn’t mean the whole sum is 0. While we can’t use any small positve real for b for dots I would still say that a point is “pointlike” and not nothing at all. A finite amount of points is not going to be more than no matter how short a line segment. But it doesn’t mean they don’t count at all. It just means that points “worth” is infinidesimal when compared to lines.
Yup. But I don’t see a good reason to apply it that way here, nor a good principled way of doing so that gives the results you want. I mean, how are you going to calculate the net utility in the “after” situation to arrive at ω+10000000 rather than ω?
no ordinal improvement [...] like a cardinality
It looks to me like it’s the other way around. Surreal integer arithmetic is much more like ordinal arithmetic than like cardinal arithmetic. That’s one reason why I’m skeptical about the prospects for applying it here: it seems like it might require imposing something like an ordering on the people whose utilities we’re aggregating.
the measure would need to be surreal
Despite my comments above, I do think it’s worth giving more consideration to using a richer number system for utilities.
[EDITED to fix an inconsequential and almost invisible typo.]
It does occur to me that while giving the people an order migth be suspicous utilities are a shorthand of preferences which are defined to be orders of preferring a over b. Therefore there is anyways going to be a conversion to ordinals so surreals should remain relevant.
I don’t think I’m convinced. Firstly, because in these cases where we’re looking at aggregating the interests of large collections of people it’s the people, not the possibilities, that seem like they need to be treated as ordered. Secondly, because having an ordering on preferences isn’t at all the same thing as wanting to use anything like ordinals for them. (E.g., cardinals are ordered too—well-ordered, even—at least if we assume the axiom of choice. The real numbers are ordered in the obvious way, but that’s not a well-ordering. Etc.)
I concede that what I’m saying is very hand-wavy. Maybe there really is a good way to make this sort of thing work well using surreal numbers as utilities. And (perhaps like you) I’ve thought for a long time that using something like the surreals for utilities might turn out to have advantages. I just don’t currently see an actual way to do it in this case.
with surreal math ω+10000000 > ω. I am doubtful on whther the two possible options would actually end in the same situation. And in a way I don’t care about how people are label, I care abuot peoples lifes. It could just be that the relationship between labels for peoples and people amounts diverge. For example in the infinite realm cardinality and ordinality diverge while in the finite realm they coincicde. It could be that your proof shows that there is no ordinal improvement but what if the thing to be cared about behaves like a cardinality?
And we can as well scale down the numbers. Given that there are infinite bad lives and infinite good lives taking one person from bad to good seems like it ought to improve mattes but technically it seems it results in the same state. I do wonder in that if I have the reals colored white for x>0 and black for 0<=x 0 chould be colored black. Now if I change the color of 0 to white I should be getting the result that now there is more white than previously and less black (even if relatively only infinidesimally so). Now you if you technically take the measure of the point 0 and the white portion of the line one might end up saying things like “measure of a single point is 0”. What I would say is that often measures are defined to be reals but here the measure would need to be surreal for it to accomodate questions about infinities. For example in normal form we could write a surreal with “a + b ω^1 + c ω^2 … ” with a,b,c being real factors. Then it woud dmake sense in that a finite countable collection would have b=0 but it doesn’t mean the whole sum is 0. While we can’t use any small positve real for b for dots I would still say that a point is “pointlike” and not nothing at all. A finite amount of points is not going to be more than no matter how short a line segment. But it doesn’t mean they don’t count at all. It just means that points “worth” is infinidesimal when compared to lines.
Yup. But I don’t see a good reason to apply it that way here, nor a good principled way of doing so that gives the results you want. I mean, how are you going to calculate the net utility in the “after” situation to arrive at ω+10000000 rather than ω?
It looks to me like it’s the other way around. Surreal integer arithmetic is much more like ordinal arithmetic than like cardinal arithmetic. That’s one reason why I’m skeptical about the prospects for applying it here: it seems like it might require imposing something like an ordering on the people whose utilities we’re aggregating.
Despite my comments above, I do think it’s worth giving more consideration to using a richer number system for utilities.
[EDITED to fix an inconsequential and almost invisible typo.]
It does occur to me that while giving the people an order migth be suspicous utilities are a shorthand of preferences which are defined to be orders of preferring a over b. Therefore there is anyways going to be a conversion to ordinals so surreals should remain relevant.
I don’t think I’m convinced. Firstly, because in these cases where we’re looking at aggregating the interests of large collections of people it’s the people, not the possibilities, that seem like they need to be treated as ordered. Secondly, because having an ordering on preferences isn’t at all the same thing as wanting to use anything like ordinals for them. (E.g., cardinals are ordered too—well-ordered, even—at least if we assume the axiom of choice. The real numbers are ordered in the obvious way, but that’s not a well-ordering. Etc.)
I concede that what I’m saying is very hand-wavy. Maybe there really is a good way to make this sort of thing work well using surreal numbers as utilities. And (perhaps like you) I’ve thought for a long time that using something like the surreals for utilities might turn out to have advantages. I just don’t currently see an actual way to do it in this case.