Infinite sums of finite terms and finite sums of infinite terms might be different and the latter are quite easy. With A= ω * 1000 + ω * −1000, B= ω * 1000 + (ω-1000) * −1000 + 1000000*1000, C= (ω-1000) * 1000 + (ω) * −1000 + 1000000*-1000, its clear that B>A>C
To my belief normal utility funtions can be scaled to remain essentially the same. That is if one explicit version gives numbers 1, 10, 100 to the options then a tenfold function that gives 1, 100 and 1000 to the same options is equally valid. I would expect this to hold in the transfinites in that a function giving 1, ω and ω * ω would be as good as one giving ω , ω * ω and ω * ω * ω.
I am not sure that surreals neccesarily invoke infinite sums and their orderings. ω can be defined without sums and it becomes a separate thing to prove for example that 1+1=2 (that is, this is a genuine claim about how addition works in relation to already existing numbers, it’s not a restatement of the definition of 2). There is the issues that just because a value is transfinite you don’t know how big it is and some problems might be sensitive to get the magnitudes right. Say that you have pascal wager options of not having a life or afterlife, having a life for another day, living one day in heaven and living indefinetely in heaven. The correctish values would be 0, 1*1 , ω * 1 and ω * ω, the fourth option being clearly better than the third rather than equally good. Also there is no natural number N so that 1 * N >= ω but 1* ω = ω. “repeatedly +1” migth only refer to the first. Surreals deals with actual infinites not infinities as a limit of finite processes. In a way both ω abd ω * ω would appear as a series of “++++++...” so decomposition into a plus ordering can’t be their distinguising mark.
Infinite sums of finite terms and finite sums of infinite terms might be different and the latter are quite easy. With A= ω * 1000 + ω * −1000, B= ω * 1000 + (ω-1000) * −1000 + 1000000*1000, C= (ω-1000) * 1000 + (ω) * −1000 + 1000000*-1000, its clear that B>A>C
To my belief normal utility funtions can be scaled to remain essentially the same. That is if one explicit version gives numbers 1, 10, 100 to the options then a tenfold function that gives 1, 100 and 1000 to the same options is equally valid. I would expect this to hold in the transfinites in that a function giving 1, ω and ω * ω would be as good as one giving ω , ω * ω and ω * ω * ω.
I am not sure that surreals neccesarily invoke infinite sums and their orderings. ω can be defined without sums and it becomes a separate thing to prove for example that 1+1=2 (that is, this is a genuine claim about how addition works in relation to already existing numbers, it’s not a restatement of the definition of 2). There is the issues that just because a value is transfinite you don’t know how big it is and some problems might be sensitive to get the magnitudes right. Say that you have pascal wager options of not having a life or afterlife, having a life for another day, living one day in heaven and living indefinetely in heaven. The correctish values would be 0, 1*1 , ω * 1 and ω * ω, the fourth option being clearly better than the third rather than equally good. Also there is no natural number N so that 1 * N >= ω but 1* ω = ω. “repeatedly +1” migth only refer to the first. Surreals deals with actual infinites not infinities as a limit of finite processes. In a way both ω abd ω * ω would appear as a series of “++++++...” so decomposition into a plus ordering can’t be their distinguising mark.