You say “removing a single unit of value from a person in the latter option … is worse by 1 unit as we would hope”, but don’t the ordering effects get very tricky?
For example, imagine everyone ever to live has a lifetime utility of 1, population size is constant, and so the utility of all future people is 1 + 1 + 1 + 1 … = ω. But if I took some living person today and zeroed out their utility, then we might have 1 + 0 + 1 + 1 … which is still ω in your formulation, no? And we’d prefer it to be ω − 1?
On the hyperreal approach 1 + 0 + 1 + 1 + … does actually equal ω−1 as desired.
This is an example of the general fact that adding zeros can change the hyperreal valuation of an infinite sum, which is a property that is pretty common in variant summation methods.
The short answer is that if you take the partial sums of the first n terms, you get the sequence 1, 1, 2, 3, 4, … which settles down to have its nth element being n−1 and thus is a representative sequence for the number ω−1. I think you’ll be able to follow the maths in the paper quite well, especially if trying a few examples of things you’d like to sum or integrate for yourself on paper.
(There is some tricky stuff to do with ultrafilters, but that mainly comes up as a way of settling the matter for which of two sequences represents the higher number when they keep trading the lead infinitely many times.)
This is really neat!
You say “removing a single unit of value from a person in the latter option … is worse by 1 unit as we would hope”, but don’t the ordering effects get very tricky?
For example, imagine everyone ever to live has a lifetime utility of 1, population size is constant, and so the utility of all future people is 1 + 1 + 1 + 1 … = ω. But if I took some living person today and zeroed out their utility, then we might have 1 + 0 + 1 + 1 … which is still ω in your formulation, no? And we’d prefer it to be ω − 1?
Thanks!
On the hyperreal approach 1 + 0 + 1 + 1 + … does actually equal ω−1 as desired.
This is an example of the general fact that adding zeros can change the hyperreal valuation of an infinite sum, which is a property that is pretty common in variant summation methods.
Thanks! I’m glad this has 1 + 0 + 1 + 1 = ω − 1, but I’m going to need to go read more to understand why ;)
The short answer is that if you take the partial sums of the first n terms, you get the sequence 1, 1, 2, 3, 4, … which settles down to have its nth element being n−1 and thus is a representative sequence for the number ω−1. I think you’ll be able to follow the maths in the paper quite well, especially if trying a few examples of things you’d like to sum or integrate for yourself on paper.
(There is some tricky stuff to do with ultrafilters, but that mainly comes up as a way of settling the matter for which of two sequences represents the higher number when they keep trading the lead infinitely many times.)