For the hyperreal approaches, ultrafilters are basically just orders over the locations of value to take limits over.
It’s worth noting that the better-behaved “finite-sum” version of the hyperreal approach outlined in Bostrom (2011) is just choosing an order to take partial sums and then limits over, and he describes Expansionism (over spacetime locations) as a special case. This is on pages 21 and 22.
You could do Expansionism over possible persons instead of spacetime locations to satisfy Pareto over persons/agents and avoid the weird issues with pulling worlds together.
The way Expansionism (over spacetime locations) works is by having a set of nested subsets of possible spacetime locations, the expanding spacetime spheres, such that:
Each location is in at least one of the subsets, and then of course in all the following subsets that contain that subset.
In any world, each subset has only finitely many locations with value in it (all but finitely many have value 0).
In math notation, the set of spacetime locations is X⊆R4, and you have a set of subsets of X, S⊂P(X), such that S is totally ordered with respect to ⊂ (nested) and
∪A∈SA=X, i.e. for every x∈X, there exists A∈S, such that x∈A.
For any world W:X→R (the assignment of spacetime locations to their utilities as real values), |{x∈A|W(x)≠0}| is finite.
1 means we cover all of spacetime in the limit and 2 allows us to take finite partial sums before taking limits (or looking at some other behaviour of these partial sums in the limit).
Strictly speaking, you don’t need the nested sets to grow uniformly in each direction (“uniform expansion”) for things to be well-defined, but it’ll probably give more intuitive answers that way.
When comparing two worlds, you’re comparing the “sequence” of partial sums in the “limit” as the set A∈S covers all of X:
∑x∈A,(W1(x),W2(x))≠(0,0)W1(x)−W2(x).
For possible persons instead of spacetime locations, just use a set of nested subsets of possible persons (instead of spacetime locations) that satisfies 1 and 2.
This seems okay and doable, as long as you can come up with such a set of nested subsets of the possible persons that you find sufficiently satisfying, and you’re able to track possible persons across worlds, over time and through space in a way you find sufficiently satisfying. You could identify persons based on characteristics when they first come into existence and then track them based on some kind of (psychological) continuity and causal connections (and make some kinds of fixes for duplication cases when there’s no fact of the matter who’s the original and who’s the duplicate). If you want to distinguish identical people who are in totally different regions of spacetime, one characteristic to use would be their initial spacetime location relative to you, maybe with substantial tolerance so that when a person first becomes conscious (if they aren’t already), it’s not too important precisely when and where that happens for identifying them across worlds, but that might lead to weird discontinuities.
For the hyperreal approaches, ultrafilters are basically just orders over the locations of value to take limits over.
It’s worth noting that the better-behaved “finite-sum” version of the hyperreal approach outlined in Bostrom (2011) is just choosing an order to take partial sums and then limits over, and he describes Expansionism (over spacetime locations) as a special case. This is on pages 21 and 22.
You could do Expansionism over possible persons instead of spacetime locations to satisfy Pareto over persons/agents and avoid the weird issues with pulling worlds together.
The way Expansionism (over spacetime locations) works is by having a set of nested subsets of possible spacetime locations, the expanding spacetime spheres, such that:
Each location is in at least one of the subsets, and then of course in all the following subsets that contain that subset.
In any world, each subset has only finitely many locations with value in it (all but finitely many have value 0).
In math notation, the set of spacetime locations is X⊆R4, and you have a set of subsets of X, S⊂P(X), such that S is totally ordered with respect to ⊂ (nested) and
∪A∈SA=X, i.e. for every x∈X, there exists A∈S, such that x∈A.
For any world W:X→R (the assignment of spacetime locations to their utilities as real values), |{x∈A|W(x)≠0}| is finite.
1 means we cover all of spacetime in the limit and 2 allows us to take finite partial sums before taking limits (or looking at some other behaviour of these partial sums in the limit).
Strictly speaking, you don’t need the nested sets to grow uniformly in each direction (“uniform expansion”) for things to be well-defined, but it’ll probably give more intuitive answers that way.
When comparing two worlds, you’re comparing the “sequence” of partial sums in the “limit” as the set A∈S covers all of X:
∑x∈A,(W1(x),W2(x))≠(0,0)W1(x)−W2(x).For possible persons instead of spacetime locations, just use a set of nested subsets of possible persons (instead of spacetime locations) that satisfies 1 and 2.
This seems okay and doable, as long as you can come up with such a set of nested subsets of the possible persons that you find sufficiently satisfying, and you’re able to track possible persons across worlds, over time and through space in a way you find sufficiently satisfying. You could identify persons based on characteristics when they first come into existence and then track them based on some kind of (psychological) continuity and causal connections (and make some kinds of fixes for duplication cases when there’s no fact of the matter who’s the original and who’s the duplicate). If you want to distinguish identical people who are in totally different regions of spacetime, one characteristic to use would be their initial spacetime location relative to you, maybe with substantial tolerance so that when a person first becomes conscious (if they aren’t already), it’s not too important precisely when and where that happens for identifying them across worlds, but that might lead to weird discontinuities.