“Are the glories of heaven worth exactly ω utility? How do we know it’s that rather than √ω or 3ω1/ω or something?”—We don’t know unless it is specified. However, it’s not a bug, but a feature.
“But there’s no obvious way to choose the ordering, and what do we do if that action that makes a million unhappy people happy also rearranges them to make the second order more natural somehow when the first was more natural before?”—Yep, this is exactly the issue I’m currently working on. But my ideas aren’t quite ready to share yet.
In order to apply surreal arithmetic to the expected utility of world-states, it seems we’ll need to fix some canonical bijection between states of the world and ordinals / surreals. In the most general case this will require some form of the Axiom of Choice, but if we stick to a nice constructive universe (say the state space is computable) then things will be better. Is this the gist of what you’re working on?
Not quite. I don’t think there’s a unique canoncial bijection—I embrace there truly being multiple countable infinities. Although I do want to insist on some regularity. And computability is relevant here, as it makes it much easier to show that certain consistent labellings exist
“Are the glories of heaven worth exactly ω utility? How do we know it’s that rather than √ω or 3ω1/ω or something?”—We don’t know unless it is specified. However, it’s not a bug, but a feature.
“But there’s no obvious way to choose the ordering, and what do we do if that action that makes a million unhappy people happy also rearranges them to make the second order more natural somehow when the first was more natural before?”—Yep, this is exactly the issue I’m currently working on. But my ideas aren’t quite ready to share yet.
In order to apply surreal arithmetic to the expected utility of world-states, it seems we’ll need to fix some canonical bijection between states of the world and ordinals / surreals. In the most general case this will require some form of the Axiom of Choice, but if we stick to a nice constructive universe (say the state space is computable) then things will be better. Is this the gist of what you’re working on?
Not quite. I don’t think there’s a unique canoncial bijection—I embrace there truly being multiple countable infinities. Although I do want to insist on some regularity. And computability is relevant here, as it makes it much easier to show that certain consistent labellings exist