I think V=∞ is logically possible when you aggregate over space and time, and I think we shouldn’t generally assign probability 0 to anything that’s logically possible (except where a measure is continuous; I think this requirement had a name, but I forget). Pascal’s wager and Dyson’s wager illustrate this.
We have reason to believe the universe is infinite in extent, and there’s a chance that it’s infinite temporally. You might claim that our lightcone is finite/bounded and we can’t affect anything outside of it (setting aside multiverses), but this is an empirical claim, so we should give it some chance of being false. That we could affect an infinite region of spacetime is also not a logical impossibility, so we shouldn’t absolutely rule it out.
Yep, we’ve got pretty good evidence that our spacetime will have infinite 4D volume and, if you arranged happy lives uniformly across that volume, we’d have to say that the outcome is better than any outcome with merely finite total value. Nothing logically impossible there (even if it were practically impossible).
That said, assigning value “∞” to such an outcome is pretty crude and unhelpful. And what it means will depend entirely on how we’ve defined ∞ in our number system. So, what I think we should do in such a case is not say V equals such and such. Instead, ditch the value function when you’ve left the domain where it works. Instead, just deal with your set of possible outcomes, your lotteries (probability measures over that set), and a betterness relation which might sometimes follow a value function but might also extend to outcomes beyond the function’s domain. That’s what people tend to do in the infinite aggregation literature (including the social choice papers that consider infinite time horizons), and for good reason.
we shouldn’t generally assign probability 0 to anything that’s logically possible (except where a measure is continuous; I think this requirement had a name, but I forget)
You’re probably (pun not intended) thinking of Cromwell’s rule.
I guess the problem is that V=∞ is nonsensical. We can talk about V→∞, but not equality.
I think V=∞ is logically possible when you aggregate over space and time, and I think we shouldn’t generally assign probability 0 to anything that’s logically possible (except where a measure is continuous; I think this requirement had a name, but I forget). Pascal’s wager and Dyson’s wager illustrate this.
We have reason to believe the universe is infinite in extent, and there’s a chance that it’s infinite temporally. You might claim that our lightcone is finite/bounded and we can’t affect anything outside of it (setting aside multiverses), but this is an empirical claim, so we should give it some chance of being false. That we could affect an infinite region of spacetime is also not a logical impossibility, so we shouldn’t absolutely rule it out.
Yep, we’ve got pretty good evidence that our spacetime will have infinite 4D volume and, if you arranged happy lives uniformly across that volume, we’d have to say that the outcome is better than any outcome with merely finite total value. Nothing logically impossible there (even if it were practically impossible).
That said, assigning value “∞” to such an outcome is pretty crude and unhelpful. And what it means will depend entirely on how we’ve defined ∞ in our number system. So, what I think we should do in such a case is not say V equals such and such. Instead, ditch the value function when you’ve left the domain where it works. Instead, just deal with your set of possible outcomes, your lotteries (probability measures over that set), and a betterness relation which might sometimes follow a value function but might also extend to outcomes beyond the function’s domain. That’s what people tend to do in the infinite aggregation literature (including the social choice papers that consider infinite time horizons), and for good reason.
You’re probably (pun not intended) thinking of Cromwell’s rule.
Yes, thanks!