Technical comments on type-1 arguments (those aiming to show there can be no probability measure). [Refer to the parent comment for the distinction between type 1 and type 2 arguments.]
I basically don’t see how such an argument could work. Apologies if that’s totally clear to you and you were just trying to make a type-2 argument. However, I worry that some readers might come away with the impression that there is a viable argument of type 1 since Vaden and you mention issues of measurability and infinite cardinality. These relate to actual mathematical results showing that for certain sets, measures with certain properties can’t exist at all.
However, I don’t think this is relevant to the case you describe. And I also don’t think it can be salvaged for an argument against longtermism.
First, in what sense can sets be “immeasurable”? The issue can arise in the following situation. Suppose we have some set (in this context “sample space”—think of the elements at all possible instances of things that can happen at the most fine-grained level), and some measure (in this context “probability”—but it could also refer to something we’d intuitively call length or volume) we would like to assign to some subsets (the subsets in this context are “events”—e.g. the event that Santa Clause enters my room now is represented by the subset containing all instances with that property).
In this situation, it can happen that there is no way to extend this measure to all subsets.
The classic example here is the real line as base set. We would like a measure that assigns measure |a−b| to each interval [a,b] (the set of real numbers from a to b), thus corresponding to our intuitive notion of length. E.g. the interval [−1,3] should have length 4.
Thus we have to limit ourselves to assigning a measure to only some subsets. (In technical terms: we have to use a σ-algebra that’s strictly smaller than the full set of all subsets.) In other words, there are some subsets the measure of which we have to leave undefined. Those are immeasurable sets.
Second, why don’t I think this will be a problem in this context?
At the highest level, note that even if we are in a context with immeasurable sets this does not mean that we get no (probability) measure at all. It just means that the measure won’t “work” for all subsets/events. So for this to be an objection to longtermism, we would need a further argument for why specific events we care about are immeasurable—or in other words, why we can’t simply limit ourselves to the set of measurable events.
Note that immeasurable sets, to the extent that we can describe them concretely at all, are usually highly ‘weird’. If you try to google for pictures of standard examples like Vitali sets you won’t find a single one because we essentially can’t visualize them. Indeed, by design every set that we can construct from intervals by countably many standard operations like intersections and unions is measurable. So at least in the case of the real numbers, we arguably won’t encounter immeasurable sets “in practice”.
Note also that the phenomenon of immeasurable sets enables a number of counterintuitive results, such as the Banach-Tarski theorem. Loosely speaking this theorem says we can cut up a ball into pieces, and then by moving around those pieces and reassembling them get a ball that has twice the volume of the original ball; so for example “a pea can be chopped up and reassembled into the Sun”.
But usually the conclusion we draw from this is not that it’s meaningless to use numbers to refer to the coordinates of objects in space, or that our notion of volume is meaningless and that “we cannot measure the volume of objects” (and to the extent there is a problem it doesn’t exclusively apply to particularly large objects—just as any problem relevant to predicting the future wouldn’t specifically apply to longtermism). At most, we might wonder whether our model of space as continuous in real-number coordinates “breaks down” in certain edge cases, but we don’t think that this invalidates pragmatic uses of this model that never use its full power (in terms of logical implications).
Immeasurable subsets are a phenomenon intimately tied to uncountable sets—i.e. ones that are even “larger” than the natural numbers (for instance, the real numbers are uncountable, but the rational numbers are not). This is roughly because the relevant concepts like σ-algebras and measures are defined in terms of countably many operations like unions or sums; and if you “fix” the measure of some sets in a way that’s consistent at all, then you can uniquely extend this to all sets you can get from those by taking complements and countable intersections and unions. In particular, if in a countable set you fix the measure of all singleton sets containing just one element, then this defines a unique measure on the set of all subsets.
Your examples of possible futures where people shout different natural numbers involve only countable sets. So it’s hard to see how we’d get any problem with immeasurable sets there.
You might be tempted to modify the example to argue that the set of possible futures is uncountably infinite because it contains people shouting all real numbers. However, (i) it’s not clear if it’s possible for people to shout any real number, (ii) even if it is then all my other remarks still apply, so I think this wouldn’t be a problem, certainly none specific to longtermism.
Regarding (i), the problem is that there is no general way to refer to an arbitraryreal number within a finite window of time. In particular, I cannot “shout” an infinite and non-period decimal expansion; nor can I “shout” a sequence of rational numbers that converges to the real number I want to refer to (except maybe in a few cases where the sequence is a closed-form function of n).
More generally, if utterances are individuated by the finite sequence of words I’m using, then (assuming a finite alphabet) there are only countably many possible utterances I can make. If that’s right then I cannot refer to an arbitrary real number precisely because there are “too many” of them.
Similarly, the set of all sequences of black or white balls is uncountable, but it’s unclear whether we should think that it’s contained in the set of all possible futures.
More importantly: if there were serious problems due to immeasurable sets—whether with longtermism or elsewhere—we could retreat to reasoning about a countable subset. For instance, if I’m worried that predicting the development of transformative AI is problematic because “time from now” is measured in real numbers, I could simply limit myself to only reasoning about rational numbers of (e.g.) seconds from now.
There may be legitimate arguments for this response being ‘ad hoc’ or otherwise problematic. (E.g. perhaps I would want to use properties of rational numbers that can only be proven by using real numbers “within the proof”.) But especially given the large practical utility of reasoning about e.g. volumes of space or probabilities of future events, I think it at least shows that immeasurability can’t ground a decisive knock-down argument.
However, rather than the argument depending too much on contingent properties of the world (e.g. whether it’s spatially infinite), the issue here is that they would depend on the axiomatization of mathematics.
The situation is roughly as follows: There are two different axiomatizations of mathematics with the following properties:
In both of them all maths that any of us are likely to ever “use in practice” works basically the same way.
For parallel situations (i.e. assignments of measure to some subsets of some set, which we’d like to extend to a measure on all subsets) there are immeasurable subsets in exactly one of the axiomatizations.
Specifically, for example, for our intuitive notion of “length” there are immeasurable subsets of the real numbers in the standard axiomatization of mathematics (called ZFC here). However, if we omit a single axiom—the axiom of choice—and replace it with an axiom that loosely says that there are weirdly large sets then every subset of the real numbers is measurable. [ETA: Actually it’s a bit more complicated, but I don’t think in a way that matters here. It doesn’t follow directly from these other axioms that everything is measurable, but using these axioms it’s possible to construct a “model of mathematics” in which that holds. Even less importantly, we don’t totally omit the axiom of choice but replace it with a weaker version.]
I think it would be pretty strange if the viability of longtermism depended on such considerations. E.g. imagine writing a letter to people in 1 million years explaining why you didn’t choose to try to help more rather than fewer of them. Or imagine getting such a letter from the distant past. I think I’d be pretty annoyed if I read “we considered helping you, but then we couldn’t decide between the axiom of choice and inaccessible cardinals …”.
Technical comments on type-1 arguments (those aiming to show there can be no probability measure). [Refer to the parent comment for the distinction between type 1 and type 2 arguments.]
I basically don’t see how such an argument could work. Apologies if that’s totally clear to you and you were just trying to make a type-2 argument. However, I worry that some readers might come away with the impression that there is a viable argument of type 1 since Vaden and you mention issues of measurability and infinite cardinality. These relate to actual mathematical results showing that for certain sets, measures with certain properties can’t exist at all.
However, I don’t think this is relevant to the case you describe. And I also don’t think it can be salvaged for an argument against longtermism.
First, in what sense can sets be “immeasurable”? The issue can arise in the following situation. Suppose we have some set (in this context “sample space”—think of the elements at all possible instances of things that can happen at the most fine-grained level), and some measure (in this context “probability”—but it could also refer to something we’d intuitively call length or volume) we would like to assign to some subsets (the subsets in this context are “events”—e.g. the event that Santa Clause enters my room now is represented by the subset containing all instances with that property).
In this situation, it can happen that there is no way to extend this measure to all subsets.
The classic example here is the real line as base set. We would like a measure that assigns measure |a−b| to each interval [a,b] (the set of real numbers from a to b), thus corresponding to our intuitive notion of length. E.g. the interval [−1,3] should have length 4.
However, it turns out that there is no measure that assigns each interval its length and ‘works’ for all subsets of the real numbers. I.e. each way of extending the assignment to all subsets of the real line would violate one of the properties we want measures to have (e.g. the measure of an at most countable disjoint union of sets should be the sum of the measures of the individual sets).
Thus we have to limit ourselves to assigning a measure to only some subsets. (In technical terms: we have to use a σ-algebra that’s strictly smaller than the full set of all subsets.) In other words, there are some subsets the measure of which we have to leave undefined. Those are immeasurable sets.
Second, why don’t I think this will be a problem in this context?
At the highest level, note that even if we are in a context with immeasurable sets this does not mean that we get no (probability) measure at all. It just means that the measure won’t “work” for all subsets/events. So for this to be an objection to longtermism, we would need a further argument for why specific events we care about are immeasurable—or in other words, why we can’t simply limit ourselves to the set of measurable events.
Note that immeasurable sets, to the extent that we can describe them concretely at all, are usually highly ‘weird’. If you try to google for pictures of standard examples like Vitali sets you won’t find a single one because we essentially can’t visualize them. Indeed, by design every set that we can construct from intervals by countably many standard operations like intersections and unions is measurable. So at least in the case of the real numbers, we arguably won’t encounter immeasurable sets “in practice”.
Note also that the phenomenon of immeasurable sets enables a number of counterintuitive results, such as the Banach-Tarski theorem. Loosely speaking this theorem says we can cut up a ball into pieces, and then by moving around those pieces and reassembling them get a ball that has twice the volume of the original ball; so for example “a pea can be chopped up and reassembled into the Sun”.
But usually the conclusion we draw from this is not that it’s meaningless to use numbers to refer to the coordinates of objects in space, or that our notion of volume is meaningless and that “we cannot measure the volume of objects” (and to the extent there is a problem it doesn’t exclusively apply to particularly large objects—just as any problem relevant to predicting the future wouldn’t specifically apply to longtermism). At most, we might wonder whether our model of space as continuous in real-number coordinates “breaks down” in certain edge cases, but we don’t think that this invalidates pragmatic uses of this model that never use its full power (in terms of logical implications).
Immeasurable subsets are a phenomenon intimately tied to uncountable sets—i.e. ones that are even “larger” than the natural numbers (for instance, the real numbers are uncountable, but the rational numbers are not). This is roughly because the relevant concepts like σ-algebras and measures are defined in terms of countably many operations like unions or sums; and if you “fix” the measure of some sets in a way that’s consistent at all, then you can uniquely extend this to all sets you can get from those by taking complements and countable intersections and unions. In particular, if in a countable set you fix the measure of all singleton sets containing just one element, then this defines a unique measure on the set of all subsets.
Your examples of possible futures where people shout different natural numbers involve only countable sets. So it’s hard to see how we’d get any problem with immeasurable sets there.
You might be tempted to modify the example to argue that the set of possible futures is uncountably infinite because it contains people shouting all real numbers. However, (i) it’s not clear if it’s possible for people to shout any real number, (ii) even if it is then all my other remarks still apply, so I think this wouldn’t be a problem, certainly none specific to longtermism.
Regarding (i), the problem is that there is no general way to refer to an arbitrary real number within a finite window of time. In particular, I cannot “shout” an infinite and non-period decimal expansion; nor can I “shout” a sequence of rational numbers that converges to the real number I want to refer to (except maybe in a few cases where the sequence is a closed-form function of n).
More generally, if utterances are individuated by the finite sequence of words I’m using, then (assuming a finite alphabet) there are only countably many possible utterances I can make. If that’s right then I cannot refer to an arbitrary real number precisely because there are “too many” of them.
Similarly, the set of all sequences of black or white balls is uncountable, but it’s unclear whether we should think that it’s contained in the set of all possible futures.
More importantly: if there were serious problems due to immeasurable sets—whether with longtermism or elsewhere—we could retreat to reasoning about a countable subset. For instance, if I’m worried that predicting the development of transformative AI is problematic because “time from now” is measured in real numbers, I could simply limit myself to only reasoning about rational numbers of (e.g.) seconds from now.
There may be legitimate arguments for this response being ‘ad hoc’ or otherwise problematic. (E.g. perhaps I would want to use properties of rational numbers that can only be proven by using real numbers “within the proof”.) But especially given the large practical utility of reasoning about e.g. volumes of space or probabilities of future events, I think it at least shows that immeasurability can’t ground a decisive knock-down argument.
As even more of an aside, type 1 arguments would also be vulnerable to a variant of Owen’s objection that they “prove too little”.
However, rather than the argument depending too much on contingent properties of the world (e.g. whether it’s spatially infinite), the issue here is that they would depend on the axiomatization of mathematics.
The situation is roughly as follows: There are two different axiomatizations of mathematics with the following properties:
In both of them all maths that any of us are likely to ever “use in practice” works basically the same way.
For parallel situations (i.e. assignments of measure to some subsets of some set, which we’d like to extend to a measure on all subsets) there are immeasurable subsets in exactly one of the axiomatizations.
Specifically, for example, for our intuitive notion of “length” there are immeasurable subsets of the real numbers in the standard axiomatization of mathematics (called ZFC here). However, if we omit a single axiom—the axiom of choice—and replace it with an axiom that loosely says that there are weirdly large sets then every subset of the real numbers is measurable. [ETA: Actually it’s a bit more complicated, but I don’t think in a way that matters here. It doesn’t follow directly from these other axioms that everything is measurable, but using these axioms it’s possible to construct a “model of mathematics” in which that holds. Even less importantly, we don’t totally omit the axiom of choice but replace it with a weaker version.]
I think it would be pretty strange if the viability of longtermism depended on such considerations. E.g. imagine writing a letter to people in 1 million years explaining why you didn’t choose to try to help more rather than fewer of them. Or imagine getting such a letter from the distant past. I think I’d be pretty annoyed if I read “we considered helping you, but then we couldn’t decide between the axiom of choice and inaccessible cardinals …”.