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 …”.
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 …”.