I do want to add as a separate point that I think Joe’s invocation of large cardinal numbers as potential amounts of ethical value is probably confused. I don’t see any reason to use cardinal numbers for cardinal rankings (although they both involve the word “cardinal”). Perhaps cardinal numbers will someday allow us to make meaningful and useful distinctions between different levels of infinite quantities of value, but I doubt it, even without knowing quite how something more grounded in extended reals (like stochastic dominance) can resolve the paradoxes here. Fundamentally, cardinal numbers are not quantities, they are names for bijection-classes of sets (except when they are finite, in which case by definition they happen to correspond to ℕ, which then includes into ℝ).
I agree that it’s unlikely that we’d ever want infinite cardinal numbers to play a utility-like role. But it’s worth noting Conway’s “surreal numbers”, which extend the familiar real numbers with a rich variety of infinities and infinitesimals, and the infinities get (handwave handwave) “as big as the infinite ordinals”. (They are definitely more ordinal-like than cardinal-like, though they’re also definitely not the exact same thing as the ordinals either.)
So in the right framework these set-theoretic exotica do function as quantities, at least if being part of an ordered field that extends the real numbers counts as “functioning as quantities”, which I think it should.
That makes sense. I wasn’t really familiar with surreal numbers, but they indeed seem to be an ordered field extending the reals, and yet also including whatever large cardinals one cares to postulate in one’s foundations.
It looks like there’s a little bit of work already on using surreal numbers for probabilities and utility values, proving a surreal version of the vNM theorem and applying it to various Pascal’s-Wager-type scenarios. This seems like a solid direction if one really does want maximal expressive power in one’s degrees of infinitude.
Of course, you could argue that a bijection class of sets is there to formalize the notion of a quantity that generalizes to infinite sets.
Indeed, 2 sets like (1,2,3) and (A,B,C) have the same cardinality, which is 3, and cardinality is IMO the generalization of quantities to infinite sets like the real numbers. Indeed, 2 sets have the same cardinality if and only if they have a bijection.
That said, cardinality is a very coarse-grained way to look at quantities, as the Turing Machine computable sets and all arithmetical sets, which include Turing uncomputable sets, are both countable:
I agree, although I’m not sure Joe himself invokes cardinals; relevant quote from the post (I think?):
imagining worlds with a “strongly Ramsey” number of people seems likely to be a total non-starter, even if one knew what “strongly Ramsey” meant, which I don’t. Still, it seems like the infinite fanatic should be freaking out (drooling?). After all, what’s the use obsessing about the smallest possible infinities?
While cardinals might not make sense here, I strongly agree with Joe that we might need to care a lot about really big infinities. If we call the utility of a finite-size positive-value system that exists forever x utility, we can imagine x2 or x3 without too much trouble (infinite copies, or infinite value per copy per finite-time-unit, or both), and this has implications for our decisions.
I do want to add as a separate point that I think Joe’s invocation of large cardinal numbers as potential amounts of ethical value is probably confused. I don’t see any reason to use cardinal numbers for cardinal rankings (although they both involve the word “cardinal”). Perhaps cardinal numbers will someday allow us to make meaningful and useful distinctions between different levels of infinite quantities of value, but I doubt it, even without knowing quite how something more grounded in extended reals (like stochastic dominance) can resolve the paradoxes here. Fundamentally, cardinal numbers are not quantities, they are names for bijection-classes of sets (except when they are finite, in which case by definition they happen to correspond to ℕ, which then includes into ℝ).
I agree that it’s unlikely that we’d ever want infinite cardinal numbers to play a utility-like role. But it’s worth noting Conway’s “surreal numbers”, which extend the familiar real numbers with a rich variety of infinities and infinitesimals, and the infinities get (handwave handwave) “as big as the infinite ordinals”. (They are definitely more ordinal-like than cardinal-like, though they’re also definitely not the exact same thing as the ordinals either.)
So in the right framework these set-theoretic exotica do function as quantities, at least if being part of an ordered field that extends the real numbers counts as “functioning as quantities”, which I think it should.
That makes sense. I wasn’t really familiar with surreal numbers, but they indeed seem to be an ordered field extending the reals, and yet also including whatever large cardinals one cares to postulate in one’s foundations.
It looks like there’s a little bit of work already on using surreal numbers for probabilities and utility values, proving a surreal version of the vNM theorem and applying it to various Pascal’s-Wager-type scenarios. This seems like a solid direction if one really does want maximal expressive power in one’s degrees of infinitude.
Of course, you could argue that a bijection class of sets is there to formalize the notion of a quantity that generalizes to infinite sets.
Indeed, 2 sets like (1,2,3) and (A,B,C) have the same cardinality, which is 3, and cardinality is IMO the generalization of quantities to infinite sets like the real numbers. Indeed, 2 sets have the same cardinality if and only if they have a bijection.
That said, cardinality is a very coarse-grained way to look at quantities, as the Turing Machine computable sets and all arithmetical sets, which include Turing uncomputable sets, are both countable:
https://en.m.wikipedia.org/wiki/Cardinal_number
I agree, although I’m not sure Joe himself invokes cardinals; relevant quote from the post (I think?):
While cardinals might not make sense here, I strongly agree with Joe that we might need to care a lot about really big infinities. If we call the utility of a finite-size positive-value system that exists forever x utility, we can imagine x2 or x3 without too much trouble (infinite copies, or infinite value per copy per finite-time-unit, or both), and this has implications for our decisions.