ℵ0 is a cardinal number, i.e. an isomorphism-class of sets
ω is an ordinal number, i.e. an isomorphism-class of well-ordered sets
∞ is an extended real number, i.e. a Dedekind cut of the rationals which fails the usual condition that the upper side is inhabited.
[Edited to add: ω is also the name of a hyperreal number, and also the name of a surreal number. That makes five different kinds of infinities on the table! Surreals include all ordinals and cardinals, and hyperreals include at least all countable ordinals.]
For the avoidance of confusion, the words “cardinal” and “ordinal” in this context are also used to mean:
Ordinal rankings can say pairwise which of two options is better
Cardinal rankings can put a number on any option, which enables assessment of lotteries in addition to ordinal ranking (by comparing two options’ numbers). There is no straightforward relationship between these uses of “ordinal” and “cardinal”.
The suggestion in the parent comment might be summarized with the slogan “for cardinal rankings, use ordinal numbers”, if that weren’t comically confusing. [Edit: actually it looks like the parent comment meant “for cardinal rankings, use hyperreals”; I’m currently leaning towards “use extended reals, unless those really can’t make the distinctions you need, in which case, use surreals.”]
However, I am not at all sure that ordinal numbers help us much here!
Although ω and ω+1 are distinct, that distinction comes from the order structure placed on their elements; as plain sets they are isomorphic. Where do we get this order structure on axiological “locations”?
Also, if each location has an ℝ-valued amount of ethical value, how could those numbers be aggregated into an overall ordinal number, when the topology of ordinals is totally disconnected?
(Disclaimer: I’m no mathematician, but I think I’m sufficiently familiar with the math in this post and comment.)
I’m quite confused about why you would use extended reals when you could use hyperreals (or surreals, which are just an extension of hyperreals). For example, my intuitions about infinite utility say we should have
∞+1 > ∞
2*∞ > ∞
∞*∞ > ∞
where “∞” is the amount of utility in some canonical infinite-utility universe and “>” means “is better than”.
Each of these works if “∞” represents an infinite hyperreal, but not if it’s just the infinite extended-real, right? There’s not just one positive and one negative possible-universe-value off the real line; it feels much more like there’s such a value for each hyperreal.
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 know ω as the smallest transfinite surreal number and atleast that is a ready source of transfinite arithmetic. Incidentally as {0,1,2,3,4,5...|} it also sounds an awful lot like the third definition. But the successor of that, {ω|} is hard to translate to isomorphisms-classes
For me ∞ is associated with “grows without limit”. In a system that allows only a handful of transfinites that might be a tempting to use as a numbers name. However in non-standard analysis integrating to ω integrates to a limit while “integrating to ∞” still exists but doesn’t refer to any number.
The successor of omega is (the equivalence class / order-type of) well-orderings that look like an infinite upward chain and then one more element above those. In general, if A is a set of ordinals that has a largest element then (A|) is the order-type of well-orderings that look like that largest element plus one more thing above it; if A is a set of ordinals that doesn’t have a largest element then (A|) is the order-type of the union of the things in A, with the understanding that if a<b with a,b in A then “corresponding” elements of a and b are identified.
(As soon as you start considering (A|B) with B nonempty, of course you are at risk of getting things that aren’t ordinals any more. But as long as you’re working with ordinals and keeping B empty, the surreal-number structure is closely related to the ordinal structure.)
It’s a while since I looked at this stuff in detail, but my recollection is that surreal numbers and nonstandard analysis don’t make very good playmates even though they are both extensions of our “usual” number systems to allow infinitesimals and infinities. One thing that makes nonstandard analysis interesting is the transfer principle (i.e., things you can say without referring to the special notions of nonstandard analysis are true about its extended number system iff they’re true about the ordinary real numbers) and so far as I know there is nothing at all like a transfer principle for the surreal numbers.
Just to spell things out a bit more:
ℵ0 is a cardinal number, i.e. an isomorphism-class of sets
ω is an ordinal number, i.e. an isomorphism-class of well-ordered sets
∞ is an extended real number, i.e. a Dedekind cut of the rationals which fails the usual condition that the upper side is inhabited.
[Edited to add: ω is also the name of a hyperreal number, and also the name of a surreal number. That makes five different kinds of infinities on the table! Surreals include all ordinals and cardinals, and hyperreals include at least all countable ordinals.]
For the avoidance of confusion, the words “cardinal” and “ordinal” in this context are also used to mean:
Ordinal rankings can say pairwise which of two options is better
Cardinal rankings can put a number on any option, which enables assessment of lotteries in addition to ordinal ranking (by comparing two options’ numbers). There is no straightforward relationship between these uses of “ordinal” and “cardinal”.
The suggestion in the parent comment might be summarized with the slogan “for cardinal rankings, use ordinal numbers”, if that weren’t comically confusing. [Edit: actually it looks like the parent comment meant “for cardinal rankings, use hyperreals”; I’m currently leaning towards “use extended reals, unless those really can’t make the distinctions you need, in which case, use surreals.”]
However, I am not at all sure that ordinal numbers help us much here!
Although ω and ω+1 are distinct, that distinction comes from the order structure placed on their elements; as plain sets they are isomorphic. Where do we get this order structure on axiological “locations”?
Also, if each location has an ℝ-valued amount of ethical value, how could those numbers be aggregated into an overall ordinal number, when the topology of ordinals is totally disconnected?
(Disclaimer: I’m no mathematician, but I think I’m sufficiently familiar with the math in this post and comment.)
I’m quite confused about why you would use extended reals when you could use hyperreals (or surreals, which are just an extension of hyperreals). For example, my intuitions about infinite utility say we should have
∞+1 > ∞
2*∞ > ∞
∞*∞ > ∞
where “∞” is the amount of utility in some canonical infinite-utility universe and “>” means “is better than”.
Each of these works if “∞” represents an infinite hyperreal, but not if it’s just the infinite extended-real, right? There’s not just one positive and one negative possible-universe-value off the real line; it feels much more like there’s such a value for each hyperreal.
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.
I know ω as the smallest transfinite surreal number and atleast that is a ready source of transfinite arithmetic. Incidentally as {0,1,2,3,4,5...|} it also sounds an awful lot like the third definition. But the successor of that, {ω|} is hard to translate to isomorphisms-classes
For me ∞ is associated with “grows without limit”. In a system that allows only a handful of transfinites that might be a tempting to use as a numbers name. However in non-standard analysis integrating to ω integrates to a limit while “integrating to ∞” still exists but doesn’t refer to any number.
The successor of omega is (the equivalence class / order-type of) well-orderings that look like an infinite upward chain and then one more element above those. In general, if A is a set of ordinals that has a largest element then (A|) is the order-type of well-orderings that look like that largest element plus one more thing above it; if A is a set of ordinals that doesn’t have a largest element then (A|) is the order-type of the union of the things in A, with the understanding that if a<b with a,b in A then “corresponding” elements of a and b are identified.
(As soon as you start considering (A|B) with B nonempty, of course you are at risk of getting things that aren’t ordinals any more. But as long as you’re working with ordinals and keeping B empty, the surreal-number structure is closely related to the ordinal structure.)
It’s a while since I looked at this stuff in detail, but my recollection is that surreal numbers and nonstandard analysis don’t make very good playmates even though they are both extensions of our “usual” number systems to allow infinitesimals and infinities. One thing that makes nonstandard analysis interesting is the transfer principle (i.e., things you can say without referring to the special notions of nonstandard analysis are true about its extended number system iff they’re true about the ordinary real numbers) and so far as I know there is nothing at all like a transfer principle for the surreal numbers.