(Hey, a note, you should probably learn to use the blockquote feature. I dunno where it is in the rich text editor if you’re using that, but if you’re using the Markdown editor you just precede the paragraph you’re quoting with a “>”. It will make your posts substantially more readable.)
Are you sure this chain of reasoning is correct?
Yes.
Consider 1/2x. For any finite number of terms it will be greater than ε, but as x approaches ω, it should approach 1/2ω.
What “terms”? What are you talking about? This isn’t a sequence or a sum; there are no “terms” here. Yes, even in the surreals, as x goes to ω, 1/(2x) will approach 1/(2ω), as you say; as I mentioned above, limits of functions of a surreal variable will indeed still work. But that has no relevance to the case under discussion.
(And, while it’s not necessary to see what’s going on here, it may be helpful to remember that if we if we interpret this as occurring in the surreals, then in the case of 1/2x as x→ω, your domain has proper-class cofinality, while in the case of this infinite sum, the domain has cofinality ω. So the former can work, and the latter cannot. Again, one doesn’t need this to see that—the partial sum can’t get within 1/ω of e even when the cofinality is countable—but it may be worth remembering.)
Why can’t the partial sum get within 1/ω of e?
Because the partial sum is always a rational number. A rational number—more generally, a real number—cannot be infinitesimally close to e without being e. (By contrast, for surreal x, 1/(2x) certainly does not need to be a real number, and so can get infinitesimally close to 1/(2ω) without being equal to it.)
You’re right that it won’t be a nice neat quotient group. But here’s an example. N_0 - N_0 can equal any integer where N_0 is a cardinal, or even +/- N_0, but in surreal numbers it works as follows. Suppose X and Y are countable infinities. Then X—Y has a unique value that we can sometimes identify. For example, if X represents the length of a sequence and Y is all the elements in the sequences except one, then X—Y = 1. We can perform the calculation in the surreals, or we can perform it in the cardinals and receive a broad range of possible answers. But for every possible answer in the cardinals, we can find pairs of surreal numbers that would provide that answer.
What??
OK. Look. I could spend my time attempting to pick this apart. But, let me be blunt, the point I am trying to get across here is that you are talking nonsense. This is babble. You are way out of your depth, dude. You don’t know what you are talking about. You need to go back and relearn this from the beginning. I don’t even know what mistake you’re making, because it’s not a common one I recognize.
Just in the hopes it might be somewhat helpful, I will quickly go over the things I can maybe address quickly:
N_0 - N_0 can equal any integer where N_0 is a cardinal, or even +/- N_0, but in surreal numbers it works as follows.
I have no idea what this sentence is talking about.
Suppose X and Y are countable infinities.
What’s an “infinity”? An ordinal? A cardinal? (There’s only one countably infinite cardinal...) A surreal or something else entirely? You said “countable”, so it has to be something to which the notion of countability applies!
This mistake, at least, I think I can identify. Maybe you should, in fact, look over that “quick guide to the infinite” I wrote, because this is myth #0 I discussed there. There’s no such thing as a unified notion of “infinities”. There are different systems of numbers, some of them contain numbers/objects that are infinite (i.e.: larger in magnitude than any whole number), there is not some greater unified system they are all a part of.
Then X—Y has a unique value that we can sometimes identify.
What is X-Y? I don’t even know what system of numbers you’re using, so I don’t know what this means.
If X and Y are surreals, then, sure, there’s quite definitely a unique surreal X-Y. This is true more generally if you’re thinking of X and Y as living in some sort of ordered field or ring.
If X and Y are cardinals, then X-Y may not be well-defined. Trivially so if Y>X (no possible values), but let’s ignore that case. Even ignoring that, if X and Y are infinite, X-Y may fail to be well-defined due to having multiple possible values.
If X and Y are ordinals, we have to ask what sort of addition we’re using. If we’re using natural addition, then X-Y certainly has a unique value in the surreals, but it may or may not be an ordinal, so it’s not necessarily well-defined within the ordinals.
If we’re using ordinary addition, we have to distinguish between X-Y and -Y+X. (The latter just being a way of denoting “subtracting on the left”; it should not be interpreted as actually negating Y and adding to X.) -Y+X will have a unique value so long as Y≤X, but X-Y is a different story; even restricting to Y≤X, if X is infinite, then X-Y may have multiple possible values or none.
For example, if X represents the length of a sequence and Y is all the elements in the sequences except one, then X—Y = 1.
Yeah, not going to try to pick this apart, in short though this is nonsense.
I’m starting to think though that maybe you meant that X and Y were infinite sets, rather than some sort of numbers? With X-Y being the set difference? But that is not what you said. Simply put you seem very confused about all this.
We can perform the calculation in the surreals, or we can perform it in the cardinals and receive a broad range of possible answers.
Are X and Y surreals or are they cardinals? Surreals and cardinals don’t mix, dude! It can’t be both, not unless they’re just whole numbers! You are performing the calculation in whatever number system these things live in.
You just said above you get a well-defined answer, and, moreover, that it’s 1! Now you’re telling me that you can get a broad range of possible answers??
If X is representing the length of a sequence, it should probably be an ordinal. As for Y… yeah, OK, not going to try to make sense of the thing I already said I wouldn’t attempt to pick through.
And if X and Y are sets rather than numbers… oh, to hell with it, I’m just going to move on.
But for every possible answer in the cardinals, we can find pairs of surreal numbers that would provide that answer.
There is, I think, a correct idea here that is rescuable. It also seems pretty clear you don’t know enough to perform that rescue yourself and rephrase this as something that makes sense. (A hint, though: The fixed version probably should not involve surreals.)
(Do surreal numbers even have cardinalities, in a meaningful sense? Yes obviously if you pick a particular way of representing surreals as sets, e.g. by representing them as sign sequences, the resulting representations will have cardinalities; obviously, that’s not what I’m talking about. Although, who knows, maybe that’s a workable notion—define the cardinality of a surreal to be the cardinality of its birthday. No idea if that’s actually relevant to anything, though.)
Even charitably interpreted, none of this matches up with your comments above about equivalence classes. It relates, sure, but it doesn’t match. What you said above was that you could solve more equations by passing to equivalence classes. What you’re saying now seems to be… not that.
Long story short: I really, really, do not think you have much idea what you are talking about. You really need to relearn this from scratch, and not starting with surreals. I definitely do not think you are prepared to go instructing others on their uses; at this point I’m not convinced you could clearly articulate what ordinals and cardinals are for, you’ve gotten everything so mixed up in your comment above. I wouldn’t recommend trying to expand this into a post.
I think I should probably stop arguing here. If you reply to this with more babble I’m not going to waste my time replying to it further.
I really appreciate the time you’ve put into writing these long responses and I’ll admit that there are some gaps in my understanding, but I don’t think you’ve understood what I was saying at all. I suspect that this is a hazard with producing informal overviews of ideas + illusion of transparency. One example, when I said equivalence classes, I really meant something like equivalence classes. Anyway, in regards to all the points you’ve raised it’d take a lot of space to respond to them all, so I think I’ll just add a link to my post when I get time to write it.
(Hey, a note, you should probably learn to use the blockquote feature. I dunno where it is in the rich text editor if you’re using that, but if you’re using the Markdown editor you just precede the paragraph you’re quoting with a “>”. It will make your posts substantially more readable.)
Yes.
What “terms”? What are you talking about? This isn’t a sequence or a sum; there are no “terms” here. Yes, even in the surreals, as x goes to ω, 1/(2x) will approach 1/(2ω), as you say; as I mentioned above, limits of functions of a surreal variable will indeed still work. But that has no relevance to the case under discussion.
(And, while it’s not necessary to see what’s going on here, it may be helpful to remember that if we if we interpret this as occurring in the surreals, then in the case of 1/2x as x→ω, your domain has proper-class cofinality, while in the case of this infinite sum, the domain has cofinality ω. So the former can work, and the latter cannot. Again, one doesn’t need this to see that—the partial sum can’t get within 1/ω of e even when the cofinality is countable—but it may be worth remembering.)
Because the partial sum is always a rational number. A rational number—more generally, a real number—cannot be infinitesimally close to e without being e. (By contrast, for surreal x, 1/(2x) certainly does not need to be a real number, and so can get infinitesimally close to 1/(2ω) without being equal to it.)
What??
OK. Look. I could spend my time attempting to pick this apart. But, let me be blunt, the point I am trying to get across here is that you are talking nonsense. This is babble. You are way out of your depth, dude. You don’t know what you are talking about. You need to go back and relearn this from the beginning. I don’t even know what mistake you’re making, because it’s not a common one I recognize.
Just in the hopes it might be somewhat helpful, I will quickly go over the things I can maybe address quickly:
I have no idea what this sentence is talking about.
What’s an “infinity”? An ordinal? A cardinal? (There’s only one countably infinite cardinal...) A surreal or something else entirely? You said “countable”, so it has to be something to which the notion of countability applies!
This mistake, at least, I think I can identify. Maybe you should, in fact, look over that “quick guide to the infinite” I wrote, because this is myth #0 I discussed there. There’s no such thing as a unified notion of “infinities”. There are different systems of numbers, some of them contain numbers/objects that are infinite (i.e.: larger in magnitude than any whole number), there is not some greater unified system they are all a part of.
What is X-Y? I don’t even know what system of numbers you’re using, so I don’t know what this means.
If X and Y are surreals, then, sure, there’s quite definitely a unique surreal X-Y. This is true more generally if you’re thinking of X and Y as living in some sort of ordered field or ring.
If X and Y are cardinals, then X-Y may not be well-defined. Trivially so if Y>X (no possible values), but let’s ignore that case. Even ignoring that, if X and Y are infinite, X-Y may fail to be well-defined due to having multiple possible values.
If X and Y are ordinals, we have to ask what sort of addition we’re using. If we’re using natural addition, then X-Y certainly has a unique value in the surreals, but it may or may not be an ordinal, so it’s not necessarily well-defined within the ordinals.
If we’re using ordinary addition, we have to distinguish between X-Y and -Y+X. (The latter just being a way of denoting “subtracting on the left”; it should not be interpreted as actually negating Y and adding to X.) -Y+X will have a unique value so long as Y≤X, but X-Y is a different story; even restricting to Y≤X, if X is infinite, then X-Y may have multiple possible values or none.
Yeah, not going to try to pick this apart, in short though this is nonsense.
I’m starting to think though that maybe you meant that X and Y were infinite sets, rather than some sort of numbers? With X-Y being the set difference? But that is not what you said. Simply put you seem very confused about all this.
Are X and Y surreals or are they cardinals? Surreals and cardinals don’t mix, dude! It can’t be both, not unless they’re just whole numbers! You are performing the calculation in whatever number system these things live in.
You just said above you get a well-defined answer, and, moreover, that it’s 1! Now you’re telling me that you can get a broad range of possible answers??
If X is representing the length of a sequence, it should probably be an ordinal. As for Y… yeah, OK, not going to try to make sense of the thing I already said I wouldn’t attempt to pick through.
And if X and Y are sets rather than numbers… oh, to hell with it, I’m just going to move on.
There is, I think, a correct idea here that is rescuable. It also seems pretty clear you don’t know enough to perform that rescue yourself and rephrase this as something that makes sense. (A hint, though: The fixed version probably should not involve surreals.)
(Do surreal numbers even have cardinalities, in a meaningful sense? Yes obviously if you pick a particular way of representing surreals as sets, e.g. by representing them as sign sequences, the resulting representations will have cardinalities; obviously, that’s not what I’m talking about. Although, who knows, maybe that’s a workable notion—define the cardinality of a surreal to be the cardinality of its birthday. No idea if that’s actually relevant to anything, though.)
Even charitably interpreted, none of this matches up with your comments above about equivalence classes. It relates, sure, but it doesn’t match. What you said above was that you could solve more equations by passing to equivalence classes. What you’re saying now seems to be… not that.
Long story short: I really, really, do not think you have much idea what you are talking about. You really need to relearn this from scratch, and not starting with surreals. I definitely do not think you are prepared to go instructing others on their uses; at this point I’m not convinced you could clearly articulate what ordinals and cardinals are for, you’ve gotten everything so mixed up in your comment above. I wouldn’t recommend trying to expand this into a post.
I think I should probably stop arguing here. If you reply to this with more babble I’m not going to waste my time replying to it further.
I really appreciate the time you’ve put into writing these long responses and I’ll admit that there are some gaps in my understanding, but I don’t think you’ve understood what I was saying at all. I suspect that this is a hazard with producing informal overviews of ideas + illusion of transparency. One example, when I said equivalence classes, I really meant something like equivalence classes. Anyway, in regards to all the points you’ve raised it’d take a lot of space to respond to them all, so I think I’ll just add a link to my post when I get time to write it.