No, the problem has nothing to do with infinitesimal probabilities. There are no infinitesimal probabilities in Oscar_Cunningham’s example, just arbitrarily small real ones. (Of course, they’re only “arbitrarily small” in the real numbers—not in the surreals!)
Thing is, you really, really can’t do limits (and thus infinite sums or integrals) in the surreals.
Just having infinitesimals is enough to screw some things up. Like the example Oscar_Cunningham gave—it seems like it should converge to 2; but in the surreals it doesn’t, because while it gets within any positive real distance of 2, it never gets within, say, 1/omega of 2. (He said it gets arbitrarily close to all of 2, 2-1/omega, and 2+1/omega^2, but really it doesn’t get arbitrarily close to any of them.)
This problem doesn’t even require the surreals, it happens as soon as you have infinitesimals—getting within any 1/n is now no longer arbitrarily close! This isn’t enough to ruin limits, mind you, but it is enough to ruin the ordinary limits you think should work (1/n no longer goes to zero). Add in enough infinitesimals and it will be impossible for sequences to converge, period.
(Edit: In case it’s not clear, here by “as soon as you have infinitesimals”, I mean “as soon as you have infinitesimals present in your system”, not “as soon as you try to take limits involving infinitesimals”. My point is that, as Oscar_Cunningham also pointed out, having infinitesimals present in the system causes the ordinary limits you’re used to to fail.)
Of course, that’s still not enough to ruin all limits ever. There could still be nets with limits; infinite sums are ruined, but maybe integration isn’t? But you didn’t just toss in lots of infinitesimals, you went straight to the surreals. Things are about to get much worse.
Let’s consider an especially simple case—the case of an increasing net. Then taking a limit of this net is just the same as taking the supremum of its set of values. And here we have a problem. See, the thing that makes the real numbers great for calculus is the least upper bound property. But in the surreals we have the opposite of that—no set of surreal numbers has a least upper bound, ever. Given any set S of surreals and any upper bound b, we can form the surreal number {S | b}; there’s always something inbetween. You have pretty much completely eliminated your ability to take limits.
At this point I think I’ve made my point pretty well, but for fun let’s demonstrate some more pathologies. How about the St. Petersburg bet? 2^-n probability of 2^n utility, yielding the infinite series 1+1+1+1+...; ordinarily we’d say this has expected value (or sum) infinity. But now we’ve got the surreals, so we need to say which infinity. Is it omega? In the ordinals—well, in the ordinals 2^-n doesn’t make sense, but the series 1+1+1+1+… would at least converge to omega. But here, well, why should it converge to omega and not omega-1? I mean, omega-1 is smaller than omega, so that’s a better candidate, right? So far this is really just the same argument as before, but it gets worse; what if we dropped that initial 1? If we were saying it converged to omega before, it had better converge to omega-1 now. (If we were saying it converged to omega-1 before, it had better converge to omega-2 now.) But we still have the same infinite series, so it had better converge to the same thing. (If we think of it as the infinite sequence (0, 1, 2, 3, 4, …) and just subtract 1 off each entry, the new sequence is cofinal with the old one, so it had better converge to the same thing also.)
Now it’s possible some things could be rescued. Limits of functions from surreals to surreals don’t seem like they’d necessarily always pose a problem, because if your input is surreals this gets around the problem of “you can’t get close enough with a set”. And so it’s possible even integration could be rescued. As I mentioned in a cousin comment, there was a failed attempt to come up with a theory of surreal integration, but that was for functions from surreals to surreals. Here we’re dealing with functions from some measure space to surreals, so that’s a bit different. Anyway, it might be possible. But I’d be very careful before assuming such a thing. As I’ve shown above, using surreals really throws a wrench into limits.
So, if you can come up with such a theory, by all means use it. But I wouldn’t go assuming the existence of such a thing until you’ve actually found it. Instead I would suggest specially constructing a system to accomplish your goals rather than reaching for something which sounds nice but is complete overkill.
Edit: And no, you can’t fix the problem by just relaxing the requirements for convergence. Then you really would get the non-uniqueness problem that Oscar_Cunningham points out. One obvious possibility that springs to mind is to break ties by least birthday; that’s a very surreal approach to things. (Don’t take the supremum of a set S, instead just take {S|}.) So 1+1+1+… really would converge to omega rather than something else, and Oscar_Cunningham’s example really would converge to 2. But it’s not clear to me that this would work nicely at all; in particular, you still have the pathology that dropping the initial “1” of 1+1+1+… somehow doesn’t cause the sum to drop by 1. Maybe something to explore, but not something to assume that it works. (I personally wouldn’t bet on it, though that’s not necessarily worth much; I am hardly an expert in the area.)
Of course, I think the best system here really is the real numbers, or rather the extended nonnegative real numbers. It only has one undifferentiated infinity, satisfying infinity-1=infinity, so we don’t have the problem that 1+1+1+1+… should converge to both infinity and infinity-1. It has the least upper bound property, so infinite sums (of positive things) are guaranteed to converge (possibly to infinity) -- this really is what forces the real numbers on us. There really is a reason integration is done with real numbers. (I for one would actually argue that utility should be bounded, but that’s an entirely separate argument.) Surreals, by contrast, aren’t just a bad setting for limits; they’re possibly the worst setting for limits.
Yeah, this is basically going to kill this, isn’t it. Oh well. Oops.
… yeah, if we’re going to use tiered values we might as well just explicitly make them program tiers, instead of bringing in a whole class’ worth of mathematical complication we don’t really need.
Well. Thanks! I can officially say I was less wrong than I was this morning.
Btw one thing worth noting if you really do want to work with surreals is that it may be more productive to think in terms of { stuff | stuff } rather than limits. (Similar to my “break ties by least birthday” suggestion.) Sequences don’t have limits in the surreals, but there is nonetheless a theory of surreal exponentiation based on {stuff | stuff}. Integration… well, it’s less obvious to me that integration based on limits should fail, but if it does, you could try to do it based on {stuff | stuff}. (The existing failed attempt at a theory of surreal integration takes that approach, though as I said above, that’s not really the same thing, as that’s for functions with the surreals as the domain.)
The extended non-negative reals really don’t do what the OP was looking for. They won’t even allow you to trade 1 life to save 10.000 lives, let alone have a hierarchy of values, some of which are tradable against each other and some of which are not.
Indeed, they certainly don’t. My point here isn’t “here is how you fix the problem with limits while still getting the things OP wanted”. My point here is “here is how you fix the problem with limits”. I make no claim that it is possible or desirable to get the things OP wanted. But yes I suppose it is possible that there may be some way to do so without completely screwing up limits, if we use a weird notion of limits.
Going back to the (extended) reals that do nothing interesting doesn’t strike me as a meaningful way of “fixing the problem with limits” in this context, when everybody knows that limits work for those… It doesn’t really fix any problem at all, it just says you can’t do certain things (namely, go beyond the (extended) reals) because that makes the problem come up.
Yes, that’s kind of my point. I’m not trying to do what the OP wanted and come up with a system of infinities that work nicely for this purpose. I’m trying to point out that there are very good reasons that we usually stick to the extended reals for this, that there are very real problems that crop up when you go beyond it, and that become especially problematic when you jump to the end and go all the way to the surreals.
I’m not trying to fix problems raised in the original post; I’m trying to point out that these are serious problems that the original post didn’t acknowledge—and the usual way we fix these is just not going beyond the extended reals at all so that they don’t crop up in the first place, because these really are serious problems. The ultimate problem here is coming up with a decision theory—or here just a theory of utility—and in that context, fixing the problem by abandoning goals that aren’t satisfiable and accepting the trivial solution that is forced on you is still fixing the problem. (Depending on just what you require, sticking to the extended reals may not be totally forced on you, but it is hard to avoid and this is a problem that the OP needs to appreciate.)
The point isn’t “this is how you fix the problem”, the point is “take a step back and get an appreciation for the problem and for what you’re really suggesting before you go rushing ahead like that”. The point isn’t “limits work in the extended reals”, the point is “limits work a lot less well if you go beyond there”. I personally think the whole idea is misguided and utilities should be bounded; but that is a separate argument. But if the OP really does want a viable theory along the lines he’s suggesting here even more than he wants the requirements that force the extended reals on us, then he’s got a lot more work to do.
Off the top of my head, if the surreals don’t allow of taking limits, the obvious mathematical move is to extend them so that they do (cf. rationals and reals). Has anyone done this?
I don’t think that’s really possible here. In general if you have an ordered field, there is a thing you can do called “completing” it, but I suspect this doesn’t really do what you want. Basically it adds in all limits of Cauchy nets, but all those sequences that stopped being convergent because you tossed in infinitesimals? They’re not Cauchy anymore either. If you really want limits to work great, you need the least upper bound property, and that takes you back to the reals.
Of course, we don’t necessarily need anything that strong—we don’t necessarily need limits to work as well as in the reals, and quite possibly it’s OK to redefine “limit” a bit. But I don’t think taking the completion solves the problem you want.
(I suppose nothing’s forcing us to work with a field, though. We could perhaps solve the problem by moving away from there.)
As for the question of completing the surreals, independent of whether this solves the problem or not—well, I have no idea whether anyone’s done this. Offhand thoughts:
You’re working with surreals, so you may have to worry about foundational issues. Those are probably ignorable though.
The surreals may already be complete, in the trivial sense that it is impossible to get a net to be Cauchy in a nontrivial manner.
Really, if we want limits for surreals, we need to be taking limits where the domain isn’t a set. Like I said above, limits of surreal functions of surreals should work fine, and it’s maybe possible to use this to get integration to work too. If you do this I suspect offhand any sort of completion will just be unnecessary (I could be very wrong about that though).
Which is the thing—if we want to complete it in a nontrivial sense, does that mean we’re going to have to allow “nets” with a proper class domain, or… uh… how would this work with filters? Yikes. Now you’re running into some foundational issues that may not be so ignorable.
Maybe it’s best to just ignore limits and try to formulate things in terms of {stuff | stuff} if you’re working with surreals.
I still think the surreals are an inappropriate setting.
(Edit: In case it’s not clear, here by “as soon as you have infinitesimals”, I mean “as soon as you have infinitesimals present in your system”, not “as soon as you try to take limits involving infinitesimals”. My point is that, as Oscar_Cunningham also pointed out, having infinitesimals present in the system causes the ordinary limits you’re used to to fail.)
And from current:
Basically it adds in all limits of Cauchy nets, but all those sequences that stopped being convergent because you tossed in infinitesimals? They’re not Cauchy anymore either. If you really want limits to work great, you need the least upper bound property, and that takes you back to the reals.
When you add infinites and infinitesimals to the reals (in the ordinary way, I haven’t worked out what happens for the surreals), then you can still have limits and Cauchy sequences, you just have to also let your sequences be infinitely long (that is, not just having infinite total length, but containing members that are infinitely far from the start). This is what happens with non-standard analysis, and there are even theorems saying that it all adds up to normality.
But I agree that surreals are not right for utilities, and that reals are (conditional on utilities being right), and that even considering just the pure mathematics, completing the surreals in some way would likely involve foundational issues.
When you add infinites and infinitesimals to the reals (in the ordinary way,
What on earth is the “ordinary way”? There are plenty of ways and I don’t know any of them to be the ordinary one. Do you mean considering the hyperreals?
(that is, not just having infinite total length, but containing members that are infinitely far from the start).
What? How does that help a sequence be Cauchy at all? If there are infinitesimals, the elements will have to get infinitesimally close; what they do at the start is irrelevant. Whether or not it’s possible for sequences to converge at all depends (roughly, I’m deliberately being loose here) on just how many infinitesimals there are.
This is what happens with non-standard analysis, and there are even theorems saying that it all adds up to normality.
I’ll admit to not being too familiar with non-standard analysis, but I’m not sure these theorems actually help here. Like if you’re thinking of the transfer principle, to transfer a statement about sequences in R, well, wouldn’t this transfer to a statement about functions from N* to R*? Or would that even work in the first place, being a statement about functions? Those aren’t first-order...
The hyperreals I’m pretty sure have enough infinitesimals that sequences can’t converge (though I’ll admit I don’t remember very well). This isn’t really that relevant to the hyperreals, though, since if you’re doing non-standard analysis, you don’t care about that; you care about things that have the appropriate domain and thus can actually transfer back to the reals in the first place. You don’t want to talk about sequences; you want to talk about functions whose domain is some hyper-thing, like the hyper-naturals. Or maybe just hyper-analogues of functions whose domain is some ordinary thing. I’ll admit to not knowing this too well. Regardless, that should get around the problem, in much the same way as in the surreals, if the domain is the surreals, it should largely get around the problem...
What on earth is the “ordinary way”? There are plenty of ways and I don’t know any of them to be the ordinary one. Do you mean considering the hyperreals?
Sorry, I think of non-standard analysis as being “the ordinary way” and the surreals as “the weird way”. I don’t know any others.
I’ll admit to not being too familiar with non-standard analysis, but I’m not sure these theorems actually help here. Like if you’re thinking of the transfer principle, to transfer a statement about sequences in R, well, wouldn’t this transfer to a statement about functions from N to R?
Yes, you get non-standard sequences indexed by N* instead of N, although what you actually do, which was the point of NSA, is express theorems about limits differently: if this is infinitesimal, that is infinitesimal.
I just thought of Googling “surreal analysis”, and it turns out to be a thing, with books. So one way or another, it seems to be possible to do derivatives and integrals in the surreal setting.
Sorry, I think of non-standard analysis as being “the ordinary way” and the surreals as “the weird way”. I don’t know any others.
Well R is the largest Archimedean ordered field, so any ordered extension of R will contain infinitesimals. The trivial way is just to adjoin one; e.g., take R[x] and declare x to be lexicographically smaller (or larger) than any element of R, and then pass to the field of fractions. Not particularly natural, obviously, but it demonstrates that saying “add infinitesimals” hardly picks out any construction in particular.
(FWIW, I think of surreals as “the kitchen sink way” and hyperreals as “that weird way that isn’t actually unique but does useful things because theorems from logic say it reflects on the reals”. :) )
Yes, you get non-standard sequences indexed by N* instead of N, although what you actually do, which was the point of NSA, is express theorems about limits differently: if this is infinitesimal, that is infinitesimal.
If I’m not mistaken, I think that’s just how you use would express limits of reals within the hyperreals; I don’t think you can necessarily express limits within the hyperreals themselves that way. (For instance, imagine a function f:R*->R* defined by “If x is not infinitesimal, f(x)=0; otherwise, f(x)=1/omega” (where omega denotes (1,2,3,...)). Obviously, that’s not the sort of function non-standard analysts care about! But if you want to consider the hyperreals in and of themselves rather than as a means to study the reals (which, admittedly, is pretty silly), then you are going to have to consider functions like that.)
I just thought of Googling “surreal analysis”, and it turns out to be a thing, with books. So one way or another, it seems to be possible to do derivatives and integrals in the surreal setting.
Oh, yes, I’ve seen that book, I’d forgotten! Be careful with your conclusion though. Derivatives (just using the usual definition) don’t seem like they should be a problem offhand, but I don’t think that book presents a theory of surreal integration (I’ve seen that book before and I feel like I would have remembered that, since I only remember a failed attempt). And I don’t know how general what he does is—for instance, the definition of e^x he gives only works for infinitesimal x (not an encouraging sign).
I’ll admit to being pretty ignorant as to what extent surreal analysis has advanced since then, though, and to what extent it’s based on limits vs. to what extent it’s based on {stuff | stuff}, though. I was trying to look up everything I could related to surreal exponentiation a while ago (which led to the MathOverflow question linked above), but that’s not exactly the same thing as infinite series or integrals...
I think you just have to look at the collection of Cauchy sequences where “sequence” means a function from the ordinals to the surreals, and “Cauchy” means that the terms eventually get smaller than any surreal.
I’d be skeptical of that assertion. Even sticking to ordinary topology on actual sets, transfinite sequences are not enough to do limits in general; in general you need nets. (Or filters.) Doesn’t mean you’ll need that here—might the fact that the surreals are linearly ordered help? -- but I don’t think it’s something you should assume would work.
But yeah it does seem like you’ll need something able to contain a “sequence” of order type that of the class of all ordinals; quantifying over ordinals or surreals or something in the “domain”. (Like, as I said above, limits of surreal-valued functions of a surreal variable shouldn’t pose a problem.)
In any case, sequences or nets are not necessarily the issue. This still doesn’t help with infinite sums, because those are still just ordinary omega-sequences. But really the issue is integration; infinite sums can be ignored if you can get integration. Does the “domain” there have sufficient granularity? Well, uh, I don’t know.
No, the problem has nothing to do with infinitesimal probabilities. There are no infinitesimal probabilities in Oscar_Cunningham’s example, just arbitrarily small real ones. (Of course, they’re only “arbitrarily small” in the real numbers—not in the surreals!)
Thing is, you really, really can’t do limits (and thus infinite sums or integrals) in the surreals.
Just having infinitesimals is enough to screw some things up. Like the example Oscar_Cunningham gave—it seems like it should converge to 2; but in the surreals it doesn’t, because while it gets within any positive real distance of 2, it never gets within, say, 1/omega of 2. (He said it gets arbitrarily close to all of 2, 2-1/omega, and 2+1/omega^2, but really it doesn’t get arbitrarily close to any of them.)
This problem doesn’t even require the surreals, it happens as soon as you have infinitesimals—getting within any 1/n is now no longer arbitrarily close! This isn’t enough to ruin limits, mind you, but it is enough to ruin the ordinary limits you think should work (1/n no longer goes to zero). Add in enough infinitesimals and it will be impossible for sequences to converge, period.
(Edit: In case it’s not clear, here by “as soon as you have infinitesimals”, I mean “as soon as you have infinitesimals present in your system”, not “as soon as you try to take limits involving infinitesimals”. My point is that, as Oscar_Cunningham also pointed out, having infinitesimals present in the system causes the ordinary limits you’re used to to fail.)
Of course, that’s still not enough to ruin all limits ever. There could still be nets with limits; infinite sums are ruined, but maybe integration isn’t? But you didn’t just toss in lots of infinitesimals, you went straight to the surreals. Things are about to get much worse.
Let’s consider an especially simple case—the case of an increasing net. Then taking a limit of this net is just the same as taking the supremum of its set of values. And here we have a problem. See, the thing that makes the real numbers great for calculus is the least upper bound property. But in the surreals we have the opposite of that—no set of surreal numbers has a least upper bound, ever. Given any set S of surreals and any upper bound b, we can form the surreal number {S | b}; there’s always something inbetween. You have pretty much completely eliminated your ability to take limits.
At this point I think I’ve made my point pretty well, but for fun let’s demonstrate some more pathologies. How about the St. Petersburg bet? 2^-n probability of 2^n utility, yielding the infinite series 1+1+1+1+...; ordinarily we’d say this has expected value (or sum) infinity. But now we’ve got the surreals, so we need to say which infinity. Is it omega? In the ordinals—well, in the ordinals 2^-n doesn’t make sense, but the series 1+1+1+1+… would at least converge to omega. But here, well, why should it converge to omega and not omega-1? I mean, omega-1 is smaller than omega, so that’s a better candidate, right? So far this is really just the same argument as before, but it gets worse; what if we dropped that initial 1? If we were saying it converged to omega before, it had better converge to omega-1 now. (If we were saying it converged to omega-1 before, it had better converge to omega-2 now.) But we still have the same infinite series, so it had better converge to the same thing. (If we think of it as the infinite sequence (0, 1, 2, 3, 4, …) and just subtract 1 off each entry, the new sequence is cofinal with the old one, so it had better converge to the same thing also.)
Now it’s possible some things could be rescued. Limits of functions from surreals to surreals don’t seem like they’d necessarily always pose a problem, because if your input is surreals this gets around the problem of “you can’t get close enough with a set”. And so it’s possible even integration could be rescued. As I mentioned in a cousin comment, there was a failed attempt to come up with a theory of surreal integration, but that was for functions from surreals to surreals. Here we’re dealing with functions from some measure space to surreals, so that’s a bit different. Anyway, it might be possible. But I’d be very careful before assuming such a thing. As I’ve shown above, using surreals really throws a wrench into limits.
So, if you can come up with such a theory, by all means use it. But I wouldn’t go assuming the existence of such a thing until you’ve actually found it. Instead I would suggest specially constructing a system to accomplish your goals rather than reaching for something which sounds nice but is complete overkill.
Edit: And no, you can’t fix the problem by just relaxing the requirements for convergence. Then you really would get the non-uniqueness problem that Oscar_Cunningham points out. One obvious possibility that springs to mind is to break ties by least birthday; that’s a very surreal approach to things. (Don’t take the supremum of a set S, instead just take {S|}.) So 1+1+1+… really would converge to omega rather than something else, and Oscar_Cunningham’s example really would converge to 2. But it’s not clear to me that this would work nicely at all; in particular, you still have the pathology that dropping the initial “1” of 1+1+1+… somehow doesn’t cause the sum to drop by 1. Maybe something to explore, but not something to assume that it works. (I personally wouldn’t bet on it, though that’s not necessarily worth much; I am hardly an expert in the area.)
Of course, I think the best system here really is the real numbers, or rather the extended nonnegative real numbers. It only has one undifferentiated infinity, satisfying infinity-1=infinity, so we don’t have the problem that 1+1+1+1+… should converge to both infinity and infinity-1. It has the least upper bound property, so infinite sums (of positive things) are guaranteed to converge (possibly to infinity) -- this really is what forces the real numbers on us. There really is a reason integration is done with real numbers. (I for one would actually argue that utility should be bounded, but that’s an entirely separate argument.) Surreals, by contrast, aren’t just a bad setting for limits; they’re possibly the worst setting for limits.
Arrgh.
Yeah, this is basically going to kill this, isn’t it. Oh well. Oops.
… yeah, if we’re going to use tiered values we might as well just explicitly make them program tiers, instead of bringing in a whole class’ worth of mathematical complication we don’t really need.
Well. Thanks! I can officially say I was less wrong than I was this morning.
Btw one thing worth noting if you really do want to work with surreals is that it may be more productive to think in terms of { stuff | stuff } rather than limits. (Similar to my “break ties by least birthday” suggestion.) Sequences don’t have limits in the surreals, but there is nonetheless a theory of surreal exponentiation based on {stuff | stuff}. Integration… well, it’s less obvious to me that integration based on limits should fail, but if it does, you could try to do it based on {stuff | stuff}. (The existing failed attempt at a theory of surreal integration takes that approach, though as I said above, that’s not really the same thing, as that’s for functions with the surreals as the domain.)
The extended non-negative reals really don’t do what the OP was looking for. They won’t even allow you to trade 1 life to save 10.000 lives, let alone have a hierarchy of values, some of which are tradable against each other and some of which are not.
Indeed, they certainly don’t. My point here isn’t “here is how you fix the problem with limits while still getting the things OP wanted”. My point here is “here is how you fix the problem with limits”. I make no claim that it is possible or desirable to get the things OP wanted. But yes I suppose it is possible that there may be some way to do so without completely screwing up limits, if we use a weird notion of limits.
Going back to the (extended) reals that do nothing interesting doesn’t strike me as a meaningful way of “fixing the problem with limits” in this context, when everybody knows that limits work for those… It doesn’t really fix any problem at all, it just says you can’t do certain things (namely, go beyond the (extended) reals) because that makes the problem come up.
Yes, that’s kind of my point. I’m not trying to do what the OP wanted and come up with a system of infinities that work nicely for this purpose. I’m trying to point out that there are very good reasons that we usually stick to the extended reals for this, that there are very real problems that crop up when you go beyond it, and that become especially problematic when you jump to the end and go all the way to the surreals.
I’m not trying to fix problems raised in the original post; I’m trying to point out that these are serious problems that the original post didn’t acknowledge—and the usual way we fix these is just not going beyond the extended reals at all so that they don’t crop up in the first place, because these really are serious problems. The ultimate problem here is coming up with a decision theory—or here just a theory of utility—and in that context, fixing the problem by abandoning goals that aren’t satisfiable and accepting the trivial solution that is forced on you is still fixing the problem. (Depending on just what you require, sticking to the extended reals may not be totally forced on you, but it is hard to avoid and this is a problem that the OP needs to appreciate.)
The point isn’t “this is how you fix the problem”, the point is “take a step back and get an appreciation for the problem and for what you’re really suggesting before you go rushing ahead like that”. The point isn’t “limits work in the extended reals”, the point is “limits work a lot less well if you go beyond there”. I personally think the whole idea is misguided and utilities should be bounded; but that is a separate argument. But if the OP really does want a viable theory along the lines he’s suggesting here even more than he wants the requirements that force the extended reals on us, then he’s got a lot more work to do.
Off the top of my head, if the surreals don’t allow of taking limits, the obvious mathematical move is to extend them so that they do (cf. rationals and reals). Has anyone done this?
I don’t think that’s really possible here. In general if you have an ordered field, there is a thing you can do called “completing” it, but I suspect this doesn’t really do what you want. Basically it adds in all limits of Cauchy nets, but all those sequences that stopped being convergent because you tossed in infinitesimals? They’re not Cauchy anymore either. If you really want limits to work great, you need the least upper bound property, and that takes you back to the reals.
Of course, we don’t necessarily need anything that strong—we don’t necessarily need limits to work as well as in the reals, and quite possibly it’s OK to redefine “limit” a bit. But I don’t think taking the completion solves the problem you want.
(I suppose nothing’s forcing us to work with a field, though. We could perhaps solve the problem by moving away from there.)
As for the question of completing the surreals, independent of whether this solves the problem or not—well, I have no idea whether anyone’s done this. Offhand thoughts:
You’re working with surreals, so you may have to worry about foundational issues. Those are probably ignorable though.
The surreals may already be complete, in the trivial sense that it is impossible to get a net to be Cauchy in a nontrivial manner.
Really, if we want limits for surreals, we need to be taking limits where the domain isn’t a set. Like I said above, limits of surreal functions of surreals should work fine, and it’s maybe possible to use this to get integration to work too. If you do this I suspect offhand any sort of completion will just be unnecessary (I could be very wrong about that though).
Which is the thing—if we want to complete it in a nontrivial sense, does that mean we’re going to have to allow “nets” with a proper class domain, or… uh… how would this work with filters? Yikes. Now you’re running into some foundational issues that may not be so ignorable.
Maybe it’s best to just ignore limits and try to formulate things in terms of {stuff | stuff} if you’re working with surreals.
I still think the surreals are an inappropriate setting.
From an ancestor:
And from current:
When you add infinites and infinitesimals to the reals (in the ordinary way, I haven’t worked out what happens for the surreals), then you can still have limits and Cauchy sequences, you just have to also let your sequences be infinitely long (that is, not just having infinite total length, but containing members that are infinitely far from the start). This is what happens with non-standard analysis, and there are even theorems saying that it all adds up to normality.
But I agree that surreals are not right for utilities, and that reals are (conditional on utilities being right), and that even considering just the pure mathematics, completing the surreals in some way would likely involve foundational issues.
What on earth is the “ordinary way”? There are plenty of ways and I don’t know any of them to be the ordinary one. Do you mean considering the hyperreals?
What? How does that help a sequence be Cauchy at all? If there are infinitesimals, the elements will have to get infinitesimally close; what they do at the start is irrelevant. Whether or not it’s possible for sequences to converge at all depends (roughly, I’m deliberately being loose here) on just how many infinitesimals there are.
I’ll admit to not being too familiar with non-standard analysis, but I’m not sure these theorems actually help here. Like if you’re thinking of the transfer principle, to transfer a statement about sequences in R, well, wouldn’t this transfer to a statement about functions from N* to R*? Or would that even work in the first place, being a statement about functions? Those aren’t first-order...
The hyperreals I’m pretty sure have enough infinitesimals that sequences can’t converge (though I’ll admit I don’t remember very well). This isn’t really that relevant to the hyperreals, though, since if you’re doing non-standard analysis, you don’t care about that; you care about things that have the appropriate domain and thus can actually transfer back to the reals in the first place. You don’t want to talk about sequences; you want to talk about functions whose domain is some hyper-thing, like the hyper-naturals. Or maybe just hyper-analogues of functions whose domain is some ordinary thing. I’ll admit to not knowing this too well. Regardless, that should get around the problem, in much the same way as in the surreals, if the domain is the surreals, it should largely get around the problem...
Sorry, I think of non-standard analysis as being “the ordinary way” and the surreals as “the weird way”. I don’t know any others.
Yes, you get non-standard sequences indexed by N* instead of N, although what you actually do, which was the point of NSA, is express theorems about limits differently: if this is infinitesimal, that is infinitesimal.
I just thought of Googling “surreal analysis”, and it turns out to be a thing, with books. So one way or another, it seems to be possible to do derivatives and integrals in the surreal setting.
Well R is the largest Archimedean ordered field, so any ordered extension of R will contain infinitesimals. The trivial way is just to adjoin one; e.g., take R[x] and declare x to be lexicographically smaller (or larger) than any element of R, and then pass to the field of fractions. Not particularly natural, obviously, but it demonstrates that saying “add infinitesimals” hardly picks out any construction in particular.
(FWIW, I think of surreals as “the kitchen sink way” and hyperreals as “that weird way that isn’t actually unique but does useful things because theorems from logic say it reflects on the reals”. :) )
If I’m not mistaken, I think that’s just how you use would express limits of reals within the hyperreals; I don’t think you can necessarily express limits within the hyperreals themselves that way. (For instance, imagine a function f:R*->R* defined by “If x is not infinitesimal, f(x)=0; otherwise, f(x)=1/omega” (where omega denotes (1,2,3,...)). Obviously, that’s not the sort of function non-standard analysts care about! But if you want to consider the hyperreals in and of themselves rather than as a means to study the reals (which, admittedly, is pretty silly), then you are going to have to consider functions like that.)
Oh, yes, I’ve seen that book, I’d forgotten! Be careful with your conclusion though. Derivatives (just using the usual definition) don’t seem like they should be a problem offhand, but I don’t think that book presents a theory of surreal integration (I’ve seen that book before and I feel like I would have remembered that, since I only remember a failed attempt). And I don’t know how general what he does is—for instance, the definition of e^x he gives only works for infinitesimal x (not an encouraging sign).
I’ll admit to being pretty ignorant as to what extent surreal analysis has advanced since then, though, and to what extent it’s based on limits vs. to what extent it’s based on {stuff | stuff}, though. I was trying to look up everything I could related to surreal exponentiation a while ago (which led to the MathOverflow question linked above), but that’s not exactly the same thing as infinite series or integrals...
I think you just have to look at the collection of Cauchy sequences where “sequence” means a function from the ordinals to the surreals, and “Cauchy” means that the terms eventually get smaller than any surreal.
I’d be skeptical of that assertion. Even sticking to ordinary topology on actual sets, transfinite sequences are not enough to do limits in general; in general you need nets. (Or filters.) Doesn’t mean you’ll need that here—might the fact that the surreals are linearly ordered help? -- but I don’t think it’s something you should assume would work.
But yeah it does seem like you’ll need something able to contain a “sequence” of order type that of the class of all ordinals; quantifying over ordinals or surreals or something in the “domain”. (Like, as I said above, limits of surreal-valued functions of a surreal variable shouldn’t pose a problem.)
In any case, sequences or nets are not necessarily the issue. This still doesn’t help with infinite sums, because those are still just ordinary omega-sequences. But really the issue is integration; infinite sums can be ignored if you can get integration. Does the “domain” there have sufficient granularity? Well, uh, I don’t know.