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.
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.