(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...
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...