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...
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...