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Or an even stronger question: the existence of the hyperreals requires the ultrafilter lemma. There is much later research showing that given the right set theory, one can show that the surreals are isomorphic to a proper-class sized ultrapower of the reals. But the surreals don't require the ultrafilter lemma at all, and yet Conway still makes that claim. It would seem to be possible to just go through the motions with taking the nonstandard derivative of, for instance, $\frac{\exp(x + 1/\omega) - \exp(x)}{1/\omega} = \exp{x}\left(\frac{\exp^{1/\omega} - 1}{1/\omega}\right) \approx \exp(x)$, treating $1/\omega$ as a formal expression satisfying the first-order properties of the reals and deriving $\exp(x)$ as the closest real number to the result, and none of this seems to have required any kind of ultrafilter to be certain that $1/\omega$ exists to begin with as it does with the hyperreals.

Or an even stronger question: do we need anything like an ultrafilter lemma for Conway's claim to be true? Unlike the hyperreals, the surreals don't require any ultrafilter at all -- which is a pretty significant achievement, really, since we are at least certain something like $1/\omega$ exists even in ZF, and that the resulting field is real-closed. It would seem to be possible to just go through the motions with taking the nonstandard derivative of, for instance, $\frac{\exp(x + 1/\omega) - \exp(x)}{1/\omega} = \exp{x}\left(\frac{\exp^{1/\omega} - 1}{1/\omega}\right) \approx \exp(x)$, treating $1/\omega$ as a formal expression satisfying the first-order properties of the reals and deriving $\exp(x)$ as the closest real number to the result, none of which seems to have required any kind of ultrafilter. (Or has it, implicitly?)

Conway seems to make clear that you could, if you wanted, use the surreal numbers for non-standard analysis. This is because they are a real-closed field and thus a model for the theory of all elementary (first-order) statements about the reals. Conway does also make the point, in the last two sentences, that he doesn't view nonstandard analysis as the ultimate application of the surreals. But, for the sake of curiosity, I'm quite interested in understanding how you could do what Conway is hinting at above.

The statement that you could use the surreals for nonstandard analysis would seem to be much stronger than just saying it's a real-closed field. For instance, the real algebraic numbers are real-closed, but that doesn't mean that for some real algebraic $x$, $\sin(x)$ is also a real algebraic number. But we do have the corresponding result for the hyperreals, which is why we can transfer functions like $\sin$ to begin with to take nonstandard derivatives. So Conway would appear to be asserting an analogous "transfer principle" exists in some form that would let some general extension of $f(x + 1/\omega)$ for arbitrary $f$ to even exist for the surreals at all, in his expression above. (Or if he doesn't mean this I don't know how else to interpret it.)

So the main question is: how would such a transfer principle work? Even in a wishy-washy non-constructible sense. If you're using an ultrapower of the reals, everything is rather straightforward: every hyperreal is a set of reals (or an equivalence class thereof), and to transfer any first-order predicate to some hyperreal, you simply ask the predicate of every real in the set and see if it's true of "most" of them (where "most" means "in the ultrafilter"). Thus you have a real-closed field, a "transfer principle," and all of that. But is there some way to get a similar result using the language of the surreals, doing something clever with birthdays, left and right sets, and so on?

Some additional details:

There has been a little bit of prior discussion about this but not quite as directly as this question is asking. But for instance see Surreal numbers vs. non-standard analysis, where it is talked about that the surreals do have a kind of transfer principle in that they are isomorphic to the proper-class sized ultrapower of the reals (with many possible isomorphisms), and thus each isomorphism can be thought of as (unconstructably) defining a corresponding way to transfer first-order properties of reals on the surreal numbers. But also this transfer principle isn't quite as "natural" in the sense that there doesn't seem to be any "standard" choice of isomorphism. The punchline of all of this was, even though there's this isomorphism in theory, there is no canonical way to say what $\sin(\omega)$ is, for instance, or even $\omega \bmod 2$.

  On the other hand, though, it doesn't seem that this is any different from the way the hyperreals work. With those, we have the same problem, which is that all of the ultrafilters are typically non-constructive. And for instance, just like there is some debate about how the $\sin$ function should transfer to the surreals, or what $\sin(\omega)$ should be, there is also no real clear answer regarding what $\sin((1,2,3,4,\dotsc))$ should be, where $(1,2,3,4,\dotsc)$ is a particular hyperreal number - because the answer depends on the ultrafilter, which determines what $(1,2,3,4,\dotsc)$ even means to begin with, or what properties it has. So it doesn't really seem like the surreals are any better or worse off than the hyperreals in this regard.

And yet Conway is making the claim that he is. So there is this subtlety here regarding what really constitutes a "transfer principle" that I am hoping to get cleared up. Basically, with the hyperreals, part of the appeal is that even though you usually can't quite hold an ultrafilter in the palm of your hand the way you'd like, you can get a good bit of the way there constructively. You know, for instance, that whatever $\sin((1,2,3,4,\dotsc))$ is, in some sense it's $(\sin(1), \sin(2), \sin(3), \sin(4), \dotsc)$. Since we don't know what the ultrafilter is we don't know exactly what properties that has, but we do know something: we know that for any ultrafilter this value will not be an integer, for instance. We can also "partly construct" an ultrafilter by constructing one that consistently assigns a membership value to all of the definable subsets of indices, and leave the undefinable ones open in a sort of choose-your-own-adventure approach, which is related to what Terry Tao calls "cheap nonstandard analysis". And then there are certain models of set theory which really do have constructible ultrafilters.

Summarizing, it seems implicit in Conway's assertion that you should also at least be able to do this kind of thing, that is to get "part of the way there" — at least have enough "transfer" for us to be sure that the expression $f(1 + 1/\omega)$ even makes sense to begin with. Perhaps part of my question is not quite so formal, but more in the direction of if it's possible to similarly get "part of the way there" but directly using the language of birthdays, left and right sets, etc., and all that stuff that makes the surreal numbers interesting, rather than sort of cheating and just using this isomorphism from the hyperreals. At least that seems like what Conway is suggesting, as it seems that it wasn't until later work by Keisler and then Ehrlich that it was proven that the proper class sized ultrapower of the reals and the surreals are isomorphic, if I interpret the above link correctly, but Conway clearly had something in mind even prior to that.

Conway here makes clear that you could, if you wanted, use the surreal numbers for non-standard analysis, because they are a real-closed field and thus a model for the theory of all elementary (first-order) statements about the reals. Conway does also make the point, in the last two sentences, that he doesn't view nonstandard analysis as the ultimate application of the surreals. But, for the sake of curiosity, I'm quite interested in understanding how you could do what Conway is hinting at above.

However, the statement that you can use the surreals for nonstandard analysis is really quite strong, and much stronger than just the field being real-closed. The real meat of the claim being made is the expression $f(x + 1/\omega)$ even exists at all. This would demand some kind of "transfer principle" for $f(x)$ to the surreals. Just being real-closed wouldn't be enough for this: the real algebraic numbers are real closed, but that doesn't mean we can use the real algebraic numbers for nonstandard analysis. But Conway says this is "of course" possible with the surreals.

So the main question is: how would such a transfer principle work?

Or an even stronger question: the existence of the hyperreals requires the ultrafilter lemma. There is much later research showing that given the right set theory, one can show that the surreals are isomorphic to a proper-class sized ultrapower of the reals. But the surreals don't require the ultrafilter lemma at all, and yet Conway still makes that claim. It would seem to be possible to just go through the motions with taking the nonstandard derivative of, for instance, $\frac{\exp(x + 1/\omega) - \exp(x)}{1/\omega} = \exp{x}\left(\frac{\exp^{1/\omega} - 1}{1/\omega}\right) \approx \exp(x)$, treating $1/\omega$ as a formal expression satisfying the first-order properties of the reals and deriving $\exp(x)$ as the closest real number to the result, and none of this seems to have required any kind of ultrafilter to be certain that $1/\omega$ exists to begin with as it does with the hyperreals.

I've tried to keep this short but there is an enormous amount of subtlety to this question, so I will go into some of that below.

Some later results have clarified the relationship between the surreals and hyperreals, so some additional detail regarding what is being asked is probably necessary.

There has been a little bit of prior discussion about this, for instance in this post, where it is talked about the much more modern result that the surreals are isomorphic to the proper-class sized ultrapower of the reals. These isomorphisms can be thought of as various ways to transfer real functions to the surreal numbers. So in one sense, the answer is yes, a transfer principle exists in theory. But the pitfall with this approach is that everything requires ultrafilters, and is non-constructive, and there is no canonical choice of isomorphism. This is very different from the way that the surreals are built, which do not require ultrafilters.

On the other hand, Conway's book was written before any of the above results were published (with possibly an exception regarding one paper of Keisler). So partly the question is informal - what did Conway have in mind? But the other part of it is to formally ask if there is some other way to do this that doesn't involve this very particular method of using these isomorphisms, or even to use ultrafilters at all. For instance, what if we don't have the ultrafilter lemma? Then the hyperreals don't necessarily exist at all, but we can still build the surreals, which don't even require choice. Even if we don't have the ultrafilter lemma, can we still just go ahead anyway and say that $\frac{f(x + 1/\omega) - f(x)}{1/\omega}$ is a well-defined expression, and look for the closest real number to it, using some other way to derive a transfer principle?

The other part of the question is admittedly a soft question, but still well worth answering. The ultrafilter construction makes it very easy to see how such a transfer principle would work. Every hyperreal is a set of reals (or an equivalence class thereof), and to transfer any first-order predicate to some hyperreal, you simply ask the predicate of every real in the set and see if it's true of "most" of them (where "most" means "in the ultrafilter"). Thus you have a real-closed field, a "transfer principle," and all of that. Conway, on the other hand, has a very interesting way of building up the surreals in his book which is somewhat agnostic to the choice of set theory, using "birthdays," "left and right sets," etc. I am curious if there is some way to interpret Conway's assertion regarding the existence of $f(x+1/\omega)$ using his own machinery for the surreals, perhaps doing something clever and inductive with the left and right sets, rather than using these later developments involving isomorphisms with the ultrapower.

The last subtlety involves a philosophical point that has sometimes been raised with the topic of surreals vs hyperreals, but it is also worth addressing. There is, for instance, some debate regarding how functions like $\sin$ and $\cos$ should be transferred to the surreals. In theory, you could say that since we have these isomorphisms to the hyperreals, which have a transfer principle, these guarantee the existence of some kind of function on the surreals with the required first-order properties. But the surreals are very tangible in a very constructive sort of way, whereas these isomorphisms are typically totally non-constructive, so there is no way to use them to see what $\sin(\omega)$ should be, if it's positive or negative, etc. On the other hand, you could raise the same philosophical issue with the hyperreals, because there also is no real answer regarding what $\sin((1,2,3,4,...))$ should be, where $(1,2,3,4...)$ is a particular hyperreal number. The answer depends on the ultrafilter, which determines what $(1,2,3,4...)$ even means to begin with, or what properties it has, or if you like, which hyperreal it's referring to.

But what you can do with the hyperreals, which is part of the appeal, is you can kind of get "part of the way there" in a totally constructive manner. You know, for instance, that whatever $\sin((1,2,3,4,...))$ is, in some sense it's $(\sin(1), \sin(2), \sin(3), \sin(4), ...)$. Since we don't know what the ultrafilter is we don't know exactly what properties that has, but we do know something: we know that for any ultrafilter this value will not be an integer, for instance, or any rational number. So, we have some idea of what the transferred sin function would have to look like on the surreals as a result, at least given that $\omega$ is some hypernatural. So even though the ultrafilter is non-constructive, you can at least get "part of the way there" in an entirely constructive manner, which is part of what makes the entire thing interesting. And of course you don't really need to know much more than these few constructive things to actually do nonstandard analysis, just kind of happily plodding along formally doing nonstandard derivatives, with the understanding that the ultrafilter handles all of the various pathological, undefinable sets of indices in some logically consistent way or another.

So the last question is if there is some way for us to do something similar with the surreals, to get "part of the way there," in this sense. That is, to at least have enough constructive "transfer" for us to play around with all of this stuff, but using the framework of the surreals rather than the hyperreals, so that we can see that $f(1 + 1/\omega)$ even makes sense to begin with and play around with it. Something like Terry Tao's "cheap nonstandard analysis", perhaps.

There has been a little bit of prior discussion about this but not quite as directly as this question is asking. But for instance see here, where it is talked about that the surreals do have a kind of transfer principle in that they are isomorphic to the proper-class sized ultrapower of the reals (with many possible isomorphisms), and thus each isomorphism can be thought of as (unconstructably) defining a corresponding way to transfer first-order properties of reals on the surreal numbers. But also this transfer principle isn't quite as "natural" in the sense that there doesn't seem to be any "standard" choice of isomorphism. The punchline of all of this was, even though there's this isomorphism in theory, there is no canonical way to say what $\sin(\omega)$ is, for instance, or even $\omega \mod 2$.

On the other hand, though, it doesn't seem that this is any different from the way the hyperreals work. With those, we have the same problem, which is that all of the ultrafilters are typically non-constructive. And for instance, just like there is some debate about how the $\sin$ function should transfer to the surreals, or what $\sin(\omega)$ should be, there is also no real clear answer regarding what $\sin((1,2,3,4,...))$ should be, where $(1,2,3,4...)$ is a particular hyperreal number - because the answer depends on the ultrafilter, which determines what $(1,2,3,4...)$ even means to begin with, or what properties it has. So it doesn't really seem like the surreals are any better or worse off than the hyperreals in this regard.

And yet Conway is making the claim that he is. So there is this subtlety here regarding what really constitutes a "transfer principle" that I am hoping to get cleared up. Basically, with the hyperreals, part of the appeal is that even though you usually can't quite hold an ultrafilter in the palm of your hand the way you'd like, you can get a good bit of the way there constructively. You know, for instance, that whatever $\sin((1,2,3,4,...))$ is, in some sense it's $(\sin(1), \sin(2), \sin(3), \sin(4), ...)$. Since we don't know what the ultrafilter is we don't know exactly what properties that has, but we do know something: we know that for any ultrafilter this value will not be an integer, for instance. We can also "partly construct" an ultrafilter by constructing one that consistently assigns a membership value to all of the definable subsets of indices, and leave the undefinable ones open in a sort of choose-your-own-adventure approach, which is related to what Terry Tao calls "cheap nonstandard analysis". And then there are certain models of set theory which really do have constructible ultrafilters.

Summarizing, it seems implicit in Conway's assertion that you should also at least be able to do this kind of thing, that is to get "part of the way there" - at least have enough "transfer" for us to be sure that the expression $f(1 + 1/\omega)$ even makes sense to begin with. Perhaps part of my question is not quite so formal, but more in the direction of if it's possible to similarly get "part of the way there" but directly using the language of birthdays, left and right sets, etc, and all that stuff that makes the surreal numbers interesting, rather than sort of cheating and just using this isomorphism from the hyperreals. At least that seems like what Conway is suggesting, as it seems that it wasn't until later work by Keisler and then Ehrlich that it was proven that the proper class sized ultrapower of the reals and the surreals are isomorphic, if I interpret the above link correctly, but Conway clearly had something in mind even prior to that.

There has been a little bit of prior discussion about this but not quite as directly as this question is asking. But for instance see Surreal numbers vs. non-standard analysis, where it is talked about that the surreals do have a kind of transfer principle in that they are isomorphic to the proper-class sized ultrapower of the reals (with many possible isomorphisms), and thus each isomorphism can be thought of as (unconstructably) defining a corresponding way to transfer first-order properties of reals on the surreal numbers. But also this transfer principle isn't quite as "natural" in the sense that there doesn't seem to be any "standard" choice of isomorphism. The punchline of all of this was, even though there's this isomorphism in theory, there is no canonical way to say what $\sin(\omega)$ is, for instance, or even $\omega \bmod 2$.

On the other hand, though, it doesn't seem that this is any different from the way the hyperreals work. With those, we have the same problem, which is that all of the ultrafilters are typically non-constructive. And for instance, just like there is some debate about how the $\sin$ function should transfer to the surreals, or what $\sin(\omega)$ should be, there is also no real clear answer regarding what $\sin((1,2,3,4,\dotsc))$ should be, where $(1,2,3,4,\dotsc)$ is a particular hyperreal number - because the answer depends on the ultrafilter, which determines what $(1,2,3,4,\dotsc)$ even means to begin with, or what properties it has. So it doesn't really seem like the surreals are any better or worse off than the hyperreals in this regard.

And yet Conway is making the claim that he is. So there is this subtlety here regarding what really constitutes a "transfer principle" that I am hoping to get cleared up. Basically, with the hyperreals, part of the appeal is that even though you usually can't quite hold an ultrafilter in the palm of your hand the way you'd like, you can get a good bit of the way there constructively. You know, for instance, that whatever $\sin((1,2,3,4,\dotsc))$ is, in some sense it's $(\sin(1), \sin(2), \sin(3), \sin(4), \dotsc)$. Since we don't know what the ultrafilter is we don't know exactly what properties that has, but we do know something: we know that for any ultrafilter this value will not be an integer, for instance. We can also "partly construct" an ultrafilter by constructing one that consistently assigns a membership value to all of the definable subsets of indices, and leave the undefinable ones open in a sort of choose-your-own-adventure approach, which is related to what Terry Tao calls "cheap nonstandard analysis". And then there are certain models of set theory which really do have constructible ultrafilters.

Summarizing, it seems implicit in Conway's assertion that you should also at least be able to do this kind of thing, that is to get "part of the way there" — at least have enough "transfer" for us to be sure that the expression $f(1 + 1/\omega)$ even makes sense to begin with. Perhaps part of my question is not quite so formal, but more in the direction of if it's possible to similarly get "part of the way there" but directly using the language of birthdays, left and right sets, etc., and all that stuff that makes the surreal numbers interesting, rather than sort of cheating and just using this isomorphism from the hyperreals. At least that seems like what Conway is suggesting, as it seems that it wasn't until later work by Keisler and then Ehrlich that it was proven that the proper class sized ultrapower of the reals and the surreals are isomorphic, if I interpret the above link correctly, but Conway clearly had something in mind even prior to that.