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The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to be varying definitions of the exponential in the surreal numbers and since I can't find any recent reference that covers them all I have little idea whether they're actually the same or not. (When I say "exponential", I mean either $e^x$ or the general $a^x$ for $a>0$; obviously one can go back and forth between these so long as $e^x$ is indeed a bijection from surreals to positive surreals.)

To wit:

1. Harry Gonshor gives one definition in his "An Introduction to the Theory of Surreal Numbers".
2. Gonshor mentions an earlier unpublished definition due to Martin Kruskal; so does Conway in the 2nd edition of "On Numbers and Games". Neither actually state this definition, but it is strictly speaking possible for someone who's never seen it to verify equivalence with it, because Conway mentions that it is inverse to a particular definition of the logarithm, which he does not explicitly state but gives enough information to deduce. Gonshor seems to suggest in his text that his definition is equivalent to Kruskal's unpublished one, but on the other hand never seems to explicitly state so.
3. Norman Alling's "Foundations of Analysis over Surreal Number Fields" looks like it might contain another definition? I'm not too clear on what he's doing, honestly, though it looks like it's restricted to non-infinite surreals...
4. Wikipedia's page gives a totally uncited definition for $2^x$. I have no idea where this might be originally from. I suppose one could substitute in other surreals for 2 to generalize this?
5. Or else one could take Wikipedia's definition and generalize it in the way one usually does when starting from $e^x$? (I should hope this agrees with definition 4!).

Note that the operation $x\mapsto \omega^x$ commonly used in the surreals is not related; though it's exponential in some sense, it's not surjective onto the positive surreals, and so definitions of a general exponential shouldn't attempt to agree with it. And of course definitions 1, 2, and 4/5 above are surjective onto the positive surreals. (Or Wikipedia claims #4/5 is, anyway.)

Edit: To avoid confusion, in what follows, I'll write $\exp_\omega x$ instead of the usual $\omega^x$, and reserve the notation $\omega^x$ for whatever that happens to be in the notion of exponentiation under discussion.

So, does anyone know to what extent these are actually equivalent? If they're not equivalent, is there agreement on which ones are the "right" definitions? (It seems like all of them have the right properties! And while it seems to be agreed that the idea behind Kruskal's definition is bad, that doesn't mean necessarily the definition itself is.) Or could anyone point me to any recent book which might clear all this up, or at least the source of Wikipedia's definition?

(I had originally intended to ask other questions about surreal exponentiation before finding that I wasn't sure what it actually was. I am hoping that whatever references people can point me to will answer my other questions as well.)

Slight update: Definitions 4 doesn't seem to agree with definition 5 (nor definition 1, see below); it would seem that definition 4 would imply $3^\omega=\omega$, while definition 5 would imply $3^\omega>\omega$. This raises the problem in that one could make more definitions by using definition 4 to define $a^x$ for some fixed $a$, and then generalizing it to $b^x$ for all $b$ via definition 5, and depending on your choice of starting $a$ -- whether $e$, 2, or something else -- you'd get different definitions of $b^x$ out. An entire proper class of distinct "exponentiation" operations! Well, perhaps not, perhaps not all starting values of $a>1$ yield an onto function -- perhaps 2 is special and it's the only one that does, though that seems pretty unlikely, and barring that, this is pretty bad regardless. Also, definition 4 seems pretty suspect as the "right" definition for another reason: If we plug in two ordinals, it looks like it will agree with ordinary ordinal exponentiation. This both disagees with Gonshor's definition (which would imply $\omega^\omega>\exp_\omega \omega$) and is suspect on its own, because we shouldn't expect to get one of the ordinary ordinal operations out of this (we use natural addition and multiplication in the surreals, not ordinary addition and multiplication). If indeed we get an ordinal operation out of this at all -- it would appear that by Gonshor's definition, $\omega^\omega$ would not even be an ordinal, instead being equal to $\exp_\omega \exp_\omega (1+1/\omega)$.

Oops: Sorry, that shouldn't be ordinary exponentiation, but rather the analogue of it based on natural multiplication. Regardless, still disagrees, still smells bad.

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• Or else one could take Wikipedia's definition and generalize it in the way one usually does when starting from $e^x$ e^x$? (I should hope these two methods agree!)this agrees with definition 4!). • ( Note that the operation$x\mapsto \omega^x$commonly used in the surreals is not related; though it's exponential in some sense, it's not surjective onto the positive surreals, and so definitions of a general exponential shouldn't attempt to agree with it. And of course definitions 1, 2, or 4 and 4/5 above are surjective onto the positive surreals. (Or Wikipedia claims #4 4/5 is, anyway.)anyway.) Edit: To avoid confusion, in what follows, I'll write$\exp_\omega x$instead of the usual$\omega^x$, and reserve the notation$\omega^x$for whatever that happens to be in the notion of exponentiation under discussion. Slight update: Definitions 4 doesn't seem to agree with definition 5 (nor definition 1, see below); it would seem that definition 4 would imply$3^\omega=\omega$, while definition 5 would imply$3^\omega>\omega$. This raises the problem in that one could make more definitions by using definition 4 to define$a^x$for some fixed$a$, and then generalizing it to$b^x$for all$b$via definition 5, and depending on your choice of starting$a$-- whether$e$, 2, or something else -- you'd get different definitions of$b^x$out. An entire proper class of distinct "exponentiation" operations! Well, perhaps not, perhaps not all starting values of$a>1$yield an onto function -- perhaps 2 is special and it's the only one that does, though that seems pretty unlikely, and barring that, this is pretty bad regardless. Also, definition 4 seems pretty suspect as the "right" definition for another reason: If we plug in two ordinals, it looks like it will agree with ordinary ordinal exponentiation. This both disagees with Gonshor's definition (which would imply$\omega^\omega>\exp_\omega \omega$) and is suspect on its own, because we shouldn't expect to get one of the ordinary ordinal operations out of this (we use natural addition and multiplication in the surreals, not ordinary addition and multiplication). If indeed we get an ordinal operation out of this at all -- it would appear that by Gonshor's definition,$\omega^\omega$would not even be an ordinal, instead being equal to$\exp_\omega \exp_\omega (1+1/\omega)$. 3 additional note on Kruskal's def The surreal numbers are sometimes introduced as a place where crazy expressions like$(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$(to use the nLab's example) make sense. The problem is, there seem to be varying definitions of the exponential in the surreal numbers and since I can't find any recent reference that covers them all I have little idea whether they're actually the same or not. (When I say "exponential", I mean either$e^x$or the general$a^x$for$a>0$; obviously one can go back and forth between these so long as$e^x$is indeed a bijection from surreals to positive surreals.) To wit: 1. Harry Gonshor gives one definition in his "An Introduction to the Theory of Surreal Numbers". 2. Gonshor mentions an earlier unpublished definition due to Martin Kruskal; so does Conway in the 2nd edition of "On Numbers and Games". Neither actually state this definition, but it is strictly speaking possible for someone who's never seen it to verify equivalence with it, because Conway mentions that it is inverse to a particular definition of the logarithm, which he does not explicitly state but gives enough information to deduce. Gonshor seems to suggest in his text that his definition is equivalent to Kruskal's unpublished one, but on the other hand never seems to explicitly state so. 3. Norman Alling's "Foundations of Analysis over Surreal Number Fields" looks like it might contain another definition? I'm not too clear on what he's doing, honestly, though it looks like it's restricted to non-infinite surreals... 4. Wikipedia's page gives a totally uncited definition for$2^x$. I have no idea where this might be originally from. I suppose one could substitute in other surreals for 2 to generalize this? Or else one could generalize it in the way one usually does when starting from$e^x$(I should hope these two methods agree!). (Note that the operation$x\mapsto \omega^x\$ commonly used in the surreals is not related; though it's exponential in some sense, it's not surjective onto the positive surreals, and so definitions of a general exponential shouldn't attempt to agree with it. And of course definitions 1, 2, or 4 above are surjective onto the positive surreals. (Or Wikipedia claims #4 is, anyway.))

So, does anyone know to what extent these are actually equivalent? If they're not equivalent, is there agreement on which ones are the "right" definitions? (It seems like all of them have the right properties!properties! And while it seems to be agreed that the idea behind Kruskal's definition is bad, that doesn't mean necessarily the definition itself is.) Or could anyone point me to any recent book which might clear all this up, or at least the source of Wikipedia's definition?

(I had originally intended to ask other questions about surreal exponentiation before finding that I wasn't sure what it actually was. I am hoping that whatever references people can point me to will answer my other questions as well.)