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 Gonshor mentions an earlier unpublished definition due to Martin Kruskal; so does Conway in the 2nd edition of "On Numbers and Games".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 Gonshor seems to suggest in his text that his definition is equivalent to Kruskal's unpublished one, but on the other hand never seems to explicitly state so.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 [old version, this definition was removed from Wikipedia soon after asking this question] 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.


1 Answer 1


There is only one official definition of surreal exponentiation in the literature, the one due to Martin Kruskal. It was rediscovered by Harry Gonshor (with hints from Kruskal) and incorporated into his book (An Introduction to the Theory of Surreal Numbers) where important results on surreal exponentiation that go beyond Kruskal's initial discoveries are also found.

Surreal exponentiation was later discussed by Lou van den Dries and myself in “Fields of Surreal Numbers and Exponentiation”, Fundamenta Mathematicae 167 (2001), No. 2, pp. 173-188; erratum, ibid. 168, No. 2 (2001), pp. 295-297. We showed:

If No(alpha) is the set of surreal number of tree-rank less than an epsilon number alpha, then No(alpha) is an elementary extension of the ordered field of reals with exp and an elementary substructure of the ordered field of surreals with Kruskal-Gonshor exponentiation.

Surreal exponentiation is also discussed on pages 31 and 32 of my just-published paper: “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 1 (2012), pp. 1-45.

Norman Alling's treatment of expX (in terms of power series) only holds when X is infinitesimal and in those those cases it coincides with the Kruskal-Gonshor definition. The definition of surreal exponentiation in Wikipedia seems dubious. In fact, a number of features of the Wikipeda article on surreal numbers are dated or misleading.

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    $\begingroup$ Good to see you on here, Phil! $\endgroup$ Feb 21, 2012 at 5:23
  • $\begingroup$ OK, thank you and upvoted! It's good to have that cleared up. I'll go ahead and actually accept an answer (i.e. probably this) either once someone answers the question of where this remaining problematic definition comes from, or else if nobody can do so after a day or two. (Unfortunately, there doesn't seem to be any way to contact the person who actually added this definition to Wikipedia, one Michael K. Edwards, since he seems to be away from Wikipedia for the time being.) $\endgroup$ Feb 21, 2012 at 7:42
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    $\begingroup$ Hey Philip, if $\alpha$ and $\beta$ are surreal numbers that happen to be ordinals, is it true that $\alpha^\beta$ is necessarily an ordinal? If so, do you know of a concrete definition of $\alpha^\beta$ in terms of well-orderings? I ask because I've been pondering how to define $\alpha^\beta$ in the spirit of the Hessenberg (sometimes called the natural) ordinal operations. (I had a look over your article for answers to these questions and didn't find any, but perhaps that is because my understanding of the surreal numbers is non-existent...) $\endgroup$ Apr 17, 2014 at 11:27
  • $\begingroup$ User18921@ Nice question! I'll have to give it some thought. $\endgroup$ Apr 17, 2014 at 22:36
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    $\begingroup$ Just noticed this question from goblin. The answer is no. For instance, $\exp(2\log\omega)=\omega^{\omega\log 2}$, and $\exp(\omega\log\omega)=\omega^{\omega^{1+1\omega}}$, neither of which are ordinals. $\endgroup$ Jul 18, 2014 at 20:56

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