# What's wrong with the surreals?

Of all the constructions of the reals, the construction via the surreals seems the most elegant to me.

It seems to immediately capture the total ordering and precision of Dedekind cuts at a fundamental level since the definition of a number is based entirely on how things are ordered. It avoids, or at least simplifies, the convergence question of Cauchy sequences. And it naturally transcends finiteness without sacrificing awareness of it.

The one "rumor" I've consistently heard is that it is hard to naturally define integrals and derivatives in the surreals, although I have yet to see a solid technical justification of that.

Are there known results that suggest we should avoid further study of this construction, or that show limitations of it?

• Regarding the difficulties wtih integrals and derivatives, see the appendix to the second edition of On Numbers and Games. I don't think that this is really relevant to your question, though. Once you have the reals you can just develop analysis as usual; you don't have to carry the surreals around as excess baggage everywhere if you don't want to. – Timothy Chow Jun 24 '10 at 1:58
• In asking a big-picture question I was looking for an equally vague-but-intuitive answer, although of course technical evidence is extremely useful! You're reference to the end of On Numbers and Games is probably the single best answer so far, or rather, Conway's epilogue itself. Thanks, Timothy. (I think I'll rephrase this as another answer). – user2498 Jun 24 '10 at 22:06
• – Ben Crowell Feb 21 '16 at 17:14
• Comments are not for extended discussion; this conversation has been moved to chat. – Todd Trimble Aug 11 '18 at 4:58

• Boyarsky, the surreals unify the transfinite ordinals with a robust theory of infinitesimals, in a single ordered field having an extremely rich structure theory. Thus, it unifies disparate number concepts arising in set theory, analysis and number theory. In the surreals, one has numbers such as $\sqrt{\frac{1}{\omega}+e^{\pi/2-\omega^2}}. How can a mathematician not be attracted to the possibilities of this? The tree was there all along, of course, so in principle there is nothing new there, but Philip uses it to provide canonical representations that he says are simplifying. – Joel David Hamkins Jun 24 '10 at 12:58 • It seems the main thing that would be needed to broaden the interest of surreals would be to provide some tools from transferring results back from the surreals back to more classical number systems (analogous to the transfer principle in nonstandard analysis). The situation here reminds me of that of generalised functions in analysis. There are many, many ways to generalise the concept of a function, but only distributions have really been successful, because there are ways to get from distributions back to classical functions, e.g. by convolving with a test function. – Terry Tao Jun 25 '10 at 0:04 • For the first 50 years after Hensel,$p$-adic analysis was championed by a few but viewed as esoteric by most; intrinsic beauty wasn't enough. Then Dwork used it to prove a Weil Conjecture and Tate invented rigid-analytic spaces to make analytic continuation possible over totally disconnected fields (with applications in number theory), and now it's a huge industry. Surreals are in need of a Dwork and Tate. (Although the set theory they require is elementary to set theorists, I conjecture it is a real obstacle for many without such expertise. Look at the history of non-standard analysis.) – Boyarsky Jun 25 '10 at 1:21 • A Boyarsky turning to Dwork for an example. Am I the only one amused by this? – KConrad Jun 25 '10 at 3:52 To me one of the more fascinating aspects of the surreals is that application by Kruskal and others to construct higher order asymptotic expansions. For example, if you want to understand the asymptotics of the function $$f(x)= {1\over 1-x}+e^{-1/x}$$ on$(0,\epsilon)$and differentiate it from$g(x)={1\over 1-x}$you look at the series" $$1+x+x^2+x^3+\dots+e^{-1/x}.$$ Kruskal and his co-authors have used surreal numbers to give an approach to these expansions and applications. This type of expansion can also be be dealt with using the transseries of Ecalle or the logarithmic-exponential series developed in model theory. This is more of an extended footnote to Nombre’s answer than an answer itself. As Nombre’s observations would suggest, I heartily agree that the algebraico-tree-theoretic simplicity hierarchy is critical to the surreals. $$\mathbf{No}$$ is not just a monster ordered field containing the reals and the ordinals. The following is a list of some recent papers on the surreals that make critical use of the simplicity hierarchy, and thereby lend credence to Nombre's observations. It is only the beginning of a new wave of work presently being done by model theorist, order algebrists and analysts that take advantage of $$\mathbf{No}$$’s simplicity-hierarchical structure. Berarducci, A. and Mantova, V. (2018): Surreal numbers, derivations and transseries, Journal of the European Mathematical Society 20, pp. 339-390. arixv:1503.00315. Berarducci, A. and Mantova, V. (forthcoming): Transseries as germs of surreal functions, Transactions of the American Mathematical Society, arXiv:1703.01995. Aschenbrenner, M., van den Dries, L. and van der Hoeven, J. (2018): Numbers, germs and transseries, Proceedings of the International Congress of Mathematicians, Rio De Janeiro, 2018, arXiv:1711.06936. Aschenbrenner, M., van den Dries, L. and van der Hoeven, J. (forthcoming): Surreal numbers as a universal $$H$$-field, Journal of the European Mathematical Society arXiv:1512.02267. Ehrlich, P. and Kaplan, E.: Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers II, The Journal of Symbolic Logic 83 (2018), No. 2, pp. 617-633, arXiv:1512.04001. Kuhlmann, S. and Matusinski, M. The exponential-logarithmic equivalence classes of surreal numbers, Order 32 (2015), no. 1, 53–68. arXiv:1203.4538. Costin, O., Ehrlich, P. and Friedman, H. (24 Aug 2015): Integration on the surreals: a conjecture of Conway, Kruskal and Norton, preprint, arXiv:1505.02478. The last paper is a rather old version of a paper now in the process of being revised and will eventually be two separate papers. Edit. May 17, 2020. The following recent paper by Elliot Kaplan and myself adds further credence to the idea that the algebraico-tree-theoretic simplicity hierarchy is of critical importance to the surreals. Surreal ordered exponential fields: (https://arxiv.org/abs/2002.07739) Abstract: In (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $$\mathbf{No}$$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $$K$$-vector space) to be isomorphic to an initial subfield ($$K$$-subspace) of $$\mathbf{No}$$, i.e. a subfield ($$K$$-subspace) of $$\mathbf{No}$$ that is an initial subtree of $$\mathbf{No}$$. In this sequel to (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), piggybacking on the just-said results, analogous results are established for ordered exponential fields. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $$(\mathbf{No}, \exp)$$. These include all models of $$T(\mathbb{R}_W, e^x)$$, where $$\mathbb{R}_W$$ is the reals expanded by a convergent Weierstrass system $$W$$. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of $$\mathbf{No}$$, which includes $$\mathbf{No}$$ itself, extend to canonical exponential functions on their surcomplex counterparts. This uses the precursory result that trigonometric-exponential initial subfields of $$\mathbf{No}$$ and trigonometric ordered initial subfields of $$\mathbf{No}$$, more generally, admit canonical sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of $$\mathbf{No}$$, to the image of the canonical map of the ordered exponential field $$\mathbb{T}$$ of transseries into $$\mathbf{No}$$, which is shown to be initial, and to the ordered exponential fields $$\mathbb{R}((\omega))^{EL}$$ and $$\mathbb{R}\langle\langle\omega\rangle \rangle$$, which are likewise shown to be initial. • Thanks for the expansion of my answer and for this relevant list of publications. There will also be a paper by Joris van der Hoeven and I about special substructures of$(\mathbf{No},\leq,\leq_s)$which will further accredit this viewpoint, but I don't know when it will be submitted. – nombre Aug 4 '18 at 22:51 • @Nombre. Looking forward to seeing the paper. – Philip Ehrlich Aug 5 '18 at 13:16 • Since you are our resident expert on the surreals, I wonder if you’ve seen this recent question: mathoverflow.net/questions/360258/… . – Emil Jeřábek May 19 '20 at 13:49 • No, Emil, I hadn't seen it, and I do not know the answer to the author's question. – Philip Ehrlich May 19 '20 at 14:05 I take advantage of this question to propose an answer in the vein of that of Philip Ehrlich, and to advertise a little bit for$\mathbf{No}$. I feel like the surreal numbers should be attractive for a several reasons, besides the recent interesting results that have been proven recently regarding them. My sense is that they fail to be in some measure because$\mathbf{No}$is mostly known as a monster model for the theory of real closed fields, albeit with a nice inductive definition. Let me recall a fundamental property (FP) of$\mathbf{No}$. The linear order$(\mathbf{No},\leq)$is equipped with a well-founded partial order$\leq_s$of simplicity, where for numbers$x,y$, we have$x\leq_s y$if as sign sequences, we have$x\subseteq y$. FP: If$L,R$are sets of surreal numbers with$L<R$(i.e.$\forall l \in L,\forall r \in R, l<r$), then there is a unique$\leq_s$-minimal surreal number$x$satisfying$L<x<R$. This, along with the fact that for$x\in \mathbf{No}$, the class of numbers$z$with$z<_s x$is a set, characterizes the class$(\mathbf{No},\leq,<_s)$up to unique isomorphism. Note that this fundamental property is a very nice and rare combination: order saturation + canonicity. That alone is a desirable quality in my opinion. The FP allows one to define easily many operations on$\mathbf{No}$, as Conway did with the field operations and the normal form, as Kruskal-Gonshor did with exponentiation, and those are only the tip of the iceberg. So$\mathbf{No}$is simply fun to play with, all the more so that easily defined operations can be very challenging to study. Basically,$\mathbf{No}$is an immense playing field for curious mathematicians interested in order-related notions, as many definitions that are only order-consistent can be thrown into existence in$\mathbf{No}$. However, the notion of simplicity has been somewhat downplayed by Conway who rather focused on the notion of earliness (and even Harry Gonshor who used it a lot didn't dedicate a symbol to it). The fundamental property is often taken as a tool designed to define the field operations, rather than a marvelous mathematical spell with various uses. This picture, in which$\mathbf{No}$is a large ordered field containing the ordinals and the real numbers, hides the truth that surreal numbers can tell many other tales. Conway himself lists a few disadvantages in On Numbers and Games, Chapter 2. One that can be dealt with quickly is that it is quite tricky to make the process stop after constructing the reals! We can cure this by adding to the construction the proviso that if$L$is non-empty but with no greatest member, then$R$is non-empty with no least member, and vice versa. This happily restricts us exactly to the reals. The remaining disadvantages are that the dyadic rationals receive a curiously special treatment, and that the inductive definitions are of an unusual character. From a purely logical point of view these are unimportant quibbles (we discuss the induction problems later in more detail), but they would predispose me against teaching this to undergraduates as "the" theory of real numbers. [Edit: I interpreted the question as, what is wrong with the construction of the real numbers via surreals? This might have been a misinterpretation. The question does, after all, literally ask what is wrong with the surreals. Obviously there is nothing wrong with the surreals. I thought this was obvious and so I assumed the question must have been something else.] • Equality is also a "defined relation" if you define the real numbers as Cauchy sequences of rationals, or even if you define the rational numbers as ordered pairs of integers, although in most cases we sweep it under the rug by talking about "equivalence classes." So I don't really see why the fact that equality of surreals is a "defined relation" should be regarded as a disadvantage. – Mike Shulman Jun 25 '10 at 5:19 • @Boyarsky: I think you vastly overstate the difficulty of dealing with classes. To an extremely good first approximation, if you were to ignore the fact that the surreals don't form a set and just blindly went ahead and pretended that they did, you would never run into any difficulties. Compare with category theory: mathematicians happily deal with basic concepts from category theory with no trouble, despite the fact that all the most common categories are proper classes. You don't need to redo anything you've learned because classes behave just like sets. – Timothy Chow Jun 26 '10 at 2:37 • @Boyarsky: It really is that easy. Every set-theoretic construction you would want to do can be done with classes. You just have to make sure not to run afoul of something like Russell's paradox, and that's something you've had to be careful of in "ordinary" set theory anyway. If you're not happy flying by the seat of your pants (which 99% of mathematicians do in the case of set theory; how many mathematicians can even list the axioms of ZFC?), then take a look at the axioms for Morse-Kelley or NBG. Don't assume that classes are difficult just because they're unfamiliar. – Timothy Chow Jun 27 '10 at 17:46 • The ordinals and cardinals also form proper classes, but these number concepts have proved enormously fruitful. For that matter, the collection of all sets is also a proper class, but we still work with sets. So why not surreals? Of course, one must respect the set/class issues, but this is neither difficult nor mysterious. – Joel David Hamkins Jun 28 '10 at 18:07 • There are mainly two main points to classes: the first is that classes, unlike sets, cannot be themselves members of other classes (they only contain elements, but they're not elements of anything). The second is that you need to be able to describe classes to use them, i.e. you need to consider definable classes only (if you're sticking to the usual set theory, ZFC, rather than someting like NBG or Morse-Kelley). Keeping those two in mind is all you need to clarify the difference between sets and classes, and to use classes correctly. – David Fernandez-Breton Jun 24 '11 at 14:46 I just read the epilogue of On Numbers and Games as suggested by Timothy Chow, and here I see very concrete references to some technical difficulties that may demystify some research in this area, although it is apparent by other answers that there is still much optimism. Some remarks based on the epilogue (written in 2000 by Conway): There is a nice definition of the surreals which does not require equality as a defined relation. It is not formally given, but I'm guessing we can think of it as a mapping from an ordinal set to signs {-,+}, each sign being a direction we take in the surreal tree. Then identity is equality. However, Conway remarks that this has two problems: 1. It forsakes the "genetic" (his word, also in quotes at first) approach of the L,R definition. I don't fully understand this, but I'm guessing he means that we're building everything on the intuition of a total ordering (and maybe a "time-of-creation" idea), and the surreals will always be identifiable with L,R sets, so why not just define them that way? 2. The sign-sequence definition requires that the ordinals are defined first. Conway goes on to discuss work by Simon Norton (a proposed definition of an integral) and Martin Kruskal. The general direction here is to define things in terms of (L,R) sets (classes??) in such a way that equal numbers (in the defined equality) give equal answers; and that classical analysis remains intact. Conway gives Norton's integral definition, which has some good properties, but fails to integrate the surreal-exponential function in accordance with classical analysis (we get ex instead of ex-1 when integrating over [0,x]). In summary, I'm choosing to interpret all of these comments and answers together (thanks to all) as: the surreals are indeed a worthwhile construction, although there is a noted lack of progress on extending calculus to work equally elegantly in a surreal-general setting. In case others are curious, here are some references (I have read none of them, yet) Conway gives in this epilogue: • The Theory of Surreal Numbers by Harry Gonshor • Foundations of Analysis over Surreal Number Fields by Norman Alling • Real Numbers, Generalizations of the Reals, and Theories of Continua by Philip Ehrlich • Gonshor's book is quite elegant and starts defining surreal numbers right away as sequences on {-,+}. Of course you need to define ordinals first, but for me that's not a disadvantage (maybe for Conway it is, since he seems to try to avoid to "restrict" himself to a particular axiomatic system such as ZFC, assuming that it's possible to even make sense of such a viewpoint). – David Fernandez-Breton Jun 24 '11 at 1:40 • This paper (arxiv.org/abs/1307.7392) gives a new definition of integration on surreals, so that$\int^x_0 e^t dt=e^x-1$. The authors suggests that further investigations into evaluation of series are needed for a more complete theory on integration. – A. Chu May 26 '17 at 13:57 One thing that might boost the stock of surreal numbers is if the genetic approach gave us a good way to define functions that mathematicians need. Here I'll fantasize a bit: What if some sort of genetic procedure were the right way to construct an admissible function (in the sense of "New upper bounds on sphere packings I" by Cohn and Elkies) that would enable one to use their Theorem 3.2 to prove optimality of the best known sphere packing in dimension 8? I hasten to add that I know of no reason why a genetic construction procedure should be helpful here. The "missing function" of Cohn and Elkies is just one example of a mathematical function that ought to exist but which we don't know how to construct. At some point, someone will probably find a construction that "comes out of left field". If surreal analysis served as "left field", more people would be interested in it. • Kind of amazing that Viazovska "came out of left field" very soon after this answer was posted! – Ryan O'Donnell Jul 30 '18 at 0:09 • (-->Feb 23 '16 at 2:35) This is what they call a wish list – Vladimir Kanovei Aug 6 '18 at 7:47 A quick search indicates that Peano axioms are not mentioned on this page. It seems reasonable to mention that there does not seem to be a good notion of natural number in the surreals that would satisfy Peano arithmetic. On the other hand, a maximal class-size surreal field is isomorphic to a suitable maximal class-size hyperreal field, so the hyperintegers can be imported to the other side, and with them all the required first-order properties. • Is it the lack of internal way to recognize the natural numbers? Because starting with the construction, it is of course easy enough to also construct a suitable set of natural numbers. – Tobias Kildetoft Feb 23 '16 at 8:48 • @TobiasKildetoft, it is not clear whether$\omega$should be even or odd. – Mikhail Katz Mar 2 '16 at 9:31 •$\omega$is not a finite number, so it is not necessary for Peano arithmetic. – Tobias Kildetoft Mar 2 '16 at 9:32 • @TobiasKildetoft,$\mathbb{R}$is sufficient for doing analysis, so in that sense the surreals are unnecessary. How do you interpret this question? – Mikhail Katz Mar 2 '16 at 9:34 • @Mikhail Katz: The (current) lack of a definable$Th(\mathbb{N})$-like integer part seems to be an important obstacle in the quest for non-standard analysis in the surreals. But the whole idea of seeing$No$as a bigger$\mathbb{R}\$ might be illusive. – nombre Jan 11 '17 at 10:09