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What is the relationship between the surreal numbers and non-standard analysis?

In particular, is there a transfer principle for surreal numbers they way there is for NSA?

A specific situation in which such a transfer principle would be useful arose in the thread Uniformizing the surcomplex unit circle ; can the surjectivity of the map $t \mapsto e^{it}$ from the reals to the complex unit circle be transferred to the surreals? Presumably, one would need a definition of the map that was in some sense first-order; what sorts of definitions count as first-order? It is not clear to me how definitions involving the two-sided bracket operation can be fit into a first-order framework.

Mike Shulman
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James Propp
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    Conway, ONAG end of Chapter 4, says "The field No is really irrelevant to nonstandard analysis." – Gerald Edgar Mar 19 '12 at 19:49
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    related: http://mathoverflow.net/questions/29300/whats-wrong-with-the-surreals –  Feb 21 '16 at 17:14
  • @GeraldEdgar, the point is not so much that the field No is irrelevant to Robinson's framework but that the field No may be of limited relevance to analysis; see my answer. – Mikhail Katz May 03 '16 at 09:04

3 Answers3

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In the final section of my paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small” (The Bulletin of Symbolic Logic 18 (2012), no. 1, pp. 1-45, I not only point out that the real-closed ordered fields underlying the hyperreal number systems (i.e. the nonstandard models of analysis) are isomorphic to initial subfields of the system of surreal numbers, but that the system of surreal numbers itself is isomorphic to the real-closed ordered field underlying what may be naturally regarded as the maximal hyperreal number system in NBG (von-Neumann-Bernays-Gödel set theory with global choice)—i.e., the saturated hyperreal number system of power On, On being the power of a proper class in NBG. It follows immediately from the latter that the ordered field of surreal numbers admits a relational extension to a model of non-standard analysis and, hence, that in such a relational extension the transfer principle does indeed hold.

By the way, by an initial subfield, I mean a subfield that is an initial subtree. Discussions of surreal numbers (including most of the early discussions) that downplay or overlook the marriage between algebra and set theory that is central to the theory overlook many of the most significant features of the theory. In addition to the paper listed above, this marriage of algebra and set theory is discussed in the following papers which are found on my website http://www.ohio.edu/people/ehrlich/

“Number Systems with Simplicity Hierarchies: A Generalization of Conway’s Theory of Surreal Numbers,” The Journal of Symbolic Logic 66 (2001), pp. 1231-1258. Corrigendum, 70 (2005), p. 1022.

“Conway Names, the Simplicity Hierarchy and the Surreal Number Tree”, The Journal of Logic and Analysis 3 (2011) no. 1, pp. 1-26.

“Fields of Surreal Numbers and Exponentiation” (co-authored with Lou van den Dries), Fundamenta Mathematicae 167 (2001), No. 2, pp. 173-188; erratum, ibid. 168, No. 2 (2001), pp. 295-297.

Philip Ehrlich
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  • Could you clarify whether the saturation property of the surreals includes a class resplendency property, by which we would get classes corresponding to the transfer of arbitrary functions and relations on the reals? (See also my comments below on the answer of Katz.) – Joel David Hamkins Jul 29 '18 at 13:41
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    @Joel. Yes, Joel. While No does not come equipped with all the relational extensions one needs to do NSA, one can induce all such relational extensions by employing an isomorphism of ordered fields from No onto an On-saturated model of say Keisler’s axioms for hyperreal fields. In fact, since every real closed field is isomorphic to an initial subfield of No, one can do the same for any hyperreal field. – Philip Ehrlich Jul 29 '18 at 14:26
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    @Joel (Continued): Thus far, there has been little cross-fertilization between the surreals and NSA, and as such there is no reason at present why one would want to use No or various subfields of No as the universe of a nonstandard model of analysis. It would simply require extra work with little gain. However, I would not rule out cross-fertilization down the line, which would make the marriage of the two systems desirable. – Philip Ehrlich Jul 29 '18 at 14:26
  • OK, good, that is what I had thought. – Joel David Hamkins Jul 29 '18 at 14:29
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    Is there in any way a "most natural isomorphism" between the surreals and hyperreals? For instance, is there some "best" way to map $\omega$ to a hypernatural, and if so, do we know whether it is prime or composite, what $\sin(\omega)$ is, etc? – Mike Battaglia Apr 26 '19 at 06:00
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    @Mike. I am not aware of the existence of a "most natural (such) isomorphism". – Philip Ehrlich Apr 29 '19 at 23:35
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Coming back to the first post. Most of modern mathematics is set-theoretic, that is, it studies sets of different kind, so that reals, real and complex functions, relations on reals, as well as a variety of more complex objects like the Hilbert space - are sets of this or another kind. In that sense, any mathematical definition is a 1-st order one, assuming there is no restriction on using the language of set theory within the common axiomatics.

Regarding the surs. The definition of them yields a certain ordered field, perhaps maximal in some well-defined sense, and nothing more. That surs are so attractive to some kind of mathematically-complying minds is, in my opinion, explainable that this still is a very rare domain where meaningful facts can be explored or observed, rather than proved. On the other hand, the students of surs I believe cannot care less about some transfer and about whether their omni-something does not satisfy some Peano axiom. After all, p-adic numbers do not satisfy Peano axioms either, but who cares.

Further it happens that the surs are isomorphic (in a class theory) to a certain nonstandard universe, defined by totally different means and towards quite different goals. This allows to enrich the surs by a variety of constructions (like the sine function) beyond their native field structure. In this case, a devoted student of surs might be interested to really figure out in some strict, well defined terms, whether a consistent sine function can be defined on surs by pure sur-means. For instance, consider a version of NBG which proves the existence of surs as a class but is not strong enough to prove the mentioned isomorphism, and prove that such a theory does not imply the existence of a consistent sur-sin. This can be very complex though.

Vladimir Kanovei
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  • I agree with this perspective. And regarding your question at the end, it is related to my question https://mathoverflow.net/q/227849/1946, asking whether we can prove even the universality of the surreal number order without the global choice principle. One is using global choice at the heart of the isomorphisms being discussed on this question, because one uses it in order to carry out the class-sized back-and-forth constructions of saturation. – Joel David Hamkins Jul 29 '18 at 17:06
  • As a surs student myself this post is pretty much on the money, I am slightly confused by the end of the first paragraph though -- isn't discussion of proper classes in NBG technically a second-order discussion, so any definitions involving proper classes would be second order in the standard axiomatics? – Alec Rhea Jul 29 '18 at 18:42
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    No it is not. 2nd order logic is a pretty peculiar thing having nothing to do with ordinary mathematics of any kind. As for theories with sets and proper classes, they are maintained in the 1st order logic, with the understanding that all objects are classes, while sets are those classes which are members of other classes. – Vladimir Kanovei Jul 29 '18 at 18:51
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    @AlecRhea This is similar to the abominably-named "second-order arithmetic" being a first-order theory: the objects it talks about are second-order in a particular sense (= sets of naturals as opposed to just naturals), but the logic it works in is first-order. Conversely, second-order Peano arithmetic talks about first-order objects (= naturals only) using second-order logic. It's the worst. – Noah Schweber Jul 29 '18 at 18:56
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    Yes exactly what I meant. Except that Peano's native axioms are 2nd order in a sense, as the induction axiom claims something about arbitrary properties or sets of naturals. One has to focus into details though. – Vladimir Kanovei Jul 29 '18 at 18:58
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    @VladimirKanovei That's a good point: second-order logic lets you quantify over subsets of the domain, so second-order objects are referred to in second-order PA. So "talks about" was a bad phrase. I should have said: a structure in genuine second-order logic is specified entirely by its first-order part, e.g. a putative model of second-order PA is just an ordered semiring of "natural numbers," with the second-order objects which we can quantify over being supplied (somehow?) by the semantics. (I know you know this, I'm trying to allay any confusion my previous comment may have caused others.) – Noah Schweber Jul 29 '18 at 19:39
  • As someone who's never studied second order logic this discussion has been enlightening, thank you both. – Alec Rhea Jul 29 '18 at 20:33
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    Is "surs" a thing? – nombre Jul 29 '18 at 21:16
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    I find the terminology of second-order arithmetic and set theory useful, in that they describe the natural language of these theories. Second-order arithmetic has a first-order sort, numbers, and a second-order sort, sets, and this clarifies a lot of the analysis. Similarly, in GBC and KM, we have the first-order objects, sets, and the second-order objects, classes, and although our theories are first-order in these sorts, the first/second-order distinction is helpful. None of the analysis uses actual second-order logic, which is really set theory anyway (undertaken in the metatheory). – Joel David Hamkins Jul 29 '18 at 23:24
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    In particular, I never use the one-sorted version of second-order set theory, by which all objects are classes and sets are those that are also elements. This way of setting up the theory makes a mess of things and serves no purpose at all, except a false economy. There really is a fundamental difference in the first and second-order sorts, and the axioms of the theory are more naturally expressed in a language that makes the distinction explicit. Similarly, I don't know of anyone who uses a one-sorted approach to second-order arithmetic, although this also would be possible in the same way. – Joel David Hamkins Jul 29 '18 at 23:27
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    However the distinction between members and non-members is not something deeply intrinsic there, it is rather similar to a distinction between say finite and infinite, countable and uncountable, Borel and non-Borel (pointsets) etc, i.e., it just separates a subuniverse of simpler objects – Vladimir Kanovei Jul 30 '18 at 03:07
  • @nombre This is the first I've heard it, but it doesn't seem like a pejorative haha. It's reminiscent of mono/epi/iso in category theory imo. – Alec Rhea Aug 01 '18 at 20:50
  • @AlecRhea Yeah, I feared this phrasing had already catched on. I think I have come to like Conway's original formulation, i.e. "numbers", although it may sound too grandiloquent and it only makes sense when the context is clear. – nombre Aug 03 '18 at 20:43
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The real question as far as "ordinary mathematics" is concerned is whether there is a set-size surreal extension of the reals useful in doing analysis, and that as a very minimum admits a sine function. As far as I know the answer is negative.

Namely, there is no transfer principle in the surreals other than the one transfered from the hyperreals. Therefore it one wishes to do analysis with anything smaller than the absolutely largest class of numbers, the surreals are not an option. For example, all real functions extend to hyperreal extensions of the real field, but even such a simple function as the sine does not extend to surreal extensions (without passing via an identification of a maximal class-size surreal field and exploiting an identification of the latter with a class-size hyperreal field and importing a hyperreal transfer principle via the identification).

Mikhail Katz
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  • How are we to reconcile your answer with Philip's? – Joel David Hamkins Jul 29 '18 at 10:43
  • @JoelDavidHamkins, I don't detect any contradiction between my answer and my friend Philip Ehrlich's. Could you be more specific? – Mikhail Katz Jul 29 '18 at 12:53
  • You say that the $\sin$ function does not extend to the surreals, but Philip seems to say that there is a transfer principle, which would provide a $\sin$ function on the surreals, which whichever desired properties of it we would want. – Joel David Hamkins Jul 29 '18 at 13:01
  • I addressed that in my answer. Whatever transfer is available over the surreals comes from the identification of the maximal surreal class with a maximal hyperreal class, as asserted in theorem 20 in Philip's paper, and also in his answer. Do you interpret his answer as saying that there is a proof of the transfer principle over the surreals that's independent of the hyperreals? I don't think such an interpretation is correct. @JoelDavidHamkins – Mikhail Katz Jul 29 '18 at 13:05
  • What I am confused about is the final sentence of your answer, where you seem to say that the sine function does not extend to the surreals, whereas the answer of Philip seems to say that it does. Are you claiming that there is no sine function on the surreals (with the expected first-order properties)? – Joel David Hamkins Jul 29 '18 at 13:34
  • Where is the claim that the sine function extends? Philip is mainly focusing on the properties of a real closed field, rather than a language suitable for analysis that would naturally include the sine function as well. Also, as the answer indicates, they were recently successful in extending the exponential function, but the sine function is still a challenge as far as I know. @JoelDavidHamkins – Mikhail Katz Jul 29 '18 at 13:35
  • He says that the surreal numbers admit a relational extension to a model of nonstandard analysis, which I take to mean that we can relations on the surreals corresponding to the transfer of any given relation on the reals, such as the graph of the sine function. – Joel David Hamkins Jul 29 '18 at 13:38
  • True, but such an extension passes via a class-size model which is being identified with a limit ultrapower model for the hyperreals. @JoelDavidHamkins – Mikhail Katz Jul 29 '18 at 13:40
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    So it seems that the sine function does extend to the surreals. But if I understand you, you seem to object to the proof of this, if one should prove it using a limit ultrapower construction? The hyperreals also are usually defined via ultrapowers, and indeed, ultrafilters are inherent in the transfer principle for the hyperreals, as I explain in https://mathoverflow.net/a/57108/1946. – Joel David Hamkins Jul 29 '18 at 13:52
  • @JoelDavidHamkins, the sine function only extends via the identification of class-size surreals with class-size hyperreals, and not by any independent construction over the surreals. Also I would guess that there is no set-size surreal field where the sine function would extend because the isomorphism in question depends crucially on the fact that the models being identified are maximal. I agree that ultrafilters are involved in the construction of the hyperreals, but I am not sure what you are trying to say. – Mikhail Katz Jul 29 '18 at 14:01
  • From my perspective, the construction is basically analogous to how one constructs the sine function in a given model of the hyperreals. (And I think maximality has nothing to do with it.) – Joel David Hamkins Jul 29 '18 at 14:26
  • @JoelDavidHamkins, Which construction are you referring to exactly? I am not familiar with results that would identify specific set-size surreals with specific set-size hyperreals. If you have in mind a definition of the sine function over the surreals without passing via the hyperreals, that would certainly be interesting. Again I am not sure what you are claiming precisely. As far as "maximality" is concerned: that was the content of theorem 20 in Ehrlich's paper, which is why I mentioned it. – Mikhail Katz Jul 29 '18 at 14:29
  • If I understand the latest comments by Philip under his answer correctly, he agrees that my comments here concerning transfer are on target. @JoelDavidHamkins – Mikhail Katz Jul 29 '18 at 14:34
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    My view is that the sine function extends to the hyperreals by means of the ultrapower construction, and the same is true for the surreals, except that it is an iterated ultrapower. This is simply a class-length analogue of the same iterated ultrapower construction that one uses when providing set-sized models of the hyperreals with a specified hierarchy of infinitesimality. – Joel David Hamkins Jul 29 '18 at 15:04
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    Let me add that I find talk about "the" hyperreals to be an inaccurate conflation of the diverse, non-isomorphic models that serve as instances. The common limit of all these fields is: the surreal numbers. – Joel David Hamkins Jul 29 '18 at 15:07
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    @Mikhail. In addition to agreeing with Joel's comment about the sin function, I'd like to add the following. You say there is no sin function that can be defined on the surreals or, at least, there is no sin function other than one that can be induced from a nonstandard model of analysis. This is false, and has been known to be false for decades. In fact, as Martin Kruskal pointed out, there is a canonical sign function: $\sin(\theta +2\pi n)=sin\theta$, where $n$ ranges over the system Oz of omnific integers of No. – Philip Ehrlich Jul 29 '18 at 15:10
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    @Mikhail (Continued): Since Oz is not a nonstandard model of arithmetic, this would not be an example of a sin function one would use if one were doing nonstandard analysis, but it is a perfectly fine sin function that is very useful for various model theoretic purposes. Moreover, there is a vast array of initial subfields of No that have sin functions based on portions of this construction. I am doing work on what I call \emp{trigonometric ordered fields} in which this and related matters are discussed. – Philip Ehrlich Jul 29 '18 at 15:12
  • The real question as far as "ordinary mathematics" is concerned is whether there is a set-size surreal extension of the reals useful in doing analysis, and that as a very minimum admits a sine function. As far as I know the answer is negative. – Mikhail Katz Jul 29 '18 at 15:14
  • @JoelDavidHamkins, I find your claim that "the common limit of all these [hyperreal] fields is: the surreal numbers" a bit of an obfuscation. Namely, the "is" should be replaced by "is isomorphic to" by an isomorphism claimed in Theorem 20 of Philip's paper (note that no proof is provided though it does exist following Keisler, and is certainly non-trivial). – Mikhail Katz Jul 29 '18 at 15:48
  • I would like to know without considering the size question can we define the sin function of surreal and more real analytic functions in surreal counterparts. the HoTT implementation of surreal does not care about the size question. – Ember Edison Jan 17 '24 at 11:51
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    Ehrlich has a recent paper where he defines the sine on suitably chosen subfields of the surreals. @EmberEdison – Mikhail Katz Jan 17 '24 at 12:06