By Alec Rhea

2019-02-06 06:41:00 8 Comments

Common folklore dictates that the Surreals were discovered by John Conway as a lark while studying game theory in the early 1970's, and popularized by Donald Knuth in his 1974 novella.

Wikipedia disagrees; the page claims that Norman Alling gave a construction in 1962 which modified the notion of Hahn series fields to realize a field structure on Hausdorff's $\eta_\alpha$-sets, and that this construction yields the Surreals when considered over all ordinals -- the article pretty explicitly characterizes Conway's construction as a popularized rediscovery more than 10 years later (!).

The given reference links to a paper from 1962 (received by editors in 1960) which seems to fit the bill, however this paper gives no indication that Alling considered his construction extended to all ordinals nor does it contain mention of proper classes that I can find.

Is there any evidence indicating that Alling considered a proper class sized version of his construction almost a decade before Conway did?

Alling published a book in 1987 which constructs the Surreals in exactly this manner*, but this is (of course) more than a decade after Conway's construction was popularized. Was this construction known to Alling earlier and only written up in book form when he realized there was a more widespread interest, or were proper-class considerations in this arena genuinely not in play before Conway?

*This is incorrect, thanks to Philip Ehrlich for pointing out my mistake. Alling's 1987 book doesn't construct the surreals as modified formal power series in the fashion of his 1962 paper; the methods involved are more subtle. On January 3, 1983 and December 20, 1982, Norman Alling and Philip Ehrlich respectively submitted papers which constructed isomorphic copies of the surreals using modified versions of the constructions in Allings 1962 paper, and these submissions eventually lead to the publication of two papers to this effect. The 1987 book mentioned above contains expansions of these works in sections 4.02 and 4.03, respectively. (see Philip's excellent answer below for a better description of events)


@Philip Ehrlich 2019-02-09 14:06:02

Let me begin by adding my voice to those who have said that Conway is the sole originator of the theory of surreal numbers, the reasons given by Andreas being critical. Moreover, based on my conversations and correspondence with Norman (Alling), with whom I did the following joint work on the surreals in the 1980s, there is no question he would concur.

(i) An Alternative Construction of Conway's Surreal Numbers, C.R. Math. Rep. Acad. Sci. Canada VIII (1986), pp. 241-46.

(ii) An Abstract Characterization of a Full Class of Surreal Numbers, C.R. Math. Rep. Acad. Sci. Canada VIII (1986), pp. 303-8.

(iii) Sections 4.02 and 4.03 of Norman Alling’s Foundations of Analysis Over Surreal Number Fields, North-Holland Publishing Co., Amsterdam, 1987.

My principal reason for writing this answer is to help clarify the relation between Conway's (1976) work, Norman’s work of 1962, and some work Norman and I did independently in the early 1980s making use of Norman’s work of 1962, and to correct the mistaken characterization of the construction of the surreals contained in Norman’s book as described by Alec in his question.

In his paper of 1962, Norman proves the existence of a real-closed field that is an $\eta_\alpha$-set of power $\aleph_\alpha$, whenever $\aleph_\alpha$ is regular and satisfies another natural set-theoretic condition, thereby providing an affirmative answer to a question of Erdös, Gillman and Henriksen that arose from their work on rings of continuous functions. To prove the result, Norman employed a construction that is a marriage of Hahn’s (1907) celebrated construction of non-Archimedean ordered fields and Hausdorff’s (1907) equally celebrated construction of $\eta_\alpha$-sets obtained from transfinite sequences of 0s and 1s. In my opinion, Norman’s result is one of the genuinely important results of the 20th-century theory of ordered algebraic systems.

On January 3, 1983 and December 20, 1982, Norman and I respectively submitted papers for publication in which we show that an isomorphic copy of Conway’s ordered field $\mathbf{No}$ could be obtained using Norman’s just-mentioned construction or simple variations thereof. Norman did this via the union of a chain $K_{\alpha + 1}$, $\alpha \in \mathbf{On}$, of real-closed fields that are $\eta_{\alpha + 1}$-sets (constructed with minor modifications as in his (1962)) and I did it more simply using Hahn’s construction in conjunction with Custa-Dutarti’s construction of successively filling in cuts (Algebra Ordinal, Rev. Acad. Ci Madrid 48 (1954), pp. 103-145), a construction Harzheim had shown leads to various $\eta_\alpha$-sets at various levels of recursion (Beiträge zur Theorie der Ordnungstypen, lnsbesondere der $\eta_\alpha$-Mengen, Math. Ann. 154 (1964), pp. 116-134). In our respective papers, we also introduced analogous constructions for families of isomorphic copies of distinguished subfields of ${\bf{No}}$ that are real-closed fields that are $\eta_\alpha$-sets. Norman’s paper appeared as Conway’s Field of Surreal Numbers, Trans. Am. Math Soc. 287, (1985), pp. 365-386, and a portion of my paper, which was initially submitted to Fund. Math., eventually appeared six years later as An Alternative Construction of Conway's Ordered Field ${\bf{No}}$, Algebra Universalis, 25 (1988), pp. 7-16. Another portion of that work appeared in my Absolutely Saturated Models, Fund. Math., 133 (1989), pp. 39-46. While I was waiting to hear back from the journal, I sent my paper to Norman, who I did not know at the time and who informed me that his paper (which I was unaware of) had been accepted for publication. Nevertheless, he believed my paper, which differed from his in various ways (including an emphasis on model theoretic connections, and the inclusion of the Custa-Dutarti cut construction which he was not familiar with), should be published, and he proposed we carry out joint work, which led to (i)-(iii) above. (i) is a treatment of the construction of the surreals based on the Custa-Dutarti cut construction, (ii) provides an axiomatiization of the surreals including its birthday structure, and (iii) is two subsections of Norman's book containing expansions of the just-said works.

As my characterization of (iii) suggests, Alec is mistaken when he asserts that in his (1987) Alling constructs the surreals as a field of formal power series (using techniques from his (1962)). What is true is that long after Norman and I introduce the surreals in that work using the Cuesta Dutari cut construction (pp. 121-127), with sums and products defined à la Conway, Norman points out (see pp. 246-247) that ${\bf{No}}$ so constructed is isomorphic to a distinguished field of formal power series. It is worth noting that while Conway was not familiar with the historical background that Norman and I drew attention to in our papers from the 1980s, Conway was aware of the relation between ${\bf{No}}$ and distinguished fields of formal power series (as is evident from Theorem 21 and the subsequent remarks from in his monograph On Numbers and Games). A simple proof of that relationship making use of ${\bf{No}}$’s simplicity hierarchy is the proof of Theorem 16, p. 1249 of the present author’s Number Systems with Simplicity Hierarchies: A Generalization of Conway’s Theory of Surreal Numbers, J. of Sym. Log. 66 (2001), pp. 1231-1258.

@Alec Rhea 2019-02-09 17:28:21

Thank you for (again) catching one of my careless errors Philip, and thank you very much for the excellent references and history. I'll edit the question to fix the mistake.

@Philip Ehrlich 2019-02-09 17:32:05

Happy to be of help, Alec.

@Carlo Beenakker 2019-02-06 07:26:14

Norman Alling's Conway's field of surreal numbers (1985) gives full credit to Conway:

Conway introduced the Field No of numbers, which Knuth has called the surreal numbers. No is a proper class and a real-closed field, with a very high level of density, which can be described by extending Hausdorff's $\eta_\xi$ condition. In this paper the author applies a century of research on ordered sets, groups, and fields to the study of No.

References to Alling's own 1962 paper appear only in passing, on page 373 and 377–378. The 1987 book referred to in the OP follows up on this 1985 exposition, which makes it clear that Alling in no way claims independence from Conway.

@Alec Rhea 2019-02-06 07:27:53

Much appreciated, it's good to know this part of my life wasn't a lie :^).

@Ben Crowell 2019-02-06 15:00:32

I've edited the WP article to reflect this answer.

@Alec Rhea 2019-02-06 17:41:59

@BenCrowell Thank you, I was going to this morning but you’ve beaten me to it! I think comments reflecting Andreas’s answer are also in order, I’ll add them to your edits.

@Andreas Blass 2019-02-06 17:13:43

If one thinks of the surreal numbers as just a proper-class sized saturated real-closed field, then I think Alling deserves much of the credit for the discovery or invention. He didn't deal with the proper-class sized case, but his hypotheses do cover the case of strongly inaccessible cardinals. A strongly inaccessible cardinal can be viewed as class-sized simply by cutting off the (cumulative hierarchy view of the) universe of sets one level after that cardinal. So I'm inclined to view the passage to proper-class size as no big deal.

But the field of surreal numbers has an important additional structure, namely what Conway calls "birthdays". In other words, this field has a very natural construction in terms of transfinitely iterated filling of Dedekind cuts. As far as I know, this construction and the fact that it produces a saturated real-closed field are entirely due to Conway.

@Gro-Tsen 2019-02-06 18:22:15

I've always been perplexed by the insistence with which some people want to consider surreals in the class-sized variant. I mean, "class-sized" just means "the smallest cardinal my model of set theory can't handle" — well, if it can't handle it, pick a larger size! And no interesting property of the surreals depends on class-size-ness and can't be stated for a cardinal $\kappa$ of large enough cofinality. So whence the class craze?

@Alec Rhea 2019-02-06 18:26:54

Thank you Andreas, I agree and I’ve edited the WP article to reflect this answer. It was also a conscious decision to use discovered over invented, but this is a piece of mathematical philosophy I was trying to rug-sweep ;).

@Alec Rhea 2019-02-06 18:34:58

@Gro-Tsen As one of the class-crazed people (see here for example…), I would be very happy to see a reference/outline of a proof for your claim about large enough cardinals capturing essential properties? In the comments on the above question nombre supports your position by pointing out that any 'model-theoretically tame' structures like its valued differential field structure will be preserved, but there still seems to be some ambiguity about higher order properties.

@Gro-Tsen 2019-02-06 21:22:11

@AlecRhea I think you need to put the question backwards: what kind of property do you think makes "class-sized" worthy of interest? The way I see it, "class-sized" just means "large enough that we don't care about" so if you try to go that far, you're rigging your own game, you should rather assume the existence of a cardinal with the properties you really want.

@Gro-Tsen 2019-02-06 21:25:36

Or, to put it differently, look at the ordinals: the class of all ordinals is a class-sized ordinal, but it isn't particularly interesting: it's the process of constructing the ordinals which is, and perhaps some particular milestones long the process (like "cardinal", "regular cardinal", etc.). The same is true of the surreals: what is interesting is the process of constructing the surreals to any given ordinal length — not the mirage of a class-sized totality.

@Gro-Tsen 2019-02-06 21:40:04

But anyway, to answer your question as to why I claim that any property of some class-sized object can be attained at some large enough ordinal, this is essentially the content of various reflection principles. Take your favorite property of the surreals, then by a reflection principle there are club-many $\alpha$ such that the same is true in $V_\alpha$, and the surreals in $V_\alpha$ are essentially the surreals of that length.

@Timothy Chow 2019-02-07 00:55:49

Just to emphasize, much of what is interesting about the surreals comes from the extra structure beyond the Field structure (what I like to call the "L|R structure"). For example there is no way to distinguish ω from ω.2 using just the Field structure. Back in 1999, Steve Simpson made a rather big fuss on the Foundations of Mathematics mailing list about the surreals being "not new," but after some back-and-forth discussion, he clarified that he meant that the structure of the surreals qua ordered Field was already known, not that the L|R structure had been previously described.

@Alec Rhea 2019-02-07 02:48:59

@Gro-Tsen Considering 'class sized' objects in a set theory makes it 'second order', and there is interesting mathematics currently taking place in second order set theory. For an example, Kameryn Williams' recent dissertation discusses an 'unrolling' and 'cutting off' process that allows one to move between second order theories of different consistency strength -- he generalizes the Marek and Mostowski result that KM $+$ class collection unrolls to a model of ZFC$^-$ to weaker set theories. I don't see how to begin these considerations without classes.

@Alec Rhea 2019-02-07 02:55:58

@Gro-Tsen Your suggestion of reflection principles is a good one, but it isn't obvious to me on the face of it that all properties we might ask about will satisfy conditions for reflection. The class of all sets obviously can't have that particular property reflect down to a set in any $V_\alpha$ unless I misunderstand the wiki page, and perhaps there are similar properties on the surreals that we can't safely reflect down without contradiction.

@Alec Rhea 2019-02-07 02:59:23

@TimothyChow It seems many people agree with you about the L|R structure being the most interesting piece, but I don't quite understand your comment about $\omega$ and $\omega.2$ -- what binary operation is $.$? Assuming it's multiplication (or any of the field operations) we definitely have $\omega\neq2.\omega$ which is part of the field structure, but perhaps $.$ is a different binary operation?

@Timothy Chow 2019-02-07 03:19:51

@AlecRhea : The dot represents ordinal multiplication. I guess maybe I phrased my comment unclearly. Conway's construction naturally gives rise to the ordinals, as as ordinals, ω and ω.2 are certainly distinguishable. However, if you throw away all the structure except the Field structure, then you can no longer reconstruct which element was ω and which element was ω.2

@Alec Rhea 2019-02-07 03:41:46

@TimothyChow What do you mean we can’t reconstruct which was which? The ordinal multiplication $\omega.2$ matches Hessenberg multiplication $\omega2=2\omega$ which in turn matches surreal multiplication, so we would have $\omega\neq\omega.2$ just using the Field structure.

@Gro-Tsen 2019-02-07 07:09:03

Yes obviously, a property such as "every ordered field embeds in the surreals" will reflect to "every ordered field in $V_α$ embeds in the $V_α$-surreals" (for club-many $α$), but "every" is a set-theoretic mirage: it only means "every in our universe" (which is the $V_κ$ for the $κ$ where we stopped making ordinals). • Note that I have the same view regarding category theory: instead of looking at class-sized categories, e.g., of (all) sets, (all) groups, etc., we should look at those of $V_α$-sets, $V_α$-groups, etc. and ask which properties of $α$ we actually need for what we care about.

@Gro-Tsen 2019-02-07 07:27:56

Regarding second-order: of course there are plenty of interesting things to say about classes from the logical point of view and how they map to models of $\mathsf{ZFC}$ of larger height, but these are fine properties of class comprehension axioms. I'm at a loss as to exactly what things you might say about surreals that require subtle class comprehension… It seems to me it's much more interesting to see how small we can make the surreals and still get nice properties (e.g., algebraic closedness) than make them "class-large" and remove such questions. This is how ordinals are studied.

@Gro-Tsen 2019-02-07 07:29:20

Anyway, I don't think the comments section of a MO answer are the right place to discuss this, so I'll refrain from making any more. You're welcome to ask a question on the subject if you can find a precise one to ask. (Of if you want general discussion, there's a "chat" but I don't know how it works. I'm also on Twitter with handle @gro_tsen.)

@Emil Jeřábek 2019-02-07 08:03:16

@AlecRhea There is a field automorphism of the sureals that takes $\omega\cdot2$ to $\omega$ (and $\omega$ to $\frac12\omega$). You can tell apart $\omega$ from $\omega\cdot2$ (in the sense that they are distinct elements, and $\omega<\omega\cdot2$), but there is no way you can just look at the field structure, and point towards $\omega$. All infinitely large elements have the same field-theoretic properties.

@Alec Rhea 2019-02-07 08:06:52

@EmilJeřábek Thank you for clarifying, Timothy’s comment makes sense now and that is interesting.

@Alec Rhea 2019-02-07 08:13:09

@Gro-Tsen I agree that we’re probably overstaying our welcome in the comments, but I’ll chew things over and message you on twitter if anything nontrivial occurs to me. Part of the problem is that I don’t know what properties to care about on the surreals since we don’t know how to/if we can do analysis on them yet, and it’s analytical properties I’d be concerned with (which are generally finer than topological properties, for example manifolds can be homeomorphic but not diffeomorphic).

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