Surreals and NSA: some foundational issues.

A. Leaving aside the whole internal machinery of surreals (with funny questions like is $\omega$ an entire number and if yes is it odd or even, simple, a factorial, etc.), it is a principal foundational achievement of surreals that they give concrete, well-defined examples of saturated fields, in particular,

1) a concrete, well-defined countably saturated rcof,

2) a concrete, well-defined set-size saturated rcof (No itself), by necessity of class size.

The saturation properties in this context were first explicitly observed probably by Ehrlich in 1980s, but they follow from the $\eta_\alpha$-properties (obvious by construction) and a result (CK, Ex 5.4.4 on p. 369, later reproved by Simpson) which likely was "of common knowledge" in model theory in 1970s if not earlier.

Note that Hausdorff studied his $\eta_\alpha$ fields most successfully as "pantachies", that is, linearly ordered subsets of a certain partial order of R^N, and with a heavy dose of the axiom of choice - which does not yeild any single, well-defined, concrete example. Thus the countably saturated rcof which emerges as a certain initial part of No is probably the closest thing to the notoriously inconsistent "infinitaire pantachie" of DuBoisReymond known so far.

(As a marginal remark, a countably saturated rcof cannot be Borel - even cannot have a Borel set as the domain and Borel relation as the order - so it cannot be too concretely well-defined.)

B. In the context of foundations of infinitesimals, the surreals have a major defect: they are just a rcof, w/o an adequate subsystem of "surnatural" numbers, which is a sine qua non for any full-scale treatment of infinitesimals.

C. Nonstandard extensions of R commonly denoted by R* do not suffer from this, of course, and moreover, along with a related system of hyperintegers N* they are equipped with the asterisk of the whole $\omega$-high superstructure over R. Yet for a long time they used to have an own foundational issue: unique, concretely defined examples of R* were not known. Such examples of OD (ordinally definable) models R* of any amount of saturation were first defined by Kanovei-Shelah (JSL, 2004) - including

(i) the full set-size saturated R* of class-size,

and in fact

(ii) an ordinal-definable full set-size saturated elementary extension of the whole set universe of ZFC considered in detail in Kanovei-Reeken, Nonstandard analysis axiomatically, Springer 2004.

D. As any two full set-size saturated rcof (both of class size, of course) are isomorphic under any suitable class theory with GC (global choice) by means af a standard application of the b&f method, No and the ground rcof structure of R* as in (i) above are isomorphic (first observed by Ehrlich). This means that, postfactum, the surreals No do indeed contain (under GC) a suitable system of surintegers and do allow the whole system of real functions and much more - simply inherited from R* as in (i) by means of the mentioned isomorphism.

This leads to the following problems of foundational importance, yet to be solved.

Problem 1. Note that both surreals and R* of type (i) are OD, well-defined ZFC classes, but the isomorphism between them is not such - it is an application of GC. So we ask: does there exist a ZFC-well-defined isomorphism between them?

Problem 2. Sharper, does there exist a ZFC-well-defined adequate system of surintegers, satiafying PA at least and preferably making (No,surintegers) to be an elementary extension of (R,N) ?

Problem 2 can be answered in the affirmative both by affirmatively answering Problem 1, and by means of an own surrealistic construction.

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    $\begingroup$ Great question! $\endgroup$ – Joel David Hamkins Apr 10 '13 at 13:30
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    $\begingroup$ The omnific integer part of No, denoted Oz, defined in theorem 16 of Ehrlich's paper ohio.edu/people/ehrlich/Unification.pdf seems to be ordinal-definable. These are just the surreal numbers whose coefficients $n_\alpha$ in their Conway names $\Sigma_{\alpha\lt\beta} \omega^{\gamma_\alpha}n_\alpha$ are integers. But I don't know if Oz satisfies PA or if you get the elementary property that you want. But they are an integer part of No, in the sense that every surreal number $x$ is between $a$ and $a+1$ for some $a\in\text{Oz}$. Could you elaborate on why this isn't what you want? $\endgroup$ – Joel David Hamkins Apr 10 '13 at 13:47
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    $\begingroup$ David, don't despair, all of nonstandard analysis can be done using the surreals or some particular initial subfield of the surreals as the underlying ordered field in a nonstandard model of analysis. On the other hand, given the present state of knowledge, it would be inconvenient to do so. After all, there are highly visible nonstandard models of analysis that are easy to work with. Of course, none of them have the intuitive appeal of the surreals. Things may change with time, but the importance of the surreals has little to do with its relation to the great theory of nonstandard analysis. $\endgroup$ – Philip Ehrlich Apr 11 '13 at 1:40
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    $\begingroup$ (Continued from above): By the way, most of the nonstandard analysts I have spoken with (including H.J. Keisler, David Ross and Mauro Di Nasso, ...) are great fans of the surreals and would welcome the discovery of useful connections. $\endgroup$ – Philip Ehrlich Apr 11 '13 at 2:09
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    $\begingroup$ >all of nonstandard analysis can be done using the surreals or some particular initial subfield of the surreals as the underlying ordered field in a nonstandard model of analysis. To be exact, changing for whatever purpose the elements of R* with some surreals. The only foundational appeal of this is that the surreal fields are uniquely defined - but the field structure is too little to work with infinitesimals (further than "aga, 1/w is an infinitesimal"). The challenge has been stated: is No capable to run the Euler sine decomposition w/o borrowing *N to replace its own non-PA omni ones. $\endgroup$ – Vladimir Kanovei Apr 11 '13 at 3:28

Installment 2.

Having addressed a historical point in the first installment of my response to Vladimir (see below), I now turn to the first of his two interesting questions. To begin with, Theorem 20 of my The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45—reads as follows:

In NBG [which I take to include global choice] there is (up to isomorphism) a unique structure (R, R*,* ) such that Axioms A–E [Keisler’s (1976) axioms for saturated hyperreal number systems] are satisfied and for which R* is a proper class; moreover, in such a structure R* is isomorphic to No. Such a structure is in fact (to within isomorphism) the unique model of Axioms A–E whose existence can be established in NBG without additional assumptions.

Vladimir’s first question regards the relation between the hypperreal number system of Theorem 20 and the full set-saturated hyperreal number system developed in his and Reeken’s important treatise on nonstandard analysis. I am skeptical that Vladimir’s first question admits an affirmative answer. In fact, I propose the following

Conjecture: Without global choice, one can not prove that the Kanovei-Reeken full set-saturated hyperreal number systems is isomorphic to the one of Theorem 20.

In NBG with global choice, No is (up to isomorphism) the unique homogeneous universal ordered field, i.e. it contains an isomorphic copy of every ordered field whose universe is a set or proper class of NBG, and every isomorphism between subfields of No, whose universes are sets, can be extended to an automorphism. The proof that No contains an isomorphic copy of every ordered field of power On uses global choice as does the proof of homogeneity. It is not clear to me how one can prove either of these results about the Kanovei-Reeken system in the Kanovei-Reeken framework. Of course, I may not be thinking creatively enough; moreover, it might be possible that Vladimir's first question would have a positive answer for some pared down version of No, but based on a private exchange with Vladimir as well as his question, it's No of Theorem 20 he has in mind.

I will address Joel’s question about the omnific integers as well as Dave’s and Emil’s informative answers in a further installment

First Installment:

I hope to return soon (perhaps after I finish my taxes) to address some of the interesting questions raised by Vladimir. In some cases I will expand on the answers I provided Vladimir in response to his recent private letter to me, responses which turned out to be incorporated into the motivation and formulation of some of his questions. At that time, I will also explain why the nonnegative portion of the omnific integers referred to by Joel is not a full model of PA and make a few points about them as well.

For the time being, I merely wish to correct misconceptions about Hausdorff’s great writings on $\eta_{\alpha}$-orderings that one might be apt to walk away with after reading Vladimir’s remarks. Since I treat these matters with some care in Section 8 of my paper,

The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45

I refer interested parties to that paper for the requisite definitions and details. Even greater detail will be found in a forthcoming work of mine entitled From du Bois-Reymond’s Infinitary Pantachie to the Surreal Numbers.

To begin with, in Hausdorff's first great paper on ordered sets of 1906, he introduces the idea of an $\eta_{\alpha}$-ordering in precisely the way we use it today and he continued to use it in the same fashion in all of his subsequent writings. The definition is given on page 132 of the original paper and on page 150 of the wonderful recent English translation by Plotkin. As I explain in my aforementioned BSL paper, Hausdorff was motivated to introduce the idea of an $\eta_{1}$-ordering to characterize the order type of his very insightful reconfiguration of Paul du Bois-Reymond's flawed conception of an infinitary pantichie. In fact, he proves:

HAUSDORFF 1 [1907]: Infinitary pantachies exist. If P is an infinitary pantachie, then P is an $\eta_{1}$-ordering of power $2^{\aleph_{0}}$; in fact, P is (up to isomorphism) the unique $\eta_{1}$-ordering of power $\aleph_{1}$, assuming (the Continuum Hypothesis) CH.

In his investigation of 1907, Hausdorff also raises the question of the existence of a pantachie that is algebraically a field, but he only makes partial headway in providing an answer. However, in 1909 he returned to the problem and provided a stunning positive answer. Indeed, beginning with the ordered set of numerical sequences of the form r, r , r, …, r, … where r is a rational number, and utilizing what appears to be the very first algebraic application of his maximal principle, Hausdorff proves the following little-known, remarkable result.

HAUSDORFF 2 [1909]. There is a pantachie H of numerical sequences of real numbers indexed over the natural numbers (with operations suitably defined) that is an ordered field. Any such pantachie is, in fact, a real-closed ordered field.

Writing before Artin and Schrier [1926], Hausdorff of course does not refer to H as real closed; but he essentially establishes H is real-closed by showing it is the union of a chain of ordered fields, each of which admits no algebraic extension to a more inclusive ordered field.

Thus, contrary to what Vladimir contends, Hausdorff does not formulate his theory of $\eta_{\alpha}$-orderings in terms of pantachies, but rather in the manner we know and love; moreover, he uses the special case of an $\eta_{1}$-ordering to characterize the order type of his pantachies.

I'm delighted to see that Vladimir has edited his remarks on Hausdorff, presumably in light of the above remarks.

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    $\begingroup$ @Joel, for a start the square root of 2 is rational! $\endgroup$ – Dave Marker Apr 11 '13 at 14:08
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    $\begingroup$ Indeed. This has also been discussed at mathoverflow.net/questions/72691 . $\endgroup$ – Emil Jeřábek Apr 11 '13 at 15:12
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    $\begingroup$ There is nothing particularly unusual about this. If you take any sufficiently saturated real-closed field $R$, it will have many different integer parts satisfying incompatible completions of arithmetic. For example, if $R$ is $\omega$-big, then every consistent extension of IOpen is satisfied in some integer part $I$ of $R$. If $R$ is countable recursively saturated, the same holds for r.e. extensions, and you can arrange that $R$ is the real closure of $I$ as a bonus. (And of course, in general there will be many different integer parts satisfying the same theory.) $\endgroup$ – Emil Jeřábek Apr 12 '13 at 15:47
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    $\begingroup$ Let $M$ be a saturated model and $T$ a theory in a expanded language that is consistent with the theory of $M$. Then there you can interpret the new symbols in $M$ so that $M$ is a model of $T$. (Exercise 4.5.35 in my book.) Let $T_0$ be a consistent extension of Open Induction. Let $R$ be a saturated RCF. Add a predicate symbol $P$ and let $T$ be the theory saying that $P$ is a model of $T$ and every element of $R$ has an integer part in $P$. This is consistent with RCF since it's true in the real closure of a model of $T_0$. So there is $P\subset R$, $(R,P)\models T$. $\endgroup$ – Dave Marker Apr 13 '13 at 14:38
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    $\begingroup$ Barwise and Schlipf showed that the same holds if $M$ is recursively saturated and $T$ is recursively axiomatizable. $\endgroup$ – Dave Marker Apr 13 '13 at 14:39

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