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The reals are the unique complete ordered field. The hyperreals $\mathbb{R}^\*$ are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded that this was because ZFC was tuned up to guarantee the uniqueness of the reals. Ehrlich wrote a long paper in 2012 (ref and link below), which I've only skimmed so far. It's mainly about the surreals $\textbf{No}$, not the hyperreals, but it seems to suggest that Robinson's idea has been carried forward successfully by people like Keisler and Ehrlich. Apparently NBG set theory has some properties that are better suited to this sort of thing than those of ZFC.

Section 9 of the Ehrlich paper discusses the relationship between $\mathbb{R}^\*$ and $\textbf{No}$ within NBG. He presents Keisler's axioms for the hyperreals, which basically say that they're a proper extension of the reals, the transfer principle holds, and they're saturated. At the end of the section, he states a theorem: "In NBG there is ... a unique structure $\langle\mathbb{R},\mathbb{R}^\*,*\rangle$ such that [Keisler's axioms] are satisfied and for which $\mathbb{R}^\*$ is a proper class; moreover, in such a structure $\mathbb{R}^\*$ is isomorphic to $\textbf{No}$."

My question is: Does this result indicate that Robinson's program has been completed successfully and in a way that would satisfy mathematicians in general that the nonuniqueness of the hyperreals is no longer an argument against NSA? It seems to me that this would depend on the consensus about NBG: whether NBG is expected to be consistent; whether it is a natural way of doing set theory with proper classes; and whether a result such as Ehrlich's theorem is likely to be true for any set theory with proper classes, or whether such results are likely to be true only because of some specific properties of NBG (in which case the nonuniqueness has only been made into a new kind of nonuniqueness). Since I know almost nothing about NBG, I don't know the answers to these questions.

One thing that confuses me here is that I thought the surreals lacked the transfer principle, so, e.g., where the hyperreals automatically inherit $\mathbb{Z}^\*$ from $\mathbb{Z}$ as an internal set, a specific effort has to be made to define the omnific integers $\textbf{Oz}$ as a subclass of $\textbf{No}$, and $\textbf{Oz}$ doesn't necessarily have the same properties as $\mathbb{Z}$ with respect to, e.g., induction and prime factorization (see Can we axiomatize Omnific Integers without the Surreal Number system? ). Would the idea be that according to Ehrlich's result, $\mathbb{Z}^\*$ would be (isomorphic to) a subclass of $\textbf{Oz}$?

I'm a physicist, not a mathematician, and if this seems inappropriate for mathoverflow, please add a comment saying so, and I'll move it to math.SE. I posted here because it relates to current research, but I'm not a competent research-level mathematician.

Philip Ehrlich (2012). "The absolute arithmetic continuum and the unification of all numbers great and small". The Bulletin of Symbolic Logic 18 (1): 1–45, http://www.math.ucla.edu/~asl/bsl/1801-toc.htm

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Since when is the non-uniqueness of the hyperreals an objection to non-standard analysis? In any case, NBG is equiconsistent with ZFC. In fact, it's a conservative extension of ZFC. There is no problem using NBG instead of ZFC, the back and forth transfer of results is mostly routine. –  François G. Dorais Feb 12 '12 at 18:24
Ehrlich doesn't give a proof of Theorem 20. However, from context, the situation appears to be similar to that of so-called monster models in model theory. These are perhaps better thought of as proper classes, but for technical reasons they are usually defined as sufficiently large saturated models. The same trick should apply to Ehrlich's model, so everything that can be proved using this proper-class hyperreals can also be proved in ZFC using a sufficiently large saturated model. Of course, Ehrlich's model is arguably aesthetically and philosophically more pleasant to work with. –  François G. Dorais Feb 12 '12 at 18:44
Incidentally, in Goldblatt's book "Lectures on the hyperreals" he mentions that $\mathbb{R}^*$ constructed in the usual way with a non-principal ultrafilter is unique up to isomorphism if the continuum hypothesis is true (!) –  Matthew Towers Feb 12 '12 at 18:47
Thanks for the helpful comments! Apparently I can't upvote because I don't have enough rep. Since Gerald Edgar thinks this is probably inappropriate for mathoverflow, I will move it to math.SE. –  Ben Crowell Feb 12 '12 at 18:54
I don't see why this should be inappropriate for MO. –  ftonti Feb 12 '12 at 21:41

4 Answers 4

As part of his question, Bell Crowell correctly observes:

"Section 9 of the Ehrlich paper discusses the relationship between R∗ and No within NBG. He presents Keisler's axioms for the hyperreals, which basically say that they're a proper extension of the reals, the transfer principle holds, and they're saturated. At the end of the section, he states a theorem: "In NBG [with global choice] there is (up to isomorphism) a unique structure ⟨R,R∗,∗⟩ such that [Keisler's axioms] are satisfied and for which R∗ is a proper class; moreover, in such a structure R∗ is isomorphic to No.""

At that time I made it absolutely clear that the first part of the result is due to H.J. Keisler (1976) and that my modest contribution is to point out the relation (as ordered fields) between R* and No. The work of Keisler and the relation of my work to it seem to be lost in the remarks of Vladimir.

Of course, attributing the result to Keisler, as I remain entirely confident I correctly did, does not diminish the subsequent important contributions of others.

Edit: Readers interested in reading the paper including the discussion of Keisler's work may go to: http://www.ohio.edu/people/ehrlich/

EDIT: Since Vladimir appears to insist in his comment below that Keisler DOES NOT discuss proper classes in 1976, I am taking the liberty to quote Keisler and some of the relevant discussion from my paper. I will leave it to others to decide if I am giving Keisler undue credit.

Following his statement of his Axioms A-D of 1976--the function axiom, the solution axiom, and the axioms the state that R* is proper ordered field extension of the complete ordered field R of real numbers--Keisler writes:

“The real numbers are the unique complete ordered field. By analogy, we would like to uniquely characterize the hyperreal structure ⟨R,R∗,∗⟩ by some sort of completeness property. However, we run into a set-theoretic difficulty; there are structures R* of arbitrary large cardinal number which satisfy Axioms A-D, so there cannot be a largest one. Two ways around this difficulty are to make R* a proper class rather than a set, or to put a restriction on the cardinal number of R*. We use the second method because it is simpler.” [Keisler 1976, p. 59]

With the above in mind, Keisler sets the stage to overcome the uniqueness problem by introducing the following axiom, and then proceeds to prove the subsequent theorem.

AXIOM E. (Saturation Axiom). Let S be a set of equations and inequalities involving real functions, hyperreal constants, and variables, such that S has a smaller cardinality than R*. If every finite subset of S has a hyperreal solution, then S has a hyperreal solution.

KEISLER 1 [1976]. There is up to isomorphism a unique structure ⟨R,R∗,∗⟩ such that Axioms A-E are satisfied and the cardinality of R* is the first uncountable inaccessible cardinal.

If ⟨R,R∗,∗⟩ satisfies Axioms A-D, then R* is of course real-closed. It is also evident that, if ⟨R,R∗,∗⟩ further satisfies Axiom E, then R* is an $\eta_{\alpha}$-ordering of power $\aleph_{\alpha}$, where $\aleph_{\alpha}$ is the power of R*. Accordingly, since (in NBG) No is (up to isomorphism) the unique real-closed field that is an $\eta_{On}$-ordering of power $\aleph_{On}$, R* would be isomorphic to No in any model of A-E that is a proper class (in NBG).

Motivated by the above, in September of 2002 we wrote to Keisler, reminded him of his idea of making “R* a proper class rather than a set”, observed that in such a model R* would be isomorphic to No, and inquired how he had intended to prove the result for proper classes since the proof he employs, which uses a superstructure, cannot be carried out for proper classes in NBG or in any of the most familiar alternative class theories.

In response, Keisler offered the following revealing remarks, which he has graciously granted me permission to reproduce.

"What I had in mind in getting around the uniqueness problem for the hyperreals in “Foundations of Infinitesimal Calculus” was to work in NBG with global choice (i.e. a class of ordered pairs that well orders the universe). This is a conservative extension of ZFC. I was not thinking of doing it within a superstructure, but just getting four objects R, R*, <*, * which satisfy Axioms A-E. R is a set, R* is a proper class, <* is a proper class of ordered pairs of elements of R*, and * is a proper class of ordered triples (f,x,y) of sets, where f is an n-ary real function for some n, x is an n-tuple of elements of R* and y is in R*. In this setup, f*(x)=y means that (f,x,y) is in the class *. There should be no problem with * being a legitimate entity in NBG with global choice. Since each ordered triple of sets is again a set, * is just a class of sets. I believe that this can be done in an explicit way so that R, R*, <*, and * are definable in NBG with an extra symbol for a well ordering of V." [Keisler to Ehrlich 10/20/02]

Moreover, in a subsequent letter, Keisler went on to add:

I did not do it that way because it would have required a longer discussion of the set theoretic background. [Keisler to Ehrlich 5/14/06]

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Philip, regarding Thm 20 in [BSL] a reference is made to your 2002-2006 exhange with Keisler - long after the whole construction of class-size saturated extensions under NBG had been already published by K&Ree 1995 if not earlier. Keisler book 1976 mentioned in your posting (even in the last version, already with a part of K&She 2004) DOES NOT discuss class-size models. But absolutely this "does not diminish the subsequent important contribution" in your further proving the "intriguing result" of facts already known or trivial, so it remains to wish you all success towards this big goal. –  Vladimir Kanovei Apr 18 '13 at 4:00
Vladimir: My response to the portion of your remark regarding Keisler is in the edited version of my answer. I will not respond to the nature of your remark regarding my work. –  Philip Ehrlich Apr 18 '13 at 14:16

Russians say the same balls from a different angle. So let me try once again and in more detail, to weed out inconsistent references and misconceptions.

Theorem 20 announced in [Ehrlich 2012] asserts, roughly, that, under axioms of NBG,

1) there is a (necessarily class-size) extension R* of R, satisfying certain (not all) typical properties of extensions considered in NSA beyond being just an elementary ext of R, and additionally full set-size saturated;

2) this extension is unique up to isomorphism

3) this extension is rcof-isomorphic to SurR


  • item 1 was established by Kanovei & Shelah (JSL 2004) in a much stronger form of a definable (OD) full set-size saturated elementary extension of the whole set universe, which exists in ZFC as a definable class - hence, unique in virtue of its unique definition;

  • in the context of NBG (with global choice) -- as in [Ehrlich BSL] -- item 1 was established much earlier, at least in Kanovei&Reeken (Studia Logica 1995, 55, 2) in a stronger form of a full set-size saturated elementary extension of the whole set universe, but the result essentially goes back to late-1970 research of Hrbacek;

  • items 2 and 3 are routine applications of back-and-forth method --- essentially, observed by Kanovei & Reeken, NSA axiomatically (2004), p. 153, Problem 4.3.18;

  • the uniqueness modulo isomorphism, trivial under NBG by the above, is an open problem in the ZFC context, first observed by Kanovei & Reeken, ibid.

Conclusion. 1) The problem of definable -- hence, unique in virtue of its unique definition -- existence of full set-size saturated models of NSA has been solved by Kanovei & Shelah 2004 in a more difficult context of ZFC and before 1995 by Hrbacek-Kanovei-Reeken in an easier context of NBG.

2) [Ehrlich BSL] has little to do with these results, of course.

3) The problem of uniqueness modulo isomorphism in ZFC is still open. $\square$

Some other remarks and problems see my earlier post on foundational problems.

And a marginal note. That NBG is a conservative extension of ZFC is irrelevant as long as one considers classes (like full set-size saturated models).

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Since my comment is too long to for a comment, I have provided an extended comment in a separate answer. –  Philip Ehrlich Apr 18 '13 at 14:46

The usefulness of the hyperreals stems from such tools as saturation and the transfer principle. These tools are available in other superfields R' of R only to the extent that one can construct morphisms between the hyperreals and R'. Even if some limit ultrapower model of $\mathbb R^*$ is maximal and unique in some weak sense, this does not really affect its applications, certainly not in physics where the most likely model to be used is the simplest one, namely $\mathbb R^{\mathbb N}$ modulo an ultrafilter on $\mathbb N$. The precise relationship to the surreals is being discussed at Surreals and NSA: some foundational issues but ultimately this discussion has little bearing on whether "Robinson's program has been completed successfully" as you put it.

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Does this result indicate that Robinson's program has been completed successfully and in a way that would satisfy mathematicians in general that the nonuniqueness of the hyperreals is no longer an argument against NSA?

in such a structure $\mathbf R^*$ is isomorphic to $\mathbf{No}$.
This isomorphism is non-constructive and non-unique.

inappropriate for mathoverflow

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The downvote was note mine. –  Ben Crowell Feb 12 '12 at 19:04
If it is inappropriate a reason should be given and we should consider closing the question. BUT I really don't see why it is inappropriate, and this answer provides no grounds for that. The only problem I see is that Ben's main question asks for an opinion, but since he asks more things concerning the details, it seems very appropriate. –  ftonti Feb 12 '12 at 21:47
Wait a minute: the isomorphism is non-unique? I thought there could be at most one isomorphism between two real closed fields. Equivalently, that the only automorphism on a real closed field is the identity morphism. –  Todd Trimble Feb 12 '12 at 22:06
@Todd: No. Although the field of reals has only the trivial automorphism, other real-closed fields can have lots of automorphisms. In fact, a classical result of Ehrenfeucht and Mostowski says that any first-order theory $T$ with infinite models also has models with lots of automorphisms. Specifically, any group that has a faithful action on a linear ordering also has a faithful action on some model of $T$. (By "faithful" I mean that the only group element that acts as the identity permutation is the identity element.) –  Andreas Blass Feb 12 '12 at 22:47
@Mike: An automorphism of a real-closed field necessarily preserves the order, since the non-negative elements are exactly the squares. –  Andreas Blass Feb 12 '12 at 23:21

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