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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 http://mathoverflow.net/questions/72691/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 wasn't sure whether to post this on mathoverflow or math.SE, since it's a question about current research, but I'm not competent as a research-level mathematician. I originally posted it there: http://mathoverflow.net/questions/88292/non-zfc-set-theory-and-nonuniqueness-of-the-hyperreals-problem-solved . Feedback there indicated I should probably move it here, so I did.

[EDIT] The following comments by François G. Dorais on mathoverflow may be helpful:

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.

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.

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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There is no additional consistency issue with NBG, it is a conservative extension of ZF. –  André Nicolas Feb 12 '12 at 19:07
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I don't understand why one would find non-uniqueness objectionable, so it's hard to say whether that would refute the objection. –  Hurkyl May 12 '12 at 13:14
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Also, as pointed out on math overflow, the usual construction is unique up to isomorphism if you include CH. I'd answer the question except I don't know for certain that $\mathbb{Z}^*$ would sit inside $\mathbf{Oz}$. –  Mark S. Jan 11 '13 at 17:23

1 Answer 1

The usefulness of the hyperreals $\mathbb{R}^*$ 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 such 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, may of which rely on the simplest model of the hyperreals, namely $\mathbb R^{\mathbb N}$ modulo an ultrafilter on $\mathbb N$. Ultimately the relationship to other superfields of $\mathbb{R}$ has little bearing on whether "Robinson's program has been completed successfully" as you put it.

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