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