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
Best Answer
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]