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.
I do not understand what the bounty on this question is for, as it seems to me that the other answers were already rather devastating. Here is a semi-reasoned technical answer.
According to G. Lolli (the paper you cite) "Sergeyev is wary of the axiomatic method because he thinks that by adopting it we would be tied to the expressive power of a language in the description of mathematical objects and concepts." Serious mathematics requires serious adherence to the generally accepted standards of mathematics. Perhaps prof. Sergeyev thinks that he can surpass the limitations of formalization by taking a non-standard route to mathematics, but I would rather suspect that route will take him backwards in time and much closer to (a bad kind of) philosophy than most mathematicians would feel comfortable with.
Regarding the formalization by G. Lolli, I see no difference between what is done in the paper and non-standard arithmetic. A grossone $G$ is axiomatized by the infinitely many axioms $0 < G$, $1 < G$, $2 < G$, ... which is exactly how one can get non-standard arithmetic going. The paper does not even mention non-standard arithmetic. This is what you get for publishing logic papers in applied math journals.
So, it looks to me that grossones are a moving target with unclear and confused mathematical content, until one actually pins them down with a precise mathematical definition, only to find out they are not new at all.
Update: it was pointed out that none of the answers has commented on the computational part of the grossone theory. I had a look at three papers, found on the infinity computer web site:
The recommended paper to start with is Sergeyev Ya.D. (2010) Lagrange Lecture: Methodology of numerical computations with infinities and infinitesimals, Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino, 68(2), 95–113. It has a lot of informal descriptions and philosophy, some illustrative examples, but nothing that would actually describe a revolutionary new way of computing. Rather, it looks like ideas that could possibly lead to re-invention of non-standard arithmetic.
Sergeyev Ya.D. (2016) The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area, Communications in Nonlinear Science and Numerical Simulation, 31(1–3):21–29. I tried this paper because the title promised that there would be a concrete result in it. There is, of course, but again the theory of computation underlying the method is not properly explained. There are examples and analogies which again hint at something like non-standard arithmetic.
Sergeyev Ya.D. (2015) Computations with grossone-based infinities, C.S. Calude, M.J. Dinneen (Eds.), Proc. of the 14th International Conference “Unconventional Computation and Natural Computation”, Lecture Notes in Computer Science, vol. 9252, Springer, 89-106. A pattern starts to emerge. Every paper contains a very long introduction to the philosophy and ideas about grossones, supported by illustrative examples, but there is no clear explanation of what is going on.
All three papers present an equational system for grossones, i.e., things like associativity, commutativity, and other equations one would expect. A smart person can use these to simplify expressions and thereby "compute" results. But a computational model requires a description of a general procedure for performing computations, whatever it is. Is there a method for normalizing expressions involving grossones? Or perhaps an abstract machine one can run? Or something else?
I suppose the infinity computer is hiding in the patent. We shall never know. And I have now wasted more time on this than 50 points of bounty are worth. If someone can point me at an actual description of a computational model (whether it be "axiomatic" or not) which is not composed of a series of analogies and good ideas, I might take another look.
Best Answer
Let us work in NBG set theory with global choice. There is, up to non unique isomorphism, a unique real-closed field that is $\kappa$-saturated for all infinite cardinals $\kappa$. Let's denote it by $\mathbf{K}$. For real-closed fields, being $\kappa$-saturated is the same as having no cut of size $<\kappa$, by which I mean an ordered pair $(L,R)$ of subsets $L,R$ of size $<\kappa$ such that $L<R$ and that there is no element between $L$ and $R$.
Since both Hahn series, Levi-Civita series and Puiseux series constructions you mention give a real-closed field as a result, and since being real-closed is preserved by increasing unions, your question reduces to the following one:
Which of those three processes ends up filling all set-sized cuts?
Let us start with the Levi-Civita one. Writing $\alpha$ for the ordinal step of the iteration process, one can see that the $\alpha$-th field $\mathbb{F}_{\alpha}$ in the construction is contained in the Hahn series field $\mathbb{L}_{<\alpha}$ of series with real coefficients and monomials in the group $\mathfrak{L}_{<\alpha}$. This is the group of formal products $\prod \limits_{\gamma<\alpha} {x_{\gamma}}^{r_{\gamma}}$ where $(r_{\gamma})_{\gamma<\alpha}$ is a family of real numbers with finite support, i.e. which is zero outside of a finite subset of $\alpha$. (Or depending on your conventions for the construction, you could replace $\alpha$ with $\alpha+1$). The group $\mathfrak{L}_{<\alpha}$ is anti-lexicographically ordered, defining a non trivial product $\prod \limits_{\gamma<\alpha} {x_{\gamma}}^{r_{\gamma}}$ to be larger than $1$ if the last non-zero exponent $r_{\gamma_0}$ is strictly negative. To see that this is the case, show that the Levi-Civita field with coefficients in $\mathbb{L}_{<\alpha}$ embeds into $\mathbb{L}_{<\alpha+1}$, and that $\bigcup \limits_{\beta<\alpha} \mathbb{L}_{<\beta}$ is naturally contained in $ \mathbb{L}_{<\alpha}$ for all non-zero limit $\alpha$.
It follows that the union $\mathbb{F}_{\infty}$ of all $\mathbb{F}_{\alpha}$'s is contained in a field $\mathbb{L}$, which is the same as $\mathbb{L}_{<\alpha}$ except the $\gamma$'s can now be arbitrary ordinals. This field is real-closed, but not at all $\kappa$-saturated for all infinite cardinals $\kappa$. For instance, there is a countable cut $(\mathbb{N},\{...,{x_0}^{-\frac{1}{4}},{x_0}^{-\frac{1}{2}},{x_0}^{-1}\})$ in $\mathbb{L}$.
So $\mathbb{F}_{\infty}$ is not isomorphic to $\mathbf{K}$.
The process with Puiseux series yields a smaller field than $\mathbb{F}_{\infty}$ which contains the previous cut, hence it is also not isomorphic to $\mathbf{K}$.
Now let's turn to the Hahn series construction. If you start with the value group $\mathbb{R}$ (i.e. group of monomials $x^{\mathbb{R}}$) and iterate by extending coefficients, then you'll still end up in $\mathbb{L}$ and contain the same cut as before. So I assume we are now taking the underlying ordered group of the stage $\alpha$ field $\mathbb{H}_{\alpha}$ as the value group for the next ordered field $\mathbb{H}_{\alpha+1}$. At limit stages, one can either take the union of the previous monomial groups as the new monomial group, or just take unions, without chancing the end result. In any case, this construction, starting with $\mathbb{H}_0=\mathbb{R}$, can be done within the field $\mathbf{No}$ of surreal numbers, where $\mathbb{H}_{\alpha+1}$ will simply be the class $\mathbb{R}[[\omega^{\mathbb{H}_{\alpha}}]]$ of surreal numbers whose Conway normal form has exponents in $\mathbb{H}_{\alpha}$. One can see that the union $\mathbb{H}$ of all such fields still contains countable cuts. For instance $(\{\omega,{\omega}^{\omega},{\omega}^{{\omega}^{\omega}},...\},\varnothing)$. In fact in $\mathbf{No}$, there are many monomials of the form $\mathfrak{m}=\omega^{a_1\pm\omega^{a_2\pm\omega^{...}}}$. If all $a_i$'s are in the field $\mathbb{H}_{\infty}$, then the simplest such monomials generate a set-sized cut over $\mathbb{H}_{\infty}$, since no iteration of the Hahn series construction gives such transfinite "$\omega$ expansions". See Denis Lemire's PhD thesis for more information.