What is... A Grossone?
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.
In my paper about the Grossone, I point out that the logic of this formalism is identical (in my version) to using $1 + x + x^2 + ... + x^G$ as a FINITE SUM with $G$ a generic positive integer. One can then manipulate the series and look at the limiting behaviour in many cases. There is no need to invoke any new concepts about infinity. This point of view may be at variance with the interpretations of Yaroslav for his invention, but I suggest that this is what is happening here.
What I did in my paper on Sergeyev’s Grossone that has been mentioned in your discussion was to present an axiomatised theory of arithmetic in the language of Peano arithmetic augmented with a new constant for Grossone. I did it because many colleagues seemed to think that Sergeyev’s approach didn’t respect the standards of acceptable mathematical exposition. In my theory I showed that Sergeyev’s axioms are provable while I argued that it respected Sergeyev’s outlook expressed in his so called (by him) postulates. The main result is that the theory is a conservative extension of Peano arithmetic, that is that it proves the same sentences in the language without Grossone; hence Sergeyev theory, if my theory is faithful to his spirit, in consistent if Peano’s arithmetic is consistent.
Moreover this should show that Sergeyev’s methods, at least as to their arithmetical part, are something different form non standard analysis. To discuss non standard analysis is not easy, since one should be precise on what one means; there are various approaches, among whom Nelson’s Internal Set Theory IST is probably the smoothest. But any theory for non standard methods should be stronger than Sergeyev’s, since these methods do not seem to be a conservative extension of the classical ones; one needs either third order logic or strong assumptions on the existence of special ultrafilters. In Sergeyev’s theory there is no transfer principle, and so on.
There is another paper you might be interested in:
F. Montagna, G. Simi and A. Sorbi, Taking the Pirahã seriously, Communication in Nonlinear Science and Numerical Simulation, 21(1–3), April 2015, 52–69.
Here, the authors go deeper into the logical use of Grossone in relation to predicative arithmetic.
Andrej Bauer asked in the comments:
"Could you comment on how your result is related to results by Kreisel on non-standard arithmetic being conservative over PA? See for example Proposition 2.3 in jstor.org/stable/2274260 and the reference to Kreisel therein? (I cannot find a direct link to Kreisel, sorry.)"
Our results are rather similar, also in the proof, by compactness and adjustment of models. But while I think that my theory can be considered a fair rendition of Sergeyev’s outlook, it is doubtful that Kreisel’s *PA can be considered a faithful framework for non standard arithmetic. The authors of the paper mentioned in the comment, Henson, Kaufmann and Keisler after briefly recalling Kreisel’s result (indeed the first in this line) go on saying that most of non standard arithmetical results depend on the use of omega1-saturated models, and discuss possible strengthening of the theory to approximate the properties of such models.
Non standard mathematics is more ambitious than Sergeyev’s computational interest, and requires in consequence stronger logical principles. It seems to me that at present there is no consensus on a formalisation of these principles, but perhaps for Nelson’s set theory.
Gabriele Lolli