Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this seems dubious but numerous articles have already appeared in refereed research journals. Thus, there are currently 23 such articles in mathscinet not to speak of numerous lectures in conferences.
In a comment accessible here Sergeyev asserts that "Levi-Civita numbers are built using a generic infinitesimal $\varepsilon$ ... whereas our numerical computations with finite quantities are concrete and not generic." Here apparently "finite" is a misprint and should be "infinite". How is this comment on the difference between Sergeyev's grossone one the one hand, and the Levi-Civita unit on the other, to be understood?
In a 2013 article, Sergeyev compares his grossone to Levi-Civita in the following terms in footnote 5: 5 At the first glance the numerals (7) can remind numbers from the Levi-Civita field (see ) that is a very interesting and important precedent of algebraic manipulations with infinities and infinitesimals. However, the two mathematical objects have several crucial differences. They have been introduced for different purposes by using two mathematical languages having different accuracies and on the basis of different methodological foundations. In fact, Levi-Civita does not discuss the distinction between numbers and numerals. His numbers have neither cardinal nor ordinal properties; they are build using a generic infinitesimal and only its rational powers are allowed; he uses symbol 1 in his construction; there is no any numeral system that would allow one to assign numerical values to these numbers; it is not explained how it would be possible to pass from d a generic infinitesimal h to a concrete one (see also the discussion above on the distinction between numbers and numerals). In no way the said above should be considered as a criticism with respect to results of Levi-Civita. The above discussion has been introduced in this text just to underline that we are in front of two different mathematical tools that should be used in different mathematical contexts. It would be interesting to have a specialist in numerical analysis comment on Sergeyev's use of the term "numerical" to explain the difference between his grossone and an infinite element of the Levi-Civita field.
Sergeyev claims that his grossone has the properties of both ordinal and cardinal numbers. Does he give a definition that would ensure such properties, or is this claim merely a declarative pronouncement?
Following the publication of an article by Sergeyev in EMS Surveys in Mathematical Sciences, the editors published the following clarification:
Statement of the editorial board
We deeply regret that this article appears in this issue of the EMS Surveys in Mathematical Sciences.
It was a serious mistake to accept it for publication. Owing to an unfortunate error, the entire processing of the paper, including the decision to accept it, took place without the editorial board being aware of what was happening. The editorial board unanimously dissociates itself from this decision. It is not representative of the very high level that we expect to see in our journal, which can be assessed from all other papers that we have published.
Both editors-in-chief have assumed responsibility for these mistakes and resigned from their position. Having said that, we add that this journal would not exist without their dedication and years of hard work, and we wish to register our thanks to them.
An interesting viewpoint of a computer scientist is developed here (as well as a related discussion of legal issues in the comments).