On the theory of grossone

Abstract

A trivial formalization is given for the informal reasonings of a series of papers by Ya. D. Sergeyev on a positional numeral system with an infinitely large base, grossone; the system which is groundlessly opposed by its originator to the classical nonstandard analysis.Keywords: nonstandard analysis, infinitesimal analysis, positional numeral system In recent years Sergeyev has published a series of papers [1][2][3][4][5] in which a positional numeral system is advanced related to the notion of grossone. ‡) Sergeyev opposes his system to nonstandard analysis and regards the former as resting on different mathematical, philosophical, etc. doctrines. The aim of the present note is to properly position the papers by Sergeyev on developing numeral systems. It turns out that a model of Sergeyev's system is provided by the initial segment {1, 2, . . . , ν!} of the nonstandard natural scale up to the factorial ν! of an arbitrary actual infinitely large natural ν. Such a factorial serves as a model of Sergeyev's grossone, thus demonstrating the place occupying by the numeral system he proposed.As the main source we have chosen [4], the latest available paper by Sergeyev, which contains a detailed description of his basic ideas.

Independence of the grossone-based infinity methodology from non-standard analysis and comments upon logical fallacies in some texts asserting the opposite

“…However, in the paper [16] published in Foundations of Science 2 and in the papers [17,22] published in two journals printed by the Institute where the authors of [17,22] work, there are numerous attacks on the ①-based methodology and its author. The paper [17] announces 'a trivial formalization of the theory of grossone' using nonstandard analysis. The paper [22] does not have any mathematical substance (not a single formula) and consists of insults and accusations starting from the title.…”

“…The paper [22] does not have any mathematical substance (not a single formula) and consists of insults and accusations starting from the title. The paper [16] in all seriousness attacks a playful note [49] regarding several ways of counting Olympic medals won by different countries and appeals to [17,22] trying again to show that the ①-based methodology can be reduced to non-standard analysis.…”

“…In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts [16,17,22] asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics.…”

A trivial formalization of the theory of grossone

“…In addition, from the illustrations of [4] it is clear that the positional system proposed admit syntactically different numerals with coincident values: for instance, 0O 1 0 ≡ 0O 1 1 , 1O 1 0 ≡ 1O 1 0O 1 0 . (The notion of the value of a term and the equivalence relation ≡ are clarified in [11].) At the same time, [4] misses not only the corresponding stipulations (easy to guess though) but also any attempts of justifying the unambiguity of the positional system, even under implicit stipulations.…”

“…To provide some justification, we briefly described in [11] one of the possible approaches to formalization of the notion of numeral as well as the corresponding algorithmic procedures.…”