Independence of the Grossone-Based Infinity Methodology from Non-standard Analysis and Comments upon Logical Fallacies in Some Texts Asserting the Opposite

Abstract

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ① (this symbol is called grossone). The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. 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 (Gutman et al. in Found Sci 22(3):539–555, 2017; Gutman and Kutateladze in Sib Math J 49(5):835–841, 2008; Kutateladze in J Appl Ind Math 5(1):73–75, 2011) 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.

Appendix

The ①-based methodology is one of the possible views on infinite and infinitesimal quantities and Mathematics, in general. It is added to other existing philosophies of Mathematics such as logicism, formalism, intuitionism, structuralism, etc. (their comprehensive analysis can be found, e.g., in Linnebo 2017; Lolli 2002; Shapiro 2001). Three postulates and an axiom that is added to axioms for real numbers form the methodological platform of the proposal. They are given below to make this paper self-contained. A special attention in the ①-based methodology is paid to the fact than numeral systems that are among our tools used to observe mathematical objects limit our capabilities of the observation. A detailed discussion on this methodological platform can be found in Sergeyev (2017).

Methodological Postulate 1

We postulate existence of infinite and infinitesimal objects but accept that human beings and machines are able to execute only a finite number of operations.

Methodological Postulate 2

We shall not tell what are the mathematical objects we deal with; we just shall construct more powerful tools that will allow us to improve our capacities to observe and to describe properties of mathematical objects.

Methodological Postulate 3

We adopt the principle ‘The part is less than the whole’ to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite).

The Infinite Unit Axiom The infinite unit of measure is introduced as the number of elements of the set, ${\mathbb {N}}$, of natural numbers. It is expressed by the numeral ① called grossone and has the following properties:

Infinity Any finite natural number n is less than grossone, i.e., $n <~\textcircled {1}$.

Identity The following relations link ① to identity elements 0 and 1

$$\begin{aligned} 0 \cdot \textcircled {1} = \textcircled {1} \cdot 0 = 0, \quad \textcircled {1}-\textcircled {1}= 0,\quad \frac{\textcircled {1}}{\textcircled {1}}=1, \quad \textcircled {1}^0=1, \quad 1^{\textcircled {1}}=1, \quad 0^{\textcircled {1}}=0. \end{aligned}$$

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Divisibility For any finite natural number n sets ${\mathbb {N}}_{k,n}, 1 \le k \le n,$ being the $n\hbox {th}$ parts of the set, ${\mathbb {N}}$, of natural numbers have the same number of elements indicated by the numeral $\frac{\textcircled {1}}{n}$ where

$$\begin{aligned} {\mathbb {N}}_{k,n} = \{k, k+n, k+2n, k+3n, \ldots \}, \quad 1 \le k \le n, \quad \bigcup _{k=1}^{n}{\mathbb {N}}_{k,n}={\mathbb {N}}. \end{aligned}$$

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