Some Paradoxes of Infinity Revisited

Abstract

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirahã, working with only three numerals (one, two, many) can help us to change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light.

Status of the Current Scientific Knowledge on Pirahã

“…The research in these articles is generally related to the question if the creation of numbers, an innate human ability, or a cultural achievement, is gained by the accumulation of knowledge, and if it is likely to be developed by language users of Pirahã in the next generations. For example, Sergeyev (2022) studied the Pirahã and Munduruku numerical systems and analyzed the idea/paradox of infinity and infinitesimals. According to the author, the numerical systems of these indigenous ethnic groups led to a new point of view on the idea of infinity.…”

“…Another significant issue in the study of Pirahã is related to the absence of ordinal and cardinal numbers in their language (Gordon 2004). The language lacks words that denote numbers and, as stated by Sergeyev (2022) and Everett (2012), it works with a few concepts -one, two or many -, to refer to small or more significant amounts (Everett 2012). "For Pirahã, all quantities larger than two are just 'many' and such operations as 2 + 2 and 2 + 1 give the same result, i.e., 'many'" (Sergeyev 2022, 7).…”

Lower and Upper Estimates of the Quantity of Algebraic Numbers

“…First of all, we mention numerous applications in local, global, and multicriteria optimization and classification (see, e.g., [4,9,10,13] and references given therein). Then, we can indicate game theory (see, e.g., [12,16]), probability theory (see, e.g., [7,[31][32][33]), fractals (see, e.g., [3,6,37]), infinite series (see [38,41,45]), Turing machines, cellular automata, and ordering (see, e.g., [11,33,36,42]), numerical differentiation and numerical solution of ordinary differential equations (see, e.g., [1,14,15,22] and references given therein), etc.…”