A constructive sequence algebra for the calculus of indications

Gangle, Rocco; Caterina, Gianluca; Tohmé, Fernando

doi:10.1007/s00500-020-05121-1

Cited by 10 publications

(6 citation statements)

References 17 publications

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“…Before we start a technical consideration, let us mention that a number of papers studying consistency of the new methodology and its connections to the historical panorama of ideas dealing with infinities and infinitesimals have been published (see [21,37,38,41,43,54,59,64]). In particular, in [59], it is stressed that it is not related to non-standard analysis.…”

Section: A New Way Of Countingmentioning

confidence: 99%

“…It is also necessary to recall that numbers that are larger than 1 are not natural, they are extended natural numbers. For instance, 1 + 2 is even but not natural, it is the extended natural, see (21). Thus, the last even natural number is 1 .…”

Section: Galileo's Paradoxmentioning

confidence: 99%

“…The introduction of the 1 -based numeral system allows us to write down the sets of natural and extended natural numbers in the form (21). By definition, the number of elements of N is equal to 1 .…”

Section: Set-multiplication Paradoxmentioning

confidence: 99%

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Some Paradoxes of Infinity Revisited

Sergeyev

2022

Mediterr. J. Math.

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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.

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Section: A New Way Of Countingmentioning

confidence: 99%

Section: Galileo's Paradoxmentioning

confidence: 99%

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Some Paradoxes of Infinity Revisited

Sergeyev

2022

Mediterr. J. Math.

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“…The dedicated web page [19], developed at the University of East Anglia, UK, contains, among other things, a comprehensive teaching manual. It should be also emphasized that numerous papers studying consistency of the new methodology and its connections to the historical panorama of ideas dealing with infinities and infinitesimals have been published (see [17,25,27,29,35,39,44]). In particular, it is stressed in [39] that this methodology is not related to non-standard analysis.…”

Section: A Theoretical Backgroundmentioning

confidence: 99%

Lower and Upper Estimates of the Quantity of Algebraic Numbers

Sergeyev

2022

Mediterr. J. Math.

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It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using $$\textcircled {1}$$ 1 -based infinite numbers is applied to measure the set A (where the number $$\textcircled {1}$$ 1 is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable or it has the cardinality of the continuum, the $$\textcircled {1}$$ 1 -based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in $$\textcircled {1}$$ 1 -based numbers.

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“…The first group discusses optimization problems and algorithms including local, global, and multi-criteria optimization (see Franchini et al 2020;Ž ilinskas and Litvinas 2020;Nesterov 2020;Posypkin et al 2020;Shao et al 2020;De Leone et al 2020;Capuano et al 2020;Lančinskas et al 2020;Sergeyev et al 2020;Crisci et al 2020;Cavallaro et al 2020;Astorino and Fuduli 2020;Candelieri et al 2020). The second group of papers (see D'Alotto 2020; Pepelyshev and Zhigljavsky 2020;Falcone et al 2020;Amodio et al 2020;Gangle et al 2020;De Leone et al 2020;Astorino and Fuduli 2020) deals with problems and algorithms using the already mentioned recent computational framework allowing one to work with different infinities and infinitesimals numerically. It can be seen from these articles that, in addition to its already known applications, the new computational methodology can be successfully used in such fields as optimization, ordinary differential equations, game theory, classification, logic, and fractals.…”

mentioning

confidence: 99%