Infinitesimals via Cauchy sequences: Refining the classical equivalence-Reference-Cited by-同舟云学术

Infinitesimals via Cauchy sequences: Refining the classical equivalence

Published:2021-01-01 Issue:1 Volume:19 Page:477-482
ISSN:2391-5455
Container-title:Open Mathematics
language:en
Short-container-title:

Author:

Bottazzi Emanuele¹,Katz Mikhail G.²

Affiliation:

1. Department of Mathematics F. Casorati, University of Pavia, Via Ferrata, 5 , 27100 , Pavia , Italy

2. Department of Mathematics, Bar Ilan University , Ramat Gan , 5290002 , Israel

Abstract

Abstract A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system. Such an approach can be seen as formalizing Cauchy’s sentiment that a null sequence “becomes” an infinitesimal. We signal a little-noticed construction of a system with infinitesimals in a 1910 publication by Giuseppe Peano, reversing his earlier endorsement of Cantor’s belittling of infinitesimals.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/math-2021-0048/pdf

Reference34 articles.

1. A. Robinson, Non-Standard Analysis, North-Holland Publishing, Amsterdam, 1966.

2. V. Kanovei and M. Reeken, Nonstandard Analysis, Axiomatically, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.

3. K. Hrbacek and M. Katz, Infinitesimal analysis without the Axiom of Choice, Ann. Pure Appl. Logic 172 (2021), no. 6, 102959, https://doi.org/10.1016/j.apal.2021.102959, https://arxiv.org/abs/2009.04980.

4. E. Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc. 64 (1948), 45–99.

5. W. Luxemburg, Nonstandard Analysis, Lectures on A. Robinson’s Theory of Infinitesimals and Infinitely Large Numbers, Mathematics Department, California Institute of Technology, Pasadena, second corrected ed., 1964.