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.
Highlights
A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system
Our approach can be seen as a formalization of Cauchy’s sentiment that a null sequence “becomes” an infinitesimal, while a sequence tending to infinity becomes an infinite number; see, e.g., Bair et al [21]
A perspective on hyperreal numbers via Cauchy sequences has value as an educational tool according to the post [22]
Summary
Robinson developed his framework for analysis with infinitesimals in his 1966 book [1]. The second approach has been carried out, for example, by James Henle in his non-nonstandard analysis [6] (based on Schmieden-Laugwitz [7]; see [8]), as well as by Terry Tao in his “cheap nonstandard analysis” [9]. These frameworks are based upon the identification of real sequences whenever they are eventually equal.. A precursor of this approach is found in the work of Giuseppe Peano In his 1910 paper [10] (see [11]) he introduced the notion of the end Commenting on Peano’s 1910 construction, Fisher notes that here Peano contradicts his contention of 1892, following Cantor, that constant infinitesimals are impossible. Peano’s 1910 article seems to have been overlooked by Freguglia who claims “to put Peano’s opinion about the unacceptability of the actual infinitesimal notion into evidence” [13, p. 145]
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