Infinitesimals via Cauchy sequences: Refining the classical equivalence

TL;DR

This paper refines the classical equivalence of Cauchy sequences to develop an infinitesimal-enriched number system, formalizing the concept of infinitesimals and revisiting historical perspectives on their legitimacy.

Contribution

It introduces a novel refinement of Cauchy sequence equivalence that formalizes infinitesimals, connecting historical and modern mathematical frameworks.

Findings

Constructs a system with infinitesimals based on refined Cauchy sequence equivalence.

Reveals a historical construction by Giuseppe Peano from 1910.

Provides a formal foundation for infinitesimals aligned with Cauchy's intuition.

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.