Two ways of obtaining infinitesimals by refining Cantor's completion of the reals

TL;DR

This paper explores two methods of refining Cantor's completion of the reals to create a continuum enriched with infinitesimals, leading to hyperreals and Fermat reals, balancing formalism and intuition.

Contribution

It introduces two novel approaches to refine the completion process, producing different types of infinitesimal-enriched continua, expanding the foundational understanding of real number constructions.

Findings

One method yields invertible infinitesimals and hyperreals.

The other produces nilpotent infinitesimals and Fermat reals.

Both approaches offer new perspectives on the structure of the continuum.

Abstract

Cantor's famous construction of the real continuum in terms of Cauchy sequences of rationals proceeds by imposing a suitable equivalence relation. More generally, the completion of a metric space starts from an analogous equivalence relation among sequences of points of the space. Can Cantor's relation among Cauchy sequences of reals be refined so as to produce a Cauchy complete and infinitesimal-enriched continuum? We present two possibilities: one leads to invertible infinitesimals and the hyperreals; the other to nilpotent infinitesimals (e.g. h nonzero infinitesimal such that h^2=0) and Fermat reals. One of our themes is the trade-off between formal power and intuition.