%0 Journal Article
%T Two ways of obtaining infinitesimals by refining Cantor's completion of the reals
%A Paolo Giordano
%A Mikhail G. Katz
%J Mathematics
%D 2011
%I arXiv
%X 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.
%U http://arxiv.org/abs/1109.3553v1