2011/09/16 by Paolo Robuffo Giordano, Giordano, Paolo, Mikhail G. Katz +2 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Mathematical and Theoretical Analysis #math.CA #math.HO #math.LO
paper · pdf · doi:10.48550/arxiv.1109.3553
31 pages, 2 figures
arxiv created 2011/09/16 · openalex publication_date 2011/09/16 · arxiv updated 2015/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 h2=0) and Fermat reals. One of our themes is the trade-off between formal power and intuition.