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

Paolo Giordano; Mikhail G. Katz

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arXiv · 1109.3553

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

Paolo Giordano·Mikhail G. Katz

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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.

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BibTeX RIS

Paolo Giordano, Mikhail G. Katz. 2011-09-16. Two ways of obtaining infinitesimals by refining Cantor's completion of the reals. https://arxiv.org/abs/1109.3553

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