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

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

2011-09-16

arxiv.org/abs/1109.3553v1

Mathematics

Logic

31 pages, 2 figures

Scientific paper

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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