  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.
