由康托实数完备化求无穷小的两种方法
Two ways of obtaining infinitesimals by refining Cantor's completion of the reals
摘要：
康托著名的关于有理数柯西序列的实连续统的构造是通过施加一个合适的等价关系来进行的。更一般地说，度量空间的完备化是从空间的点序列之间类似的等价关系开始的。康托的关系柯西序列的实数可以细化，以产生一个柯西完整和无穷小丰富的连续？我们提出两种可能性：一个导致可逆无穷小和超实数;另一个导致幂零无穷小（例如h非零无穷小使得h^2=0）和费马实数。我们的主题之一是形式力量和直觉之间的权衡。
英文摘要：
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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