HOMOMORPHISMS BETWEEN RINGS WITH INFINITESIMALS AND INFINITESIMAL COMPARISONS

EMANUELE BOTTAZZI

【摘要】我们研究了Reeder的一个论证，该论证表明P.G的实数环延拓•R中的幂零无穷小要小于A.R.对实数的非标准延拓*R中的任何无穷小超实数。我们的方法包括研究两个规范的保序同态，它们分别在•R和*R中取值，其值域是“非非标准”分析框架下实数的Henle扩展。在Henle环中存在一个非零元素，它在•R中被映射到0，而在∗R中它被视为一个非零的无穷小，这表明在*中存在的一些无穷小要比在•R中存在的无穷小小。我们认为，与Reeder的结论明显矛盾的原因仅仅是在•R中存在幂零元。

ABSTRACT We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano’s ring extension of the real numbers•R are smaller than any infinitesimal hyperreal number of Abraham Robinson’s nonstandard extension of the real numbers*R. Our approach consists in the study of two canonical order-preserving ho-momorphisms taking values in•R and *R, respectively, and whose domain is Henle’sex tension of the real numbers in the framework of “non-nonstandard” analysis. The ex-istence of a nonzero element in Henle’s ring that is mapped to 0 in•R while it is seen as a nonzero infinitesimal in*R suggests that some infinitesimals in *R are smaller than the infinitesimals in•R. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in•R.

http://dx.doi.org/10.30970/ms.52.1.3-9​dx.doi.orgHomomorphisms Between Rings with Infinitesimals and Infinitesimal Comparisons​arxiv.org