具有幂零无穷小量的Cantor-Dedekind连续统的推广
A Generalization of the Cantor-Dedekind Continuum with Nilpotent Infinitesimals
摘要：
我们介绍了一个推广的康托-戴德金连续显式无穷小。这些无穷小被用作遵守与广义连续统的其他元素相同的基本规则的数，这与莱布尼茨最初的直觉一致，但有一个重要的区别：它们的乘积为零，正如荷兰神学家伯纳德·尼乌文蒂特（Bernard Nieuwentijt）所坚持的那样，反对莱布尼茨的观点。起点是阴影的概念，并从它定义不可分割性（中心概念）和单子。点的单子具有全局-局部性质，因为尽管它们是无限维的实仿射空间，具有与整个广义连续统相同的基数，但它们是长度为0的闭区间。单子和阴影（最初定义为点），然后扩展到任何子集的新的连续统，他们的研究揭示了有趣的结果，保存在集合论和拓扑学领域。所有这些概念都不依赖于新的连续统中极限的定义，但利用它们我们得到了微分学的基本结果。最后，我们给出了两个例子，说明如何的整体局部性质的单子的实数可以应用到某些奇异性的微分处理。
英文摘要：
We introduce a generalization of the Cantor-Dedekind continuum with explicit infinitesimals. These infinitesimals are used as numbers obeying the same basic rules as the other elements of the generalized continuum, in accordance with Leibniz's original intuition, but with an important difference: their product is null, as the Dutch theologian Bernard Nieuwentijt sustained, against Leibniz's opinion. The starting-point is the concept of shadow, and from it we define indiscernibility (the central concept) and monad. Monads of points have a global-local nature, because in spite of being infinite-dimensional real affine spaces with the same cardinal as the whole generalized continuum, they are closed intervals with length 0. Monads and shadows (initially defined for points) are then extended to any subset of the new continuum, and their study reveals interesting results of preservation in the areas of set theory and topology. All these concepts do not depend on a definition of limit in the new continuum; yet using them we obtain the basic results of the differential calculus. Finally, we give two examples illustrating how the global-local nature of the monad of a real number can be applied to the differential treatment of certain singularities.
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