A Generalization of the Cantor-Dedekind Continuum with Nilpotent Infinitesimals

Abstract: 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.