On Poincaré′s Remark and a Kind ofNonstandard Measure Defined on *(Rn)

Abstract:

      Here concerned is a certain kind of non-standard measure defined on the n-dimensional Euclidean space ^*(Rⁿ), which (with n = 1) can be used to show that any standard linear point-set or the usual ordered field R of real numbers is of measure zero. The proposition just mentioned is basically consistent with Poincaré's famous remark which renders a deep insight into the paradoxical structural nature of Cantor's continuum consisting precisely of all distinct real numbers.