Infinitesimal Calculus

Abstract The controversy surrounding the infinitesimals, obstructed the development of the Infinitesimal Calculus. Postulating infinitesimals, only revealed Logic’s inapplicability to Mathematical Analysis, where claims must be proved, and unproven claims are ignored. Recently we have shown that when the Real Line is represented as the infinite dimensional space of all the Cauchy sequences of rational numbers, the hyper-reals are spanned by the constant hyper-reals, a family of infinitesimal hyper-reals, and the associated family of infinite hyper-reals. The infinitesimal hyper-reals are smaller than any real number, yet bigger than zero. The reciprocals of the infinitesimal hyper-reals are the infinite hyper-reals. They are greater than any real number, yet strictly smaller than infinity.