Grossone Innity Computing: Foundations

bers and numerals; infinite sets. This tutorial introduces a new methodology (different from non-standard analysis theories) allowing one to execute numerical (not symbolic) computa-tions with finite, infinite, and infinitesimal numbers (see surveys [1, 2] and a semi-popular introduction in [3]). These numerical computations can be exe-cuted on a new type of a computational device – the Infinity Computer (see patents [4]). The new approach is based on the principle ‘The part is less than the whole ’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The new methodology examines in detail the difference between mathemat-ical tools used to describe mathematical objects and the objects themselves. In particular, it is stressed that numerals we use to write down numbers (finite, infinite, and infinitesimal) are among our tools and, as a result, they strongly influence our capabilities in studying numbers. A variety of numeral systems in-cluding numerals used by Cantor and numerals introduced in [1–3] are compared from several points of view. The concept of accuracy of mathematical languages and its importance for a number of theoretical and practical issues regarding computations is discussed. Numerous examples and applications are given. Th