Computations with Grossone-Based Infinities

In this paper, a recent computational methodology is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses as a computational device the Infinity Computer (patented in USA and EU) working numerically with infinite and infinitesimal numbers that can be written in a positional system with an infinite radix. On a number of examples dealing mainly with infinite sets and Turing machines with different infinite tapes it is shown that it becomes possible to execute a fine analysis of these mathematical objects. The accuracy of the obtained results is continuously compared with results obtained by traditional tools used to work with mathematical objects involving infinity.

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