Infinite Computations and the Generic Finite

Abstract

This paper introduces the concept of a generic finite set, and points out that a consistent and significant interpretation of the grossone notation of Yarolslav D. Sergeyev is that of a generic natural number. This means that the grossone is not itself a natural number, yet it can be treated as one and used in the generic expression of finite sets and finite formulas, giving a new power to algebra and algorithms that embody this usage. We further consider the concept of infinity in categories. An object A in a given category C is said to be infinite relative to that category if and only if there is a injection from A to A in C that is not a surjection. In the category of sets this recovers the usual notion of infinity. In other categories, an object may be non-infinite (whence finite) while its underlying set (if it has one) is infinite. The computational methodology due to Yarolslav D. Sergeyev for executing numerical calculations with infinities and infinitesimals is considered from this categorical point of view.