Infinite computations and the generic finite

摘要

This paper introduces the concept of a generic finite set and points out that a consistent and significant interpretation of the grossone, ①① notation of Sergeyev is that ① takes the role of a generic natural number. This means that ① 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. In this view,N={1,2,3,…,①-2,①-1,①}is not an infinite set, it is a symbolic structure representing a generic finite set. We further consider the concept of infinity in categories. An object A in a given category C is infinite relative to that category if and only if there is a injection J:A⟶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 (finite) while its underlying set (if it has one) is infinite. The computational methodology due to Sergeyev for executing numerical calculations with infinities and infinitesimals is considered from this categorical point of view.