  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.
