The notion of "Unimaginable Numbers" in computational number theory

Abstract: Literature considers under the name \emph{unimaginable numbers} any positive integer going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathematical definition. This simply means that research in this topic must always consider shortened representations, usually involving \emph{recursion}, to even being able to describe such numbers.\par\medskip One of the most known methodologies to conceive such numbers is using \emph{hyperoperations}, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. The most important notations to represent such hyperoperations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work's introduction.\par\medskip Within this work we will give an axiomatic set for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorithmic nature of representations and the increased computation capabilities of computers give the perfect field to develop further the topic.\par\medskip After the introduction, we will give axioms and generalizations for the up-arrow notation; in the subsequent section we consider a representation via rooted trees of the \emph{hereditary base-$n$ notation} involved in Goodstein's theorem, which can be used efficiently to represent some defective unimaginable numbers, and in the last section we will analyse some methods to compare big numbers, proving specifically a theorem about approximation using scientific notation.