Beyond Knuth's notation for unimaginable numbers within computational number theory

Literature considers under the name "unimaginable numbers" any positive integer going beyond physical application. One of most known methodologies to conceive such numbers is using hyper-operations, that a sequence binary functions dened recursively starting from usual chain: addition - multiplication exponentiation. The important notations represent hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work's introduction. Within work we will give an axiomatic setup for topic, then try nd on one hand other ways unimaginable numbers, well applications computer science, where algorithmic nature representations increased computation capabilities computers perfect eld develop further exploring some possibilities effectively operate with big numbers. In particular, axioms generalizations up-arrow notation and, considering representation via rooted trees hereditary base-n notation, determine cases effective bound related "Goodstein sequences" Knuths notation. Finally, also analyze methods compare proving specically theorem about approximation scientic hyperoperation bounds Steinhaus-Moser