A New Perspective on Diagonalization and Computability

This article reexamines the classical diagonal argument underlying the claim of the existence of non-computable functions, based on binary encodings of functions N → {0, 1}. We clarify why the diagonal construction does not yield a new non-computable function in the finite case. The argument is then reconsidered within the recently introduced grossone-based computational paradigm, which allows numerical computations with different infinite and infinitesimal quantities. From this perspective, the function constructed by Turing can be interpreted not as non-computable, but as belonging to a class of binary functions that remain inaccessible to classical approaches while becoming observable and computable within the grossone-based framework.