Constructive Limits of Cantor's Diagonal Method: Countability, Enumerability, and the Impossibility of Exhausting the Continuum

Abstract

Cantor's diagonal method is traditionally used to prove the uncountability of the set of all infinite binary sequences. This paper analyzes the expressive limits of this method. It is shown that under any constructive application -- including generalizations with computable permutations and infinite hierarchies of diagonal extensions -- the resulting set remains countable. Thus, the method demonstrates the incompleteness of countable coverage but is unable to generate an uncountable set. This highlights its limitations as a constructive tool and reveals the boundary between constructive enumerability and the completeness of the continuum.