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

TL;DR

This work investigates whether Cantor's diagonal method, when used constructively, can generate uncountable sets. By formalizing the countable set $X$ of infinite binary sequences and its diagonally derived counterparts $Y$, it shows that iterative diagonal extensions produce only countable unions $X_\infty = \bigcup_n X_n$. Consequently, the method cannot exhaust the continuum $|\mathbb{B}^{\mathbb{N}}| = 2^{\aleph_0}$ and cannot construct an uncountable set, even under infinite hierarchical diagonalizations. The results emphasize a clear boundary between constructive enumerability and the full continuum, highlighting non-computability as a pervasive feature of $\mathbb{B}^{\mathbb{N}}$ and informing the interplay between constructive mathematics and axiomatic set theory such as $\mathrm{ZFC}$.

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.