The Uncountability of the Unit Interval

TL;DR

This paper compiles an extensive collection of proofs demonstrating the uncountability of the real numbers in the unit interval, showcasing various methods from classical to modern mathematical techniques.

Contribution

It provides the most comprehensive collection of proofs for the uncountability of the unit interval, including classical, measure-theoretic, game-theoretic, algebraic, and analytical approaches.

Findings

Multiple proofs illustrating uncountability

Diverse mathematical techniques showcased

Enhanced understanding of the uncountability concept

Abstract

For any particularly interesting theorem one proof is never enough. Instead, the first proof sets the challenge to find a more elegant method that illuminates subtle features of the math, is simpler to understand, or even avoids using controversial subjects. In this paper we consider a subject that has attracted the attention of many mathematicians: the uncountability of the real numbers in the unit interval. We present the most exhaustive collection of proofs of this fact that we know. These range from Cantor's three published proofs, including his famous diagonalization method, to more recent proofs that employ measure theory, game theory, algebra, and analysis.