Cantor versus Cantor

TL;DR

This paper investigates extending Cantor's arguments on uncountability to the rationals, highlighting the need to verify certain conditions in Cantor's set theory to ensure its consistency.

Contribution

It demonstrates that extending Cantor's uncountability arguments to the rationals is generally possible unless specific restrictive conditions are met, emphasizing the importance of these conditions for consistency.

Findings

Extensions are possible under certain conditions

Conditions must be verified for consistency

Implications for Cantor's set theory

Abstract

This paper examines the possibilities of extending Cantor's two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain restrictive conditions are satisfied, both extensions are possible. It is therefore indispensable to prove that those conditions are in fact satisfied in Cantor's theory of transfinite sets. Otherwise that theory would be inconsistent.