  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.
