  For more than a century, Cantor's theory of transfinite numbers has played a
pivotal role in set theory, with ramifications that extend to many areas of
mathematics. This article extends earlier findings with a fresh look at the
critical facts of Cantor's theory: i) Cantor's widely renowned Diagonalization
Argument (CDA) is fully refuted by a set of counter-examples that expose the
fallacy of this proof. ii) The logical inconsistencies of CDA are revisited,
exposing the short-comings of CDA's implementation of the method of proof by
contradiction. iii) The denumerability of the power set of the set of the
natural numbers, P(N), is substantiated by a proof that takes full account of
all the infinite subsets of N. Such a result confirms the denumerability of the
set of the real numbers, R, and with it the countable nature of the continuum.
iv) Given that the denumerable character of (probably) all infinite sets makes
their comparison in terms of one-to-one correspondences a rather pointless
exercise, a new concept of relative cardinality is introduced which facilitates
a quantitative evaluation of their different magnitudes.
