Cantor’s Countability Concept Contradicted

Cantor’s famous proof of the uncountability of real numbers is shown to apply to the set of natural numbers as well. Independently it is proved that the uncountability of the real numbers implies the uncountability of the rational numbers too. Finally it is shown that Cantor’s second diagonalization method is inapplicable at all because it lacks the diagonal. Hence, the conclusion that the cardinal number C of the continuum be larger than aleph0 is invalid. As a consequence, there remains no evidence for the existence of different infinities denoted by so−called transfinite cardinal numbers. The continuum hypothesis is not only undecidable but meaningless.