  It is shown that the pillars of transfinite set theory, namely the
uncountability proofs, do not hold. (1) Cantor's first proof of the
uncountability of the set of all real numbers does not apply to the set of
irrational numbers alone, and, therefore, as it stands, supplies no distinction
between the uncountable set of irrational numbers and the countable set of
rational numbers. (2) As Cantor's second uncount-ability proof, his famous
second diagonalization method, is an impossibility proof, a simple
counter-example suffices to prove its failure. (3) The contradiction of any
bijection between a set and its power set is a consequence of the impredicative
definition involved. (4) In an appendix it is shown, by a less important proof
of Cantor, how transfinite set theory can veil simple structures.
