On Cantor's important proofs

Abstract

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.