A severe inconsistency of transfinite set theory
  Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well ordered. (3) In a finite set of real numbers the maximum below a given limit can always be determined. (4) Any two different real numbers are separated by at least one rational number. These theorems are applied to map the irrational numbers into the rational numbers, showing that the set of all irrational numbers is countable. 
W. Mueckenheim