  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.
