  Whatever other beliefs there may remain for considering Cantor's diagonal
argument as mathematically legitimate, there are three that, prima facie, lend
it an illusory legitimacy; they need to be explicitly discounted appropriately.
  The first, Cantor's diagonal argument defines a non-countable Dedekind real
number; the second, Goedel uses the argument to define a formally undecidable,
but interpretively true, proposition; and the third, Turing uses the argument
to define an uncomputable Dedekind real number.
