6/25    Does anyone actually have a direct proof as to why the Irrational
        Numbers are countable, (one that doesn't start with "assume they
        are countable.."  Thx.
        \_ No, it's not possible.  "Uncountable" means "can't be put into
           one-to-one correspondence with the natural numbers"; the only way
           to prove an impossibility like that is to assume the opposite and
           derive a contradiction.  Mathematicians who reject the idea of
           proof by contradiction also reject the idea of uncountable sets.
           See http://en.wikipedia.org/wiki/Constructivism_(mathematics
           \_ Interesting- i are these folk discredited? I ask because there
              are an awful lot of proofs (including the halting problem,
              not the version that is directly from cantor's diagonal tho)
              that always start out with "assume this is true..."
              \_ I really don't think you'll find a proof that irrational
                 numbers are countable.
        \_ they're not countable.