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. |