In article <firstname.lastname@example.org>, "Peter Webb" <webbfamily@DIESPAMDIEoptusnet.com.au> wrote:
> > > > Since by definition, "listability" = "countability", Cantor's proof of > > unlistability proves uncountability. > > Really? Where did you get that from? > > The computable Reals cannot be listed. > > Therefore according to you they are uncountable. > > But they aren't. > > Maybe your definition needs a little work?
Consider the set of computable numbers, S.
According to http://en.wikipedia.org/wiki/Countable_set, one definition of such a set, S, being countable is that there is a injective function from S to N, which is equivalent to there being a surjective function from N to S.
Since you object to there being any bijection from N to any superset of S, you must equally be rejecting any surjection from N to S and rejecting any injection from S to N, since from any such bijection such a surjection is easily derived.
So in what sense do you claim that the the set S of computable numbers is countable?
It is certainly not in any sense that I am aware of.