In article <firstname.lastname@example.org>, "Peter Webb" <webbfamily@DIESPAMDIEoptusnet.com.au> wrote:
> "Tim Little" <email@example.com> wrote in message > news:firstname.lastname@example.org... > > On 2010-06-18, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: > >> I made no premise. > > > > Sure you did: you assumed that no list of computable numbers can > > exist. You also assumed an incorrect definition of "computable". > > > > No, I assumed that a list of all computable numbers can exist. Then I gave a > simple algorithm which forms a computable number which is not on the list. I > therefore proved that no list of all computable numbers can exist. > > It is *exactly* the same as Cantor's proof that the Reals cannot be listed. > > It is of interest because it is known that the computable numbers are > countable. Therefore the property "cannot be listed" is *not* the same as > the property "is uncountable". > > Cantor's diagonal proof does *not* show the Reals are uncountable; it just > proves the much weaker statement that "the Reals cannot be listed".
Given the axiom of choice, as in ZFC, any countable set must be, at least theoretically, listable, though such a listing need not be computable.
And if countable, for infinite sets, does not mean bijectable with N (or listable), what does it mean?