> In article <firstname.lastname@example.org>, > Shmuel (Seymour J.) Metz <email@example.com> wrote: > > > In <firstname.lastname@example.org>, on 11/02/2012 > > at 05:48 AM, "Peter Webb" <webbfamilyDIEspamDie@optusnet.com.au> > > said: > > > > > The proof moves from the assertion that there can be no list of > > > Real numbers to the assertion they are not countable. These are > > > not quite the same thing. There are countable sets (eg computable > > > numbers) which cannot be explicitly listed. > > > > You are confusing constructability with existence. Cantor's results > > are not about constructability. > > > > > Given that the inability to list a set even in principle is not > > > proof that it is uncountable, how does the proof go from 'there > > > is no list' to 'the set is uncountable'. > > > > Because the definition of uncountable is that such a list does not > > exist. > > Or, even more directly, countability requires that such a list does > exist.
But the fact that no such list can be constructed is not evidence the set is uncountable, merely that it is not r.e.. I can use Cantor's form of proof to show that there can be no list of all computable Reals, but they are certainly countable.