> In <email@example.com>, 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. >
Obviously they are. Somebody constructs a list of purported Reals, and you show a Real is missing. It actually commences with a constructed list.
I can't produce a list of all computable numbers, but that doesn't mean they are uncountable. Cantor showed that the Reals are not recursively enumerable, not that they are uncountable.
> > 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.
The proof actually says that such a list cannot be constructed. It starts with a hypothetical list that has already been constructed. No constructed list, no anti-diagonal. No anti-diagonal, no proof.