In article <103cc56f-746e-482f-9315-0c9e8eaf45e1@c10g2000yqi.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 22 Jun., 21:05, "Mike Terry" > <news.dead.person.sto...@darjeeling.plus.com> wrote: > > > > > Ln = > > > > > An > > > ... > > > A2 > > > A1 > > > A0 > > > L0 > > > > > where L0 is thelist of all rationals. > > > > > This list Ln contains a countable set of numbers > > > > ..correct, {An, ...A0, L0(0), L0(1), ...L0(n),...} > > is obviously countable. [L0(k) is the k'th element in list L0] > > > > > but the set of its > > > diagonals is not listable, because An is not in the list. > > > > The "set of its diagonals" = {An}. A list has just one diagonal. Every set > > of one element is listable. Like Sylvia I must be misunderstanding what you > > mean. (But I'm not misunderstanding what you actually say. :-) > > To spell it out clearly: The set of all diagonals (including or > exluding all rationals - that does not matter) > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > that are constrcuted according to my prescription cannot be listed > although it is countable.
Except in WM's world, a set is listable (can be a bijective image of the naturals) if and only if it is countable (there is abijection with the naturals).
WM is very careful NOT to give his own definition of listable and countable in which there are countable sets which are not listable. > > If we use Cantor's definiton of "countable", then the set > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > is uncountable.
Cantor's definition says a set is countable if there is a surjection from the naturals onto that set. But such a surjection IS a list. > > If we use the definition that a subset of a countable set is > countable
That is not a definition but a theorem, at least if countable includes all finite sets.
, then the set > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > is countable. > > > > > > > > > This is wrong. An obvious listing is (A0, A1, ...) > > The set > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > cannot be listed.
You just listed it, since every-double-open-ended list can easily be single-open-end listed. > > > > > there exists a countable set M, such that > > If L is a Cantor-list, then > > (anti-?)diagonal of L belongs to M. > > > > That is so obviously false that its banal. > > No it is not.
It is outside of WM's world, such as in FOL+ZFC.
> If there exists a Cantor-list, i.e., that what Cantor > really understood by the term list, then it is a list of *defined* > reals. Again, this holds only in WM's world, and not in such standard set theories as FOL+ZFC.
And then its anti-diagonal is a defined real too. Then exists a > countable set M, namely the set of all defined reals, that is > countable. Nevertheless it cannot be listed.
If it cannot be listed, how does WM know that it satisfies any standard definition of countability?
