On 5 Mai, 10:02, Virgil <vir...@ligriv.com> wrote:
> > > Thus (reals countable) <==> (reals listable). > > > > But Cantor has shown that there is no complete list of real possible as > > > every listing of reals necessarily omits some reals. > > > And counterfactually he has assumed that there is a complete list of > > natural numbers. > > If there isn't then the set of naturals is not countable either.
Correct. > > > > > > > My list is constructed such that all possible squences of digits 1 are > > > > already in the list. No longer sequence is possible > > Except for the concatentation of all those listed finite sequences, > which is clearly longer than any of them.
But not longer than itself. And itself is potentially finite, i.e. finite but not fixed. > > > > Unless your list claims to list all reals, it has no effect on the > > > validity of the Cantor diagonal argument, which remains valid in spite > > > of it. > > > It is true: If the naturals are complete somewhere, the reals cannot > > be complete there. Alas this theorem is void since the naturals cannot > > be complete anywhere. > > Then the set of naturals is also not a countable set.
That is obvious by the fact that "countability" is nonsense. You can count up to every n, but you cannot count all. The correct terminus is "infinite".