On 7 Feb., 15:56, William Hughes <wpihug...@gmail.com> wrote: > On Feb 7, 3:25 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > <snip> > > >... a subset S of the countable set F of finite words bijects with > > the set D of definable numbers
by definition. > > Nope. Every D corresponds to some finite word.
No, D is a set or at least a collection. A definable number is an element of D.
> However, S, > the collection of all the correspondences, may not be a subset > of F (subsets must be computable).
Need not be a subset. It is sufficient to know that there are not more than countably many correspondences, because the upper limit would be "all elements of F".
> Hence we know D is > sub-countable, but we do not know D is countable.
You intermingle elements D and sets D in a really refreshing way. Nevertheless, we know that all correspondences together cannot surpass aleph_0.
According to this definition the real numbers are subcountable. All paths of the Binary Tree cross every finite level in a finite number of nodes. Aleph_0 is not reached at any finite level - and there is nothing, in particular no path, beyond every level. Nevertheless there is no list of all paths.