On 13 Jan., 13:15, Zuhair <zaljo...@gmail.com> wrote:
> > > Notice also that one can have a COUNTABLE tree (i.e. a tree that has > > > countably many paths and nodes) that has finite paths > > > indistinguishable from the finite paths of the complete binary tree by > > > labeling of their nodes. > > > I noticed that already some years ago. > > What mean nothing more than saying that we have Countably many FINITE > paths of the complete binary infinite tree, nothing less nothing more.
It means that all nodes and edges are constructed with countably many finite paths. It means that if you want to distinguish an infinite paths from all finite paths you cannot do so by nodes or edges. And that's a lot.
> However the complete infinite binary tree does have paths that are not > finite, and those are Uncountably many.
But unfortunately they are not defined by nodes. Every node that you may want to use to distinguish an infinite path from all finite paths is already occupied by a finite path. If you wish to prove that infinite paths are more than the union of finite paths, than you will fail. If you agree that infinite paths are only unions of finite paths, then you agree to potential infinity or limitis that are not defined by nodes - and that excludes uncountability.