On 10 Dez., 09:07, Virgil <vir...@ligriv.com> wrote: > In article > <5a2d9b2e-c558-446a-908f-1a5f24d3f...@r14g2000vbd.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 10 Dez., 06:32, Zuhair <zaljo...@gmail.com> wrote: > > > On Dec 10, 12:21 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > If so what is the proof that ALL reals belong to that tree? > > > There is no proof because the Binary Tree contains all reals between 0 > > and 1 by definition. > > What definition is that? I know of no such definition.
All real numbers that have binary representation starting with 0. are in the Binary Tree and are in the interval [0, 1]. No one is missing in any of these systems.
> Every binary sequence is in any complete infinite binary tree.
Of course. But you had forgotten that above? > > > Cantor's proof (concerning the binaries with bits w and m) shows that > > not all that are in the tree can be in the list. > > Right for once!
> > This proof is given by constructing the whole Binary Tree by countably > > many actually infinite paths (i.e. finite paths with infinite endings > > like 010101... or 000... or 111... or 001001001... or any desired > > ending that I do not publish). My proof rests upon your inability to > > find out what paths are missing. > > And Cantor's counterproof says that no such list contains all paths and > provides an unambiguous way of determining from any such list ( which > exists as a consequence of WM's claim of countability) any finite number > of the uncountably many paths which are missing.
But his proof is taken as evidence that the Binary Tree contains uncountably many paths *that are defined by nodes only* (as is every path in the Binary Tree). And this result is false.
> > Find a path that I have not used! > > List the ones you have used
I have constructed all paths that can be defined by nodes in the Binary Tree, because no node is missing in my construction.
> (and until you have listed them , or at > least prove you can list them, your claim of countability is > notstablished) and then it is trivial to find others.
Wrong. The notion of countability is self-contradictory. There is no chance to recognize more than countably many paths by nodes.