In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
Binary strings are not numbers but are, in a sense, numerals. Wm really should learn the difference if he intends to be a mathematician some day. > > > Every binary sequence is in any complete infinite binary tree. > > Of course. But you had forgotten that above?
But WM's binary trees are not complete ones as they do not have enough paths to be complete. > > > > > 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 ending that I do not publish).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.
Not outside of Wolkenmuekenheim!
Cantor's diagonal argument establishes that any attempt to count all binary sequences must fail. Thus they are collectively uncountable.
If WM wishes to claim that the set of all binary seqeunces is countable, he must show us a listing of all such sequences which is immune to Cantor's argument.
Absent that listing, WM is wrong to deny that uncountability. > > > > 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.
Since you have not listed them, nor show that they can all be included in any such a complete listing, your claim of countability is unfounded. > > > (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.
The definition of countability is not self-contradictory and has successfully been applied to any number of sets, including, for example, the set of rationals. That WM does not like it is not a mathematically valid argument against it.
There are two valid tests for countability in general use: (1) seeking a surjection from |N to the set in question, or (2) seeking an injection from the set into |N. In either case, finding the desired mapping proves countability of he set in question.
So unless WM can find either a surjection from |N to |E or an injection from |R to |N, and present it here, the uncountability of |R, and of the set of paths in the complete infinite binary tree, remains here as proven by Cantor.
Any other form of argument against such uncountability is irrelevant and invalid.
But I have no doubt that WM will persist in his invalid irrelevancies. --