On 9 Dez., 10:41, Zuhair <zaljo...@gmail.com> wrote: > On Dec 9, 11:35 am, WM <mueck...@rz.fh-augsburg.de> wrote:> On 9 Dez., 05:55, Zuhair <zaljo...@gmail.com> wrote: > > > > You need to prove that the set of all paths is countable, and so far > > > you didn't present a proof of that. > > > The set of all finite paths is countable. Therefore it is not possible > > to define an infinite path by adding nodes to any finite path. All > > nodes to be added are already in finite paths. Therefore, by following > > the nodes of a path, you never define an infinite path. It is > > interesting that practically everybody not yet brainwashed can > > understand that. > > Every node is reachable by a finite path, that's correct. But that is > irrelevant > here, we are speaking here about the number of all "path"s in the > Binary Tree > and not about the number of all nodes. we know that the number of all > nodes > is countable, the question is: is the number of all paths (finite and > infinite) > is countable?
So it has become obvious now, that is not possible to define "all paths" by nodes. Only the finite paths can be defined by their nodes. How can you define all paths if not by nodes? > > > Hence, there remains only the possibility to define the infinite path > > by a finite definition. But there are only countably many. > > Yes, correct. > > > What is that contradiction, nobody is assuming that all infinite paths > are definable?
The contradiction is that an undefinable path cannot be put into any order or well-order. > > > But there is not even a contradiction. More precisely: There would be > > a contradiction, if there wer uncountably many diagonal possible. But > > in fact, no infinite sequence of paths defines a diagonal-path unless > > the sequence has a finite definition such that every path is known. > > No that is not necessary really, the infinite sequence of paths can be > indefinable
Again: A real number is a quantity and must be in trichotomy with all other reals. An undefinable number can neither be in trichotomy nor can it be pit into a well-ordering. It simply is not a number. Butr Cantor's ideas are too nice to be given up when faced with reality of real numbers.
It is a shame that someone defends the concept of "undefinable number", unthinkable thought, anusable use, - and nevertheless claims to be a logican and mathematician!