On 9 Dez., 21:16, Zuhair <zaljo...@gmail.com> wrote: > On Dec 9, 10:59 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > On 9 Dez., 20:37, Zuhair <zaljo...@gmail.com> wrote: > > > > > Only a countable subset can be represented by the Binary Tree. The > > > > reason is that no path is really actually infinite. > > > > Then you are not addressing what Cantor was speaking about, he is > > > speaking about reals represented by ACTUALLY infinite sequences (paths > > > in your case). It is clear that the set of all reals represented by > > > FINITE sequences is countable, but those are just a very small subset > > > of the set of all reals. > > > > If one assumes Actual infinity, then it is easy to recover the > > > diagonal path from any bijection between the reals and the set of all > > > paths of the infinite binary tree, and this will be a path that is not > > > present in the tree of course. > > > Then you are wrong from the scratch. Every real number has a > > representation by an infinite sequence (= infinite path of nodes in > > the tree). > > Why you don't just prove that statement.
Because it has been proven by Cantor. Can you imagine any diagonal number that has not a representation in one and the same Binary Tree?
Please don't completely forget your ability to think when adhering to the "experts" who require "proofs" (of that silly kind they prefer) for 2 + 2 = 4 and theorems like that.
> Of course this is a clear > retreat from what you've just said before, where you said that no path > is actually infinite, anyhow.
That is the reason why Cantor is wrong. But the Binary Tree that I assume has an actual infinity of levels. Every path has an actual infinity of nodes. Nevertheless I construct this actually infinite Binary Tree by means of a countable set of paths.
> Possibly you are referring to potential > infinity when you say infinite path of nodes. But by argument of > potential infinity you cannot have something called "infinite" path of > nodes, all what you can have is 'finite' paths. And again clearly what > you are addressing is something quite different from what Cantor is > speaking about. Cantor is speaking about Reals that are represented by > ACTUALLY INFINITE paths.
Nevertheless, every node lies at a finite distance from the root node. This is just the Tree that I construct!
> And so far nobody have succeeded to > demonstrate any contradiction involved with this concept.
It is ridiculous! I have shown you the contradiction. You escaped into the completely nonsensical idea, saying: "You will need uncountably many infinite binary trees to recover all the reals."
You cannot recognize that I have a contradiction if I can force you to utter such nonsense?