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). But as my proff shows I construct the whole Binary Tree by > countably many paths. There are not more nodes available to add > further paths. > > >You will need uncountably many infinite > > binary trees to recover all the reals. > > That is purest nonsense. And it has nothing to do with Cantor's > diagonal which is of course an infinite sequence of digits > corresponding to a path in the Binary Tree. > Yes corresponding to an ACTUAL infinite path in the Binary Tree, which is something that you already refuse to address.