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. Of course this is a clear retreat from what you've just said before, where you said that no path is actually infinite, anyhow. 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. And so far nobody have succeeded to demonstrate any contradiction involved with this concept. It is remains a standing possibility that mathematics must deal with.