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.
> What you are not getting is that uncountability of the reals is a > PROVED issue, it is proved in very weak fragments of second order > arithmetic that are PROVED to be consistent. I don't know if you > really get what I'm saying here.
If you were right, then ZFC is inconsistent. Even the very weak fragment is inconsistent. > > However on the other hand still you can get countable models of those > theories where the set of all reals can be defined. So both > countability of the reals and uncountability of reals are open > possibilities and can be spoken about by consistent discourses. So > both are pieces of mathematics. > Everything depends on the model you are working in.
The real numbers as we teach them are at issue. Everything else may be left to the "experts" or fools of matheology.