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.
Hence, there remains only the possibility to define the infinite path by a finite definition. But there are only countably many.
>I think using Cantor's argument > one can prove that for any countably many paths of the Binary Tree > there is a path that is not among them and thus establish a proof of > uncountability of those paths.
Then there is a contradiction. But you dismiss every contradiction? Therefore my proof mus be invalid???
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. As you know an infinite sequence cannot be defined by listing its terms. You need a finite definition to define the sequence and its diagonal. Again we reach the conclusion: There are only countably many finite definitions. > > Math is discourse about "possible" form. Uncountability of the reals > is PROVED in very weak fragments of ZFC, actually in PREDICATIVE > fragments of second order arithemtic, which are PROVED to be > consistent. This provide a discourse about form, thus uncountability > of the reals is a possibility! thus it is mathematical, since reals > can be interpreted as forms in the set hierarchy.
See above. People have not taken into account that only a *defined* sequence gives a defined diagonal. There are only countably many finite definitions.