On 15 Jun., 00:19, William Hughes <wpihug...@hotmail.com> wrote:
> Your claim is that "no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path."
In fact that is not possible. >
> > > > There are two statements: > > > > 1) Path p can be distinguished from every path of P. > > > > 2) Path p cannot be distinguished from every path of P.
> > > You agreed that constucting the tree does not change > > > path p or any element of P > > > That is correct. It is a basic of mathematics. > > So either you are completly incoherent or you > now agree that under the assumption that > actual infinity exists path p can be distinguished > from every path of P.
But it turns out that this assumption cannot be satisfied. > > > Your claim explicitly mentions distinguishing p from P by > "one or many or infinitely many nodes". > Answer yes or no > > you are able to distinguish p from P by subsets of digits
I will try to explain it for you again:
We have a list of a countable set P of terminating paths p_n. By the diagonal method we can distinguish p from every p_n (if actual infinity exists).
Now we write the paths p_n in slightly different form: The beginning zeros of all paths are written only once, the following 1 or 0 also are written only once each and so on. Note, we have not done anything else but writing the list in slightly different form, saving some ink. In particular we have not added any new path.
This yields the complete binary tree. You know that you are unable to distinguish any binary sequence representing a real of the unit interval from every path of the tree.
This proves that you, contrary to the assumption, have been unable at the beginning too, because there the same set of paths was presented to you. You only believed that you were able. Cantor's diagonal method is falsified.