On 15 Jun., 16:01, William Hughes <wpihug...@hotmail.com> wrote: > On Jun 15, 9:23 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > 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." > > You are trying a proof by contradiction of > "Infinite paths exist" > > We have
> If [actually]infinite paths exist > there is a path p that can be distinguished > from every path of P. > > We need > > If infinite paths exist path > there is no path p that can be distinguished > from every path of P.
No we need and have in fact: There is no path p that can be distinguished from every path of P. > > Your putative proof > > you are not able to distinguish p from P (by means of digits). > > (Note that you have agreed that distinguish p from P > means distinguish p from every element of P)
only if actually infinite path exist. > > I agreed, and pointed out that you distinguish p from every element of > P by > subsets of digits.
That is wrong once P has been used to construct the tree. Proof: You cannot distinguish p from the tree, because the tree would not be different if p and only paths with tails consisting of p were used to construct it.
> > > 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. > > Correct, you can get every node in the binary tree without > using every subset of nodes in the binary tree.
Wrong. P contains every subset of nodes of the binary tree. There is nothing else. > > > You know that you are unable to > > distinguish any binary sequence representing a real of the unit > > interval from every path of the tree. > > Correct the tree contains every subset of nodes, > however P ( a list of subsets of nodes) > does not contain every subset of nodes
As my proof shows, P contains every subset of nodes.
> You can distinguish p from every element of P by > subsets of nodes. > Wrong. The tree contains nothing more than the elements of P. This gets clear from the fact that you only write the list in somewhat compressed form.