On 2 Jul., 03:40, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > Sorry, no dreaming involved. You did show that you could cover each node by > countably many paths. You did *not* show that the set of all paths was > exhausted by that process.
Proof by your failure to find a path that is not in the tree.
The tree is constructed by terminating paths with tails 000... appended, but the same tree can also be constructed by terminating paths with tails 010101.. or pi appeded. If you, by looking at the completely constructed tree only, can find a path that is not in the tree and that was not used to construct the tree, then you are right. But you cannot. Therefore your assertion concerning missing paths is wrong. > > > I show that the number of paths that can be identified by sequences of > > nodes or bits is countable. > > Not at all, because you did not show a bijection.
I did use the countable set of terminating paths (extended by a special kind of tails). And you cannot identify a path that I did not use, if you don't know from what kind of paths the tree was constructed. This proves that there is nothing further in R that is related to sequences of nodes only.