On 19 Jun., 15:42, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > If every terminating path of the form 111...1000... is used, then > > every node is covered by the last 1. > > But that still does not show that all nodes are covered by a countable set of > paths. It only shows that every node is covered by a set of paths that > contains a countable subset, not that the set itself is countable.
It is easy to construct a bijection between a countable set of paths and all nodes. These nodes can even be defined by the paths that lead to them. That shows that all nodes of the tree can be covered by a countable set of paths.
> Yes, they are in the tree, so you have to prove that they are also used in the > covering, which you have not done. You do not have to show that each node > is covered, you have to show that with your covering you use each path. You > did show the former.
If every node is covered, then there remains no path to be covered. > > > > Or can you find a node that is *not* covered by a path that after some > > > node to the right? Can you find a node that is covered by a path that > > > alternates going left and right that is not covered by a path that after > > > some node always goes to the right? > > > > I see that every such alternating path with all its nodes is in the > > tree, after construction. > > This is not an answer to my question. The path is in the tree, but does it > contain a node that is not covered by a path that after some node always goes > to the right?
No. From that we can obtain that the number 1/3 is also in a list of terminating rationals. There does not exist a sequence 0.010101... that is longer than *every* finite sequence of that form.
> > We can state: The number of nodes is countable. The number of paths > > required to cover all nodes is countable too. > > Right. But the number of required paths is the minimal number of paths needed > to cover all the nodes. That is *not* the number of all paths.
If all nodes are covered then all path that can be distinguished by nodes, i.e., all reals that can be distinguished by digits, are there.
> Again: > consider the set of paths where each path after some point always goes to > the right. This set of paths covers each node. Consider the path that > alternates going right and left continuously. Can you point to a node in > this alternating path that is not covered by one of the paths in the earlier > set? But you did agree slightly higher up that that path was in the tree.
All parts of that path that really exist, really are in the tree.
> > That is a ridiculous requirement in an informal discussion, in > > particular if not mathematics is concerned. > > Oh. So in an informal discussion you can tell nonsense whatever you like > and when challenged you do not need to back it up?
I have read that there is a dogma of the catholic church stating that the existence of God can be proved by means of scientific reasoning. I do not remember the source. Does that turn this information into nonsense? (Except that its contents is nonsense.)