In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 13 Jan., 13:15, Zuhair <zaljo...@gmail.com> wrote: > > > > > Notice also that one can have a COUNTABLE tree (i.e. a tree that has > > > > countably many paths and nodes) that has finite paths > > > > indistinguishable from the finite paths of the complete binary tree by > > > > labeling of their nodes. > > > > > I noticed that already some years ago. > > > > What mean nothing more than saying that we have Countably many FINITE > > paths of the complete binary infinite tree, nothing less nothing more. > > It means that all nodes and edges are constructed with countably many > finite paths. It means that if you want to distinguish an infinite > paths from all finite paths you cannot do so by nodes or edges. And > that's a lot.
One does not distinguish one infinite path from all others by anything less that an infinite set of nodes. Actually any infinite subset of the infinite set of nodes from any infinite path is enough to distinguish it from all others, but no finite subset of its node set is sufficient. > > > However the complete infinite binary tree does have paths that are not > > finite, and those are Uncountably many. > > But unfortunately they are not defined by nodes.
No infinite path is "distinguished" by any finite set of its nodes from ALL others.
Note that a finite set of nodes from any path tells us no more than the last node in that set about which path can pass through them all.
For any finite set of nodes in any path there are uncountably many paths through each and every one of those nodes, all of the uncountably many paths passing through the last of those finitely many nodes!
> Every node that you > may want to use to distinguish an infinite path from all finite paths > is already occupied by a finite path.
But no infinite set of nodes occurs in such a finite path, and to distinguish any one path by its nodes from all others requires an infinite set of its nodes
> If you wish to prove that > infinite paths are more than the union of finite paths
In a Complete Infinite Binary Tree, all paths are of the same infinite length, one cannot have any finite paths in a CIBT.
If you are referring to finite initial segments of such infinite paths, you will have to use some other word or phrase, since the word "path" is taken.
> If you agree that infinite paths are only unions of finite > paths
There are no finite paths in a Complete Infinite Binary Tree.
Perhaps you might call any finite initial section of a path a subpath. --