In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 11 Jun., 15:36, William Hughes <wpihug...@hotmail.com> wrote: > > On Jun 11, 9:19 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 11 Jun., 13:00, William Hughes <wpihug...@hotmail.com> wrote: > > > > > > > Because paths cannot be distinguished without nodes. > > > > > > Proves nothing. We know that a countable > > > > number of elements can distinguish an uncountable > > > > number of subsets. A countable number of nodes can > > > > distinguish an uncountable number of paths. > > > > > No that is provably wrong. All nodes are used up by a countable number > > > of paths, e.g., all paths ending in a tail of zeros. Therefore no > > > possibility exists to construct or to distinguish by one or many or > > > infinitely many nodes of the tree another path. > > > > Your claim is that "no possibility exists" > > > > Nope. In any tree, any node that is not a leaf node > > can contribute to more than one path. The possibility > > exists to construct another path using nodes > > which are not leaf nodes. > > If you have a tree that is (the nodes of which are) completely covered > by a set of paths then you cannot show any node that will distinguish > a further path from that given set of paths.
In a maximal infinite binary tree, no one node is, not even any finite set of nodes, is enough to separate any one path from all others, it takes infinitely many nodes to do that.
If WM's trees are not like that then they no maximal either, meaning that there are paths in maximal trees that WM's trees exclude.
> But if you cannot > distinguish a path from a given set of paths, then it belongs to that > set. One can distinguish any path from any set of paths not containing it by a set of nodes if one is allowed sets with enough nodes to do the job, but not if, as WM would have it, one is restricted to too few nodes.
In finite trees, the only single node that can separate any one path from all others is its leaf node, but in infinite trees there are no leaf nodes.
In a maximal infinite binary tree, any infinite subset of any path's set of nodes suffices to separate it from any set of paths not containing it.
But no finite set of its nodes, or any other nodes, suffices to separate a path from all sets of paths not containing it..