In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 26 Mrz., 03:10, "Ross A. Finlayson" <ross.finlay...@gmail.com> > wrote: > > > Two different paths, finite or infinite, have at least one node not in > > common. > > If the path of 1/pi is not in the Binary Tree, it is impossible to > find that path in the Binary Tree
It is obviously impossible to find any path in a tree when that path is not in that tree.
But WM cannot prove that that path cannot occur in a COMPLETE Infinite Binary Tree, because it does.
If that path is not in some tree, it is because the tree is incomplete.
> notwithstanding the fact that for > every rational approximation q of 1/pi, there is a better one q' (with > a node belonging to 1/pi but not to q). The result remains that q' is > not 1/pi.
If no binary sequence for 1/pi exists in WM's Complete Infinite Binary Tree, there must be a first node at which the infinite binary sequence for 1/pi from every path of length n in the tree.
I.e., there must be a first binary digit at which the binary representataion of 1/pi differs from every binary sequence.
And thus there must be a finite binary approximation of 1/pi which does not appear in WM's tree.
> > > > And they have at most countably many nodes not in common, there are > > only countably many nodes. > > > That is provable. > > > > So, are there uncountably many nodes? > > No: > > 0 > 1,2 > 3,4,5,6 > ... > > > Because, there's a rational for > > each node, and the rationals are countable. There are only countably > > many different paths, from a path. The paths are rooted. > > > > Seems rather muddled. > > Why. It is obvious that only countably many different steps will > construct the Binary Tree and that no step will be able to distinguish > more than one path from the body already constructed.
WEven though there are only countably many nodes, there are UNcountably many sets of nodes, and paths are SETS of nodes.
So countability of the set of nodes does not imply countability of the set of paths, but tends to contradict it.
Thus, once again, WM's ability to keep thing straight is found wanting, and his arguments found flawed. --