In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> Am Donnerstag, 6. Februar 2014 10:19:54 UTC+1 schrieb Virgil: > > > > > > The finitely defined real number r = Sum_(n in |N) 1/2^(n!), in base 2, > > > > is not anywhere in WM's "rationals-complete list" > > No it is not since it cannot be represented by digits.
It IS representable by digits in base 2 by having a 1 in each n! place and 0's elsewhere.
What exactly does WM mean by "represented by digits"? Rationals do not all have a representation "by digits" in any one base
> You will never distinguish this > number *by digits* from all terminating rationals.
I already did! > > > or in his > > pseudo-binary tree, but does represent a path in any Complete Infinite > > Binary Tree > > > There is no path in the Complete Infinite Binary Tree that defines this > number r.
There is in any Complete Infinite Binary Tree. In is only in WM's incomplete trees that it may be overlooked.
> I have proved this by constructing a Binary Tree
WM's trees are all incomplete.
> from a set of > paths that contain all nodes of r but does not contain that path itself.
One must construct the tree first and then that tree defines its paths.
WM is trying to do things backwards by patching together a bunch of what he calls paths (which cannot be paths until after the tree is completed) and calling it a tree.
A standard model for a Complete Infinite Binary Tree , as Ben said, is the set of naturals as the set of nodes, with (n,2*n), n in |N, linking n as the parent node with 2*n as its left child node and (n, 2*n+1) linking n with its right child node, 2*n+1.
This tree exists as a tree prior to any definition of pathhood within it, so that tree-existence is entirely independent of and prior to path-existence.
Thus, as usual, WM's approach to trees is directly backwards. --