Re: Cantor and the binary tree

In article <1117619936.210622.223600@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
> >
> > > And if they are paths in the tree, then all nodes of
> > > them are existing within the tree.
> >
> > Wrong! Not every path will be associated with a node by WM's "last right
> > turn" rule, because not every path has a last right turn.
> >
> > Can WM find the LAST right turn in the infinite path that alternates
> > left and right turns? That is the path corresponding to 1/3, and it has
> > no last right turn. So there is also no associated node by WM's "last
> > right turn" rule.
>
>
> If the binary representation of 1/3 (isomorphic to the path in
> question) does exist, then it has to be distinguishable from any other
> real number. If it is distinguishable from any other real number, then
> it has a node which differs from all other paths of the tree.

I can think of no interpretion which makes this anything but nonsense.

Every node of a maximal binary tree can be taken as the root of a tree
starting at that node and that subtree will also be a maximal binary
tree.

So that unless there is only one branch per node from some node onward,
what WM requires cannot happen.

And binary trees always have two branches, if any , from any node.



> This node is it, which 1/3 is mapped upon.

"This node" does not exist.

> If this distinguished node
> does not exist, then there are other numbers which have same paths and
> are isomorphic to 1/3.

WRONG! Given any node on any path in any maximal binary tree, there are
infinitely many paths passing through that node, as it is itself the
root node of a maximal binary tree, with infinitely many paths rooted at
it.

>
> To have shown this fact and the necessarily opposite opinion of set
> theorists is sufficient for me.
>
> Therefore, there is no reason to continue this discussion. My purpose
> has been reached. Our positions are clear. As they can remain between
> men of perhaps moderate but certainly not below-average intelligence,
> we see mathematics in a very deplorable state.

And mathematicians, with better reason, see WM in a deplorable state.
.

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