Re: Cantor and the binary tree



Virgil wrote:

> > Double the number of nodes. The set of nodes remains countable. One of
> > the nodes is always related to a path turning right, the other one to a
> > path turning left. Your idea does no longer apply.
>
> Yes it does, since doubling the number of nodes must double the number
> of paths as well, unless WM wants all those new nodes to be sitting out
> in the open with no tree at all to shelter them.

Yes, every split point is equipped with two nodes. The set of these
nodes remains countable. Every edge gets bijected to such a node. Every
bunch (= set) of paths which has its own edge also has its own node.
Paths which do never separate in the tree do never separate by their
binary representation. They are not different paths, not different
representations, not different numbers.

Irrational numbers do not exist. All numbers are countable.


> Is "bunch" a new techical word? How does a "bunch" of paths differ from
> a set of paths?

bunch of paths is the same as set of paths.
>
> And how do "separated" paths differ from other paths?

They have at least one different edge.
>
>
> > The nodes are denoted by capitals, the
> > paths by small letters. Start at the root A. The edge A-->B contains
> > all paths.
>
> Wrong already, since from any node, including the root node, there are
> TWO branches to its two child nodes. So neither branch can only contain
> more than "half" of the paths.

In order to get rid of he term nodes + 1, I assgned one node to the
root, see below.
>
> > Node A is mapped on one of them. Call that path a.
> > The set of all path splits off at node B. That subset which does not
> > contain a must contain at least one path. Call that path b. Node B is
> > mapped on path b.
> > The set of these path splits off at the next node, say C. That subset
> > which does not contain path b must contain at least one path. Call that
> > path c. Node C is mapped on path c.
> > The same is valid for the other subset containing a. There is no subset
> > possible without a node.
> >
> > A
> > /
> > B
> > / \
> > C D
> > / \ / \
> >
> > This mechanism remains as long as a subset of paths contains at least
> > one path.
> > There cannot exist any subset of paths without one node.

Make it easier: The edges form a countable set. Bunches of paths which
are not separated, i.e., which have not at least one edge of their own
in the tree, do not exist as individuals. The corresponding reals are
ot existing.

Regards, WM

.

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