Re: Cantor and the binary tree

mueckenh@xxxxxxxxxxxxxxxxx wrote:
> imaginatorium@xxxxxxxxxxxxx wrote:
> > mueckenh@xxxxxxxxxxxxxxxxx wrote:
> >
> > > We have a law yielding all possible infinite strings of bits.
> > > We have a law connecting each spring-off, i.e., the source of a newly
> > > separated path with a node B:
> > > /
> > > B
> > > /\
> >
> > I don't understand this. As far as I can see, each node 'B' has two
> > branches from it, to left and to right, and neither identifies a single
> > path. Rather, each one leads to another (unending, infinite) subtree.
>
> One path enters that node, such that one path is newly created by the
> node.

One? Created? This is all very woolly. At each node, the set of paths
coming in divides into two. You can easily pick one out and associate
it with this node. But how can you be sure that all the other paths
will get accounted for? To do this, for an arbitrary node you have to
be able to tell me where it is deemed to "separate off", thus
associating that node with the particular path.

You can't. Sorry, I've only been watching this thread in passing - I
imagine a number of people have been through all this already. You have
an agenda, so I don't think I have the perseverance to go through it
over and over again.

> > On the contrary, I raise it. In maths, if the answer to any question is
> > obvious, it can be dealt with quickly. Anyway, we could identify with
> > this node the path going right, and subsequently always going left. In
> > this way we can identify every node with a single path, showing that
> > there cannot be more nodes than paths, but not showing the reverse
> > (because we haven't mapped every path to a node in this way).
>
> Where should the other paths come into being? Where should they be
> separated, if not n a node?

What does "be separated" mean? Have you actually defined it?

Brian Chandler
http://imaginatorium.org

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