Re: Cantor and the binary tree



imaginatorium@xxxxxxxxxxxxx wrote:
> 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 need not define which of the two paths going out is the new one and
which is the old one coninued. All we need to know ist that the number
of paths has increased by 1. The numbr of paths is equal to the number
of path-origins (that is a branching like that described above).
>
> 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?

Separated means separately visible from the others. Some words must be
talen for granted even in math definitions.

Se theory requires: The number of paths is larger than the number of
path-origins. That is not counterintuitive. That is wrong.

Regards, WM

.

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