Re: Cantor and the binary tree



*** T. Winter wrote:

> > I don't know how many bits the number 1/3 has.
>
> Infinitely many, and that is not a natural number.

But each level is enumerated by a natural number. Or do you think that
1/3 has any bit, 0 or 1, at a position which cannot be enumerated by a
natural number? Hence every node can be enumerated. Therefore your
answer does not hit the point.
>
> > But if it is a number
> > then it has a path in my tree. And if it has a path in my tree then it
> > has a node to be mapped on.
>
> Right for the first, wrong for the second.

The nodes _are_ the path. As long as the path stretches we do never run
out of nodes. Therefore the argument, that there is no last node, is
meaningless. There are always nodes enough.


>
> > You must know, there are infinitely many
> > nodes in my tree.
>
> That does not matter, an infinite (i.e. unending path) as that which
> 1/3 corresonds to does not have a node on it that corresponds to 1/3.

If there is a path (= set of nodes) which 1/3 corresponds to, then
there is always the node required.

But I see, the advocates of set theory must insist on the idea that the
paths run into a domain without nodes. It is strange, because the paths
are the nodes. So you must insist that there are strings of bits
without bits. It is a mystery because the strings cannot exist without
bits.

But it is impossible to argue against strings of bits without bits.

Regards, WM

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