Re: Cantor Confusion


*** T. Winter schrieb:

> There is no difference between enumerating all the rational numbers in
> the well-known scheme, starting at the corner of an infinite square
> matrix

There is. You can assign numbers to edges that terminate at nodes. And
*that* numbering is different.

I can assign numbers to edges that start at nodes. I can also assign
numbers to nodes.

It numbers only those edges that are finitely
far away from the root. But that way you do not number all edges in the
infinite tree, because that contains edges that are *not* finitely far
away from the root.

That is an interesting remark. The paths are isomorphic to the binary
representations of real numbers. So you claim that there are positions
in the binary representation of the real numbers that are infinitely
far from the decimal point. O course such positions cannot be
enumerated or indexed by natural indexes. This means, they are not
defined at all.

> > Indeed. To prove countability you do not need transfinite induction.
> > But my remark "you are wrong" was to your statement:
> > "The mapping N -> {edges} has been established".
> > See J. H. Conway, On Numbers and Games, for a clear exposition about the
> > difference between the union of all finite trees and the infinite tree.
>
> Whatever Conway may say, there is no difference between enumerating all
> rational numbers in the well-known scheme, starting a the corner of an
> infinite square matrix, and the edges of the tree. In both cases you
> have a system with a limit which is countably infinite.

There is a difference, see above. You can only number edges that are
finitely far away from the root, but in that way you will only number
edges that terminate at nodes finitely far away from the root. As in
the edges there are nodes infinitely far away from the root you will
never number the edges that terminate there.

Let us enumerate the nodes.

The situation with the rationals is quite different, because in the
matrix *each* rational is finitely far away from the root.

Each node is finitely far from the root. (Does Conway really tell what
you reproduce here?)

> Perhaps you see a difference between the union of all finite numbers n
> and the set N?

Apparently you are not interested in what Conway did write, otherwise you
would understand how ridiculous this comment is.

I have experienced worse opinions. But let us finish with the polemics
(if possible), because we are at the most important point. Please think
over your argument:
1) Do you say that the nodes cannot be enumerated?
2) Do you agree that this implies: There are bit positions infinitely
far from the decimal point (or how this point may be called for binary
numbers).

Regards, WM

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