Re: An uncountable countable set


*** T. Winter schrieb:

In article <1155674426.389672.172370@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter schrieb:
> > In article <1155640812.322879.187040@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> > >
> > > *** T. Winter schrieb:
> > >
> > > > > Nonsense. Cantor invented the list and constructed the first one.
> > > >
> > > > Quote, please, especially for the latter remark.
> > >
> > > =DCber eine elementare Frage der Mannigfaltigkeitslehre.
> > > [Jahresbericht der Deutsch. Math. Vereing. Bd. I, S. 75-78 (1890-91).]
> >
> > Do you have an URL? Is it in his "Werke"? Otherwise I have no access
> > to it.
>
> Werke, page 278. In particular the list is on p. 279.

That is not a specific list, but an arbitrary list.

It is a list, the first one.

Anyhow, that assumed list can be enumerated by definition.

That definition is the following: If for any line a natural number can
be determined, uniquely, then the lines are countable.


> > > Are you really believing that the number of edges was uncountable?
> > > Take any edge you like. You can attach a natural number to it. That and
> > > nothing else is the definition of countability.
> >
> > That is not the definition of countability. The definition of countability
> > is that you can attach a (different) natural number to each and every
> > edge.
>
> Yes, of course. And what is your problem with the edges (how do they
> differ from the lines of Cantor's list)?

The lines are countable (by assumption). But pray assign natural numbers to
each edge in your tree. I will even allow you to assign finite sequences of
digits 0 (go left) and 1 (go right) to your tree (the mapping to natural
numbers is quite standard in that case). What is the finite sequence of
digits assigned to the edge that leads to 1/3? You claim that the edges
are countable, so you should have an answer to my question.

The sequence of edges is countable but it is not finite!
1/3 has the sequence of nodes 0.010101... in binary notation.

0.
/1 \2
0 1
/3 \4 /5\6
0 1 0 1
/7\8/9\10........

The edges of 1/3 are enumerated 1, 4, 9, 20, ... . For any edge a
natural number can be determined, somewhat cumbersome, but it can be
done.


And before you mutter: I just assign 0 to it and change all other assignments
so that it fits. Be prepared that I will ask you to give such a sequence
for another number. I will not tell you which, but if your mapping does
not map that other number to a finite sequence, your mapping is shown
incomplete. And this works *regardless* the mapping you are using.
Your mapping really should contain each and every real number before I
consider further.

It does - obviously.

Regards, WM

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