Date: Dec 21, 2009 8:23 AM
Author: Dik T. Winter
Subject: Re: Another AC anomaly?

In article <9ad63bf1-549b-4822-bf86-839dd7c64d58@j4g2000yqe.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 18 Dez., 15:19, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > > > No, it is a matter of convention. In mathematics
> > > > a, b, c, ..., z
> > > > means a, b, c, continue this way until you reach z. But starting
> > > > {1}, {1, 2}, {1, 2, 3}
> > > > and going on you never reach
> > > > {1, 2, 3, ...}

> > >
> > > That is true. Therefore it does not exist.

> >
> > That you can not get there step by step does not mean that it does not
> > exist.

>
> That you cannot get step by step to 1/0 does not mean that it does not
> exist?


Indeed. On the projective line (that precedes Cantor by quite some time as
far as I know) it does exist.

> > > However, see Cantor,
> > > collected works, p 445:
> > > 0, 1, 2, 3, ... w_0, w_0 + 1, ..., gamma, ...,
> > > He seems to reach far more.

> >
> > Right, he uses a convention that is no longer used.

>
> Wrong, it is used presently, for instance by myself.


But you are not a mathematician.

> > > No. But the union contains two paths.
> >
> > Wrong. If we look at the paths as sets, they are sets of nodes. Their
> > union is a set of nodes, not a set of paths. And as a set of nodes we
> > can form from them seven different paths.

>
> Wrong. The nodes of two paths give exactly two paths.


Darn, the paths 0.000 and 0.100 contain the following nodes:
0.000 = {0., 0.0, 0.00, 0.000} and 0.100 = {0., 0.1, 0.10, 0.100}
where a node is named by the path leading to it. Their union contains
the following paths:
0., 0.0, 0.00, 0.000, 0.1, 0.10, 0.100
and I count seven.

> > > Let every finite path of every infinite path be mapped on the elements
> > > of omega. That was simple.

> >
> > By your statements infinite paths do not exist. But pray give such a
> > mapping. Until now you have only asserted that such a mapping exists
> > without showing that.

>
> Do you accept the mapping from omega on SUM{k = 1 to n} 3*10^-k yields
> the infinite decimal expansion of 1/3?


*What* mapping? Do you mean from n in omega -> SUM...? In that case the
infinite decimal expansion of 1/3 is unmapped.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/