Date: Dec 22, 2009 9:47 AM
Author: Dik T. Winter
Subject: Re: Another AC anomaly?

In article <b484e377-9dc2-424b-80c3-2912165f636c@a32g2000yqm.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 21 Dez., 14:23, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > > 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.

>
> But does the projective line exist?


Exist in what sense? But it is a concept from projective geometry, but
perhaps you think that is also nonsense? I may note that the point at
infinity was developed by Kepler and Desargues in the 17th century.

> > > > > 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.

>
> Do you think so?


Yes, I think so and your inability to provide definitions and proper
proofs shows it.

> I studied mathematics and a university council
> appointed me to teach mathematical lessons.


Well, the same did hold for my father, but he never considered himself
a mathematician, but a physicist.

> > > > 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.

>
> The nodes of two maximal paths give two maximal paths. You count
> initial segments.


I did not count segments, I counted paths. You stated there are only
two paths in the union, I count seven paths. Or are those seven things
suddenly not paths?

> > > > 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.

>
> The sum of all finite segments is not the infinite path. Interesting.


You have stated that your paths are finite, so I consider the mentioning
of an infinite path strange. Moreover, you are now talking about sums,
not about paths. But *what* n in omega maps to the infinite decimal
expansion? You have not stated that. And if you think the infinite
expansion is mapped you should be able to state an element of omega
that maps to that infinite path.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/