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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

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mueckenh@rz.fh-augsburg.de

Posts: 18,076
Registered: 1/29/05
Re: Another AC anomaly?
Posted: Dec 21, 2009 12:15 PM
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On 21 Dez., 14:23, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> In article <9ad63bf1-549b-4822-bf86-839dd7c64...@j4g2000yqe.googlegroups.com> WM <mueck...@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.


But does the projective line 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.


Do you think so? I studied mathematics and a university council
appointed me to teach mathematical lessons. Have you better insights
than they? Or have you only a different definition of mathematics? Do
you mean that I am not a matheologian? Then you are right!
>
>  > >  > 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.


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


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

Regards, WM


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11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil

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