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Re: Another AC anomaly?
Posted:
Dec 3, 2009 4:11 PM
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On 3 Dez., 16:27, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> In article <27aee2d6-5966-4b62-a026-fea13e0ba...@h2g2000vbd.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> ...
> > > > If the limit set is "assumed" by the sequence of sets, if for instance
> > > > N is given by unioning all naturla numbers, then tghe limit set has to
> > > > have the limit cardinality.
> > >
> > > The limit set is not necessarily "assumed" by the sequence of sets (if by
> > > that you mean that there is a set in the sequence that is equal to the
> > > limit set, if you mean something else I do not understand it at all).
> > > When you mean with your statement about N:
> > > N = union{n is natural} {n}
> > > then that is not a limit. Check the definitions about it.
> >
> > It is a limit. That is independent from any definition.
>
> It is not a limit. Nowhere in the definition of that union a limit is used
> or mentioned.
1) N is a set that follows (as omega, but that is not important) from
the axiom of infinity. You can take it "from the shelf".
2) N is the limit of the sequence a_n = ({1, 2, 3, ...,n})
3) N is the limit, i.,e. the infinite union of singletons {1} U {2}
U ...
This is fact.
But if (3) is correct, then N must also be the limit of the process
described in my
http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie 22
without and *with* the intermediate cylinder.
Then the cylinder must be empty in the limit, and cardinality of the
limit set in the cylinder must be 0. That, however, is wrong, simply
because the cylinder is never empty.
Regards, WM
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