Virgil
Posts:
9,012
Registered:
1/6/11
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Re: Matheology � 200
Posted:
Jan 27, 2013 5:25 PM
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In article
<68e81e84-7fd2-4682-ab9d-d94ac9c65665@v7g2000yqv.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 27 Jan., 14:04, William Hughes <wpihug...@gmail.com> wrote:
> > On Jan 27, 9:40 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 27 Jan., 01:55, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > On Jan 26, 10:51 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > The set of indexes is 1 + the decadic
> > > > > logarithm of L.
> >
> > > > Nope, 1+ the decadic logarithm of L is the cardinality
> > > > of the set of indexes.
> >
> > > I know.
> >
> > So do not make silly statements
> > about what the set of indexes is.
> >
> > You are taking the limit of
> >
> > 1+decadic logarithm of L
> >
> > You are not taking the limit of sets
> > but the limit of cardinalites of sets.
>
> You are wrong. Look it up, in case you have forgotten. For instance
> here
> http://www.hs-augsburg.de/~mueckenh/GU/GU12c.PPT#403,25,Folie 25
Don't ever look up anything in a source by WM unless you can agree with
WM's WMytheology.
>
> The sets are {2, 1}, {2}, {4, 3, 2}, {4, 3}, {6, 5, 4, 3}, {6, 5,
> 4}, ...
> This is a sequence of sets. If the digits in their order are
> interpreted as a number (the usual and most natural way to do), then
> the limit is oo.
As far as I can see, each n in |N will occur in only finitely many of
the sets in that sequence, so should not be the limit be the empty set?
Both the limsup and liminf for that sequence of sets works out to {}.
http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#Sequences_
of_sets
> The cardinality of the indexes of this limit in
> analysis is aleph_0.
>
> The sequence of cardinalities is 2, 1, 3, 2, 4, 3, ... The limit of
> this sequence is aleph_0 too.
>
> > The limit you calculate is not a limit set, nor the
> > cardinality of a limit set.
>
> Analysis shows that the cardinality of the digits is 1 + logn. This
> does not break down for n = oo.
Since we are talking about a sequence of sets, not a sequence of
numbers. "1+log(n)" is irrelevant.
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