Virgil
Posts:
4,661
Registered:
1/6/11
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Re: Matheology � 200
Posted:
Jan 26, 2013 6:08 PM
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In article
<992831a0-de7b-4739-9035-1a277b97d17c@r14g2000yqe.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 26 Jan., 23:24, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <fde5d8dc-6b0f-44ec-aaec-a585f1bb5...@f6g2000yqm.googlegroups.com>,
> >
> >
> >
> >
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 26 Jan., 13:06, William Hughes <wpihug...@gmail.com> wrote:
> > > > On Jan 26, 12:52 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > On 26 Jan., 12:31, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > > > On Jan 26, 9:24 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > > > Matheology 200
> >
> > > > > > > We know that the real numbers of set theory are very different
> > > > > > > from
> > > > > > > the real numbers of analysis, at least most of them, because we
> > > > > > > cannot
> > > > > > > use them. But it seems, that also the natural numbers of analysis
> > > > > > > 1,
> > > > > > > 2, 3, ... are different from the cardinal numbers 1, 2, 3, ...
> >
> > > > > > > This is a result of the story of Tristram Shandy, mentioned
> > > > > > > briefly in
> > > > > > > 077 already, who, according to Fraenkel and Levy ["Abstract Set
> > > > > > > Theory" (1976), p. 30] "writes his autobiography so pedantically
> > > > > > > that
> > > > > > > the description of each day takes him a year. If he is mortal he
> > > > > > > can
> > > > > > > never terminate; but if he lived forever then no part of his
> > > > > > > biography
> > > > > > > would remain unwritten, for to each day of his life a year
> > > > > > > devoted to
> > > > > > > that day's description would correspond."
> >
> > > > > > > This result is counter-intuitive,
> >
> > > > > > Correct. But counter-intuitive does not mean contradictory.
> > > > > > Outside of Wolkenmeukenheim, the limit of cardinalites is not
> > > > > > necessarily equal to the cardinality of the limit.-
> >
> > > > > Obviously you have not yet understood?
> > > > > In my proof the cardinality of the limit in set theory and the
> > > > > cardinality of the limit in analysis are different.
> >
> > > > Nope In analysis you take the cardinalities
> > > > of a sequence of sets, i.e. take a sequence of numbers,
> > > > and calculate a limit. However, this limit is not the
> > > > cardinality of a limit set. In anylysis you calculate
> > > > the limit of the cardinalities not the cardinality of
> > > > the limit.-
> >
> > > In order to correct your mistake, here are the details. In my proof we
> > > have:
> > > 1) The limit of the cardinals in set theory: aleph_0
> > > 2) The cardinality of the limit in set theory: 0
> > > 3) The limit of the number of digits in analysis: oo
> >
> > Then those "numbers of digits" cannot be cardinal numbers, but real
> > numbers.
> >
> > > 4) The number of digits of the limit in analysis: oo
> >
> > Then those "numbers of digits" cannot be cardinal numbers, but real
> > numbers.
>
> On the contrary. The digits form a set of positions indexed by natural
> ordinal numbers. The number of the set of ordinal is a cardinal
> number.
But "oo" is a part of the extended reals, not a cardinal. The cardinal
is aleph_0, the ordinal is omega, but neither of then is the limit of
any real function.
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