Date: Jan 26, 2013 4:51 PM
Subject: Re: Matheology § 200
On 26 Jan., 16:18, William Hughes <wpihug...@gmail.com> wrote:
> On Jan 26, 1:54 pm, 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
> Note that the cardinality of the limit is
> not equal to the limit of the cardinals.
That is the most ridiculous nonsense that is inherent in matheology,
but, of course, it is required to maintain it. If there is any
meaningful limit of a sequence of sets, then this limit is a set and
as such has a cardinality. If both diverge, then we have the proof
that the limit set is not a meaningful notion.
Nevertheless, I did not use this sover fact in order to show you, that
not even nonsense is sufficient to maintain set theory.
> > 3) The limit of the number of digits in analysis: oo
> This is the limit of the cardinals.
This is the improper analytical limit L of the sequence of numbers.
This are real numbers, in the opinion of some cranks like Greinacher
they differ from cardinals. You seem to be of different opinion. By
the way it is astonishing if not surprising how many different
opinions matheologians entertain.
> Since the limit of the cardinals does not equal the
> cardinality of the limit, there is no reason the number
> of digits in the cardinality of the limit should be the
> same as the number of digits in the limit of the cardinals
Perhaps that would be possible in set theory, but not in analysis. In
analysis we have the set of digits of the limit L and we have the set
of indexes of these digits. The set of indexes is 1 + the decadic
logarithm of L. Never heard about that fact? But even if you don't
know, be sure every mathematician knows.