Date: Jan 26, 2013 5:33 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 200

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.

Note that Greinacher's objection is simply wrong because for every
natural number we can set the same number as an ordinal number. Then
we have a set of ordinal numbers. And that set has with certainty,
according to set theory, a cardinal number.

Regards, WM