
Re: Matheology § 200
Posted:
Jan 26, 2013 10:18 AM


On Jan 26, 1:54 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > On 26 Jan., 13:06, William Hughes <wpihug...@gmail.com> wrote: > > > > > > > > > > > On Jan 26, 12:52 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > On 26 Jan., 12:31, William Hughes <wpihug...@gmail.com> wrote: > > > > > On Jan 26, 9:24 am, WM <mueck...@rz.fhaugsburg.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 counterintuitive, > > > > > Correct. But counterintuitive 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.
> 3) The limit of the number of digits in analysis: oo
This is the limit of the cardinals.
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.

