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.