
Matheology § 200
Posted:
Jan 26, 2013 3:24 AM


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, but set theory needs the feature of completeness for the enumeration of all rational numbers. If not all could be enumerated, the same cardinality of Q and N could not be proved.
However recently a formal contradiction with the corresponding limit of real analysis could be shown: http://planetmath.org/?op=getobj&from=objects&id=12607 and here http://www.hsaugsburg.de/medium/download/oeffentlichkeitsarbeit/publikationen/forschungsbericht_2012.pdf on p. 242  244 The limit of remaining unwritten days is infinite according to analysis whereas Fraenkel's story is approved by set theory.
Nevertheless, matheologians deny every contradiction. One of them, Michael Greinecker (as a selfproclaimed watchdog, and bouncer in MathOverflow http://meta.mathoverflow.net/discussion/1296/crankposttoflagasspam/#Item_0 an interbreeding of Tomás de Torquemada and Lawrenti Beria) stated: "there is no contradiction. Just a somewhat surprising result. And there is no a apriory reason why one should be able to plug in cardinal numbers in arithmetic formulas for real numbers and get a sensible result." This means the finite positive integers differ significantly from the finite positive cardinals or, as Cantor called them, the finite positive integers. Well, maybe, sometimes evolution yields strange results. But if they differ, how can set theory any longer be considered to be the basis of analysis?
Regards, WM
For recent paragraphs of matheology look here: http://www.hsaugsburg.de/~mueckenh/KB/Matheology.pdf

