Date: Jan 26, 2013 3:24 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 200



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, 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.hs-augsburg.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 self-proclaimed watchdog, and bouncer in
MathOverflow
http://meta.mathoverflow.net/discussion/1296/crank-post-to-flag-as-spam/#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.hs-augsburg.de/~mueckenh/KB/Matheology.pdf