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