Re: [OFFTOPIC] Goedel and the mathematic (was: Re: Copyright from the lcs-projekt!? [dwarf@polaris.net: Re: First cut at testing and validation]
On Mon, Aug 17, 1998 at 03:51:14PM -0400, Steve Dunham wrote:
> Marcus Brinkmann <Marcus.Brinkmann@ruhr-uni-bochum.de> writes:
>
> > It's a pity that it is not so easy. Instead Goedels theorem is pretty
> > weired, and you can't give any easy example of such a theorem.
>
> IIRC, Goedel used the old "this statement is false" trick.
Yes, in a strange way, because you have to make mathematic saying things
about mathematic itself. So you assign a three digit number to every
character, and code theorems in numbers. 1=1 could be 001 666 001, and so
on. You could do this with ascii, too. Then you put this numbers into other
theorems, and then you are doing some other magic, and you get there. Not
that I understand it in detail yet. But this is how it works.
> > maybe an example is the theorem, that there is no set with a
> > cardinality between the cardinality of the set of real numbers and
> > the set of all subsets of the set of real numbers (is somebody still
> > reading here :). At least it was proved that this theorem can
> > neither be proved nor disproved.
>
> It was proved that this axiom was independent of the rest of the
> axioms of set theory (by Cohen circa 1926, I believe - I skimmed the
> article a few years ago). IIRC, the idea here was to construct two
> different set theories, one of which contained the normal axioms plus
> that continium hypothesis and the other which contained the normal
> axioms plus the negation of the continium hypothesis. The
> construction is done inside set theory without the additional
> hypothesis.
Ah, if it is independently it is not an example of Goedels theorem, I think.
Fermat's theorem was my favourite candidate, but now it is proofed... well,
I give up, I don't know an example from praxis.
> (Some details could be incorrect. The main focus of my study was
> Group Theory and Algebra, not set theory or logic. Anyways, this
> stuff, while profoundly interesting to me, is off topic, and perhaps a
> bit dry.)
:) We are talking about math, aren't we :)
It is a very interesting topic, but nothing serious in the day work of a
mathematician. Thank you for the clarification,
Marcus
--
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