Re: request for ideas

From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 08/24/04 

Date: Tue, 24 Aug 2004 14:30:03 +0000 (UTC)

cafeinst@msn.com (Craig Feinstein) wrote in message news:<cgd62a$no8$1@news.ks.uiuc.edu>...
> I am planning to write a paper which surveys mathematical results that
> show that the old "axiom->proof->theorem" way of doing mathematics
> does not always yield complete information about mathematics. The
> prime example of this (which started it all) is Godel's Incompleteness
> Theorem, but there has been a lot of work in this area since then.
>
> For instance, Gregory Chaitin has an incompleteness theorem which
> shows conclusively that a certain number which he calls Omega, which
> is really the probability that a computer program halts (defined in a
> way that makes sense), is a random number - which implies that there
> is no finite axiom system that can yield all of the bits of Omega. He
> concludes from all of his work that sometimes one has to simply
> perform experiments in mathematics and form conclusions from the
> experiments without being absolutely certain that the conclusions are
> correct.
>
> It is these types of very original ideas that I am looking for to put
> in my paper, that there are some problems out there that are so
> difficult for us to get a grip on that we might have to approach them
> like a chemist approaches chemistry, never being 100% sure that his or
> her theories are always correct.
>
> Anyone who knows of results like these or has done work in this area
> or has original ideas is welcome to respond to me on usenet or if you
> want, you can email me directly.

Have you looked into Super Omegas? Might be slightly relevant to your inquiry.

http://www.idsia.ch/~juergen/kolmogorov.html

Best Regards, 

--
Eray Ozkural

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