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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, 17 Aug 1998, Marcus Brinkmann wrote:

} 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.

Almost there!  He used a more complicated numbering system called Godell
numbering, and using this showed that you can always costruct a theorem G
theat syas "The theorem with the number n cannot be proven," such that
this theorem in fact has the number n.  The negation of this is ~G, "The
theorem with the number n can be proven," but if this negation can be
proven, then G (since it has the number n) can be proven.  Thus if ~G is
proveable, G is proveable, a contradiction, so in fact G cannot be proven.
But this means that G is in fact true, so the number system is not
complete.

What does this mean with respect to the original topic?  I think (and
could very well be wrong) the claim was you cannot expect Debian to have
a license system that is complete, in the sense that it can't have a set
of axiomatic licenses which handle themselves and all other packages.
This may be correct, since the key features to Goedell's Theorem, and
Russell's and related paradoxes is that you must be able to have
self-reference and universality.  For instance, if the GPL applied to
itself, people would be able to rewrite the license.  So you might in fact
need to have some sort of non-free core of Debian which contains just the
licenses and nothing else.  It at least requires some thought.

--
f00f
ashinn1@umbc.edu


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