FOM: Goedel: truth and misinterpretations
Martin Davis
martin at eipye.com
Wed Oct 25 16:17:51 EDT 2000
At 09:43 PM 10/25/00 +0200, Kanovei wrote:
>It is understandable that any non-0 set of natural numbers has a
>concrete element, say the least number.
>But clearly by "concrete" I meant something defined not as
>"the least polynomial of some kind", be it even of degree 1000^1000,
>but really meaningful mathematical statement, like CH, here
>"meaningful" means that it expresses a mathematical property of
>some well defined mathematical meaning. From your answer it is not clear
>is your P of that kind.
I refer you to Yuri Matiyasevich's book HILBERT'S TENTH PROBLEM, MIT Press
1993,
page 70. (In the original Russian edition, page 64). There you will find an
entirely explicit system of equations that is "universal". This implies
that given any suitable theory, by choosing appropriate values for
parameters, a system will be obtained that has no solutions but such that
this fact is unprovable in the given system.
To be clear: for a theory like ZFC the values of the parameters will be
very large. But the algorithm for computing them is perfectly explicit.
Martin
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
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http://www.eipye.com
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