FOM: Goedel: truth and misinterpretations

[Martin Davis]

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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