FOM: Goedel: truth and misinterpretations

[Kanovei]

> From martin at eipye.com Wed Oct 25 20:51:03 2000
> X-Sent: 25 Oct 2000 18:49:20 GMT
> 
> At 07:28 PM 10/25/00 +0200, Kanovei wrote:
> >These sentences are obtained as follows.
> >There are two sets of sentences, say X \subseteq Y,
> >and we know that X is r.e. but Y is not r.e., hence, the
> >difference Y - X is infinite, and the "sentences" are just
> >those in the difference. There is no one concrete sentence
> >there, all we know is that they do exist in plentitude.
> 
> This is just not true. For any suitable theory (e.g. any axiomatizable 
> consistent extension of Robinson's Q) one can exhibit an explicit 
> polynomial P with integer coefficients such that the equation P=0 has no 
> solutions in natural numbers, but that fact is not provable in the given 
> theory. In fact, one can even manage this so that the only change from one 
> theory to another is in the value of a single parameter. Of course, P will 
> not be very pretty and some of the numbers will be very large.
 
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.  

V.Kanovei