FOM: Goedel: truth and misinterpretations
Kanovei
kanovei at wmwap1.math.uni-wuppertal.de
Wed Oct 25 15:43:09 EDT 2000
> 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
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