Bruno Marchal wrote:
>
> On 10 Jun 2009, at 02:20, Brent Meeker wrote:
>
>
>
>> I think Godel's imcompleteness theorem already implies that there must
>> be non-unique extensions, (e.g. maybe you can add an axiom either that
>> there are infinitely many pairs of primes differing by two or the
>> negative of that). That would seem to be a reductio against the
>> existence of a hypercomputer that could decide these propositions by
>> inspection.
>
>
> Not at all. Gödel's theorem implies that there must be non-unique
> *consistent* extensions. But there is only one sound extension. The
> unsound consistent extensions, somehow, does no more talk about
> natural numbers.
OK. But ISTM that statement implies that we are relying on an intuitive notion
as our conception of natural numbers, rather than a formal definition. I guess I
don't understand "unsound" in this context.
>
> Typical example: take the proposition that PA is inconsistant. By
> Gödel's second incompletenss theorem, we have that PA+"PA is
> inconsistent" is a consistent extension of PA. But it is not a sound
> one. It affirms the existence of a number which is a Gödel number of a
> proof of 0=1. But such a number is not a usual number at all.
Suppose, for example, that the twin primes conjecture is undecidable in PA. Are
you saying that either PA+TP or PA+~TP must be unsound? And what exactly does
"unsound" mean? Does it have a formal definition or does it just mean
"violating our intuition about numbers?"
Brent
>
> An oracle for the whole arithmetical truth is well defined in set
> theory, even if it is a non effective object.
>
> Bruno
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>
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Received on Wed Jun 10 2009 - 11:00:52 PDT