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 
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?"


> An oracle for the whole arithmetical truth is well defined in set  
> theory, even if it is a non effective object.
> Bruno
> > 

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