On 10 Jun 2009, at 20:00, Brent Meeker wrote:

> 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.

You are right. We have to rely on our intuition. After Gödel we know  
that even our use of formal system has to be based on our intuition of  
the natural number, and we don't have any fixed and complete  
formalization of the natural numbers.

> I guess I
> don't understand "unsound" in this context.

Unsound means false in the structure (N, +, *). We can define this  
"formally" in a theory which is richer than PA, like ZF set theory. Of  
course we can't define "sound" for ZF. Intuition just cannot be  
avoided. Today we can understand how machine can develop intuition,  
despite this cannot be formalized.

>> 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?

Yes. That is why, unlike in Set Theory, nobody seriously doubt about  
the excluded middle principle in the structure (N,+,*).

> And what exactly does
> "unsound" mean?

It really means false in the structure (N,+,*).

> Does it have a formal definition or does it just mean
> "violating our intuition about numbers?"

As I said, you can formalize the notion of soundness in Set Theory.  
But this adds nothing, except that it shows that the notion of  
soundness has the same level of complexity that usual analytical or  
topological set theoretical notions. So you can also say that  
"unsound" means violation of our intuitive understanding of what the  
structure (N,+,*) consists in. We cannot formalize in any "absolute  
way" that understanding, but we can formalize it in richer theories  
used everyday by mathematicians.



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