[FOM] Truth theories and the conservativity argument

[praatika@mappi.helsinki.fi]

henri galinon <henri.galinon at libertysurf.fr> wrote:

> Well, I think your right, and it seems that incompletness says that  
> it will never happen.  On the other hand it seems that we have a  
> clear concept of what natural numbers are (this is not to say that we  
> know all there is to know about them), and it seems also that PA (and  
> extensions) is not faithfull or complete (in a sense) relative to  
> that knowledge.

There are two issues. One is that no first order theory (not even the 
complete one) can rule out non-standard models, with non-standard 
("infinite") numbers. Nevertheless, we seem to understand the notion of 
natural number. The difference, I think, is in our intuitive grasp of the 
notion of finiteness. The other issue is whether all we really know/can 
know about natural numbers is exhausted by some formal system, when its 
symbols are interpreted standardly. I happen to believe that the answer is 
positive (assuming that the question is at all meaningful, which I 
sometimes doubt). But these two issues seem to be different, and 
independent. 


> My tentative remark was : there is something important we now about  
> the natural numbers, namely that the w-rule is valid : had we proved  
> for each number n that phi(n), w e would have proved that for all  
> natural numbers phi.   This knowledge (the validity the w-rule) does  
> not help in proving new theorems from old ones, but it seems most  
> important in guiding us when we want to know wether the theory we're  
> working in/on is a theory of the natural numbers or not.
>   For example, consider again the difference between formal theories  
> and arithmetic. Groups are just whatever is specified by the axioms  
> of group-theory. Add some axioms to it and you have another theory,  
> ring-theory, say. Things are really different for arithmetic. We may  
> write down PA as a theory of arithmetic, but still we can add some  
> axioms to it and still be doing arithmetic, while adding some others,  
> this is no more arithmetic in any straighforward sense : when working  
> in PA+non-Con(PA), that is in an w-inconsistent theory, you're not  
> doing arithmetic anymore.( We can't anymore read "Ax phi(x)" as "for  
> every natural numbers phi", or so it seems).

Yes, but here we are talking about w-consistency, which is just a kind of 
soundness statement, which can be easily formalized, though obviously not 
provable in PA.


> Take whatever finitary-ruled r.e. system S of arithmetic, and you'll  
> have that  for some phi, S proves phi(0), phi(1), etc. and does not  
> prove that for all natural numbers phi(0), while *we*, looking at the  
> system from outside and through a representation of it (an emulation  
> of the so-called "semantic" point of view I suppose ?), we can see  
> that the conclusion is implied. Thus it seems that from outside, the  
> finite-proving-system will always seems defective. Is it right ? And  
> if it is, what is the significance of this ?

I am afraid that you are here re-inventing something like Lucas-Penrose 
argument against mechanism. The problem is that we can *not* always see 
that "for all natural numbers x, phi(x)"; for example, if S is a very 
complicated and strong system, and the generalization in question is the 
consistency statement for S. Focusing on relatively weak PA gives a 
misleading impression that we can. 

You can read more about this in my paper on the philosophical relevance of 
incompleteness theorems (Revue Internationale de Philosophie no 234, 4-
2005; a penultimate draft also in my homepage. see p. 9-

> ok, thanks. And by the way, let me add that it is a paper of Parsons  
> ( "Informal axiomatization, Formalization and the concept of truth",  
> maybe the one you're talking about ?) that reminded me of  
> formalization being a process. I've been perplexed since.

I think Parsons made the point I mentioned in "Sets and Classes"... 
 
> By faithfull arithmetic, I had in mind something like PA, minus  
> induction, plus w-rule (not just complete arithmetic with usual rules).
> There, since for each natural number we can prove Sent(x)---> (true 
> (x) or true(not-x), it seems that the w-rule gives you the  
> generalization.


Right, I see... But do you think that simple complete theory is not 
faithful? (I am not sure if I've got you notion of faithfulness...)

Best, Panu


Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy 
University of Helsinki
Finland


E-mail: panu.raatikainen at helsinki.fi
 
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm