Bruno Marchal wrote:
> I said to Brent,
> 
> 
> Le 31-août-07, à 11:00, Bruno Marchal a écrit :
> 
>> So, no, I don't see why you think my objection is a non-sequitur. It
>> seems to me you are confusing arithmetic and Arithmetic, or a theory
>> with his intended model.
> 
> 
> Brent, rereading your post I think there is perhaps more than one 
> confusion. I cannot really be sure, because your wording "arithmetic" 
> is ambiguous.
> 
> Let me sum up by singling out three things which we should not be 
> confused:
> 

OK.  I don't think I'm confused about them.
 
> 
> 1) A theory about numbers/machines, like PA, ZF or any lobian machine.  
> (= finite object, or mechanically enumerable objet)
> 
> 2) Arithmetical truth (including truth about machine).   (infinite and 
> complex non mechanically enumerable object)

Is this the set of all (countably infinite) true propositions about the natural 
numbers?  Is the existence of this set a matter of faith?

> 
> 3) A meta-theory of PA (that is a theory about PA)  (again a 
> mechanically enumerable object)
> 
> 
> 
> Only a meta-theory *about* PA, can distinguish PA and arithmetical 
> truth. But then Godel showed that sometimes a meta-theory can be 
> translated in or by the theory/machine. Rich theories/machine have 
> indeed self-referential abilities, making it possible for them to guess 
> their limitations. 

Are there not infinitely many Godel numbering schemes.

>By doing so, such machines infer the existence of 
> something transcendenting <transcending> (if I can say) themselves.
> 
> OK?

OK.

Brent Meeker

> 
> Bruno
> 
> 
> 
> http://iridia.ulb.ac.be/~marchal/
> 
> 
> > 
> 
> 


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