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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Received on Fri Aug 31 2007 - 12:07:30 PDT