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:
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)
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. By doing so, such machines infer the existence of
something transcendenting (if I can say) themselves.
OK?
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Aug 31 2007 - 05:40:22 PDT