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/ > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
