Bruno Marchal wrote: > Hi, > > I have come back from Bergen (it was very nice) and I have read the > last posts and I will make some comments in order. > > Peter D. Jones said some time ago, after I said that I will identify > "(digital) machines" with number; he said: > > "You can't". > > Of course I can. This is a key point, and it is not obvious. But I can, > and the main reason is Church Thesis (CT). Fix any universal machine, > then, by CT, all partial computable function can be arranged in a > recursively enumerable list F1, F2, F3, F4, F5, etc.
I have never denied that you can count or enunumerate machines or algorithms, i.e. attach unique numerical labels to them. The problem is in your "Fix any universal machine". Given a string of 1's and 0s *wihout* a universal machine, and you have no idea of which algorithm (non-universal machine) it is. Two things are only identical if they have *all* their properties in common (Leibniz's law). But *none* of the propeties of the "machine" are detectable in the number itself. (Yopu can also count the even numbers off against the odd numbers , but that hardly means that even numbers are identical to odd numbers!) > It is the list of > the Fi, which has this fundamental and amazing property that it is > close for the diagonalization operation. I have explain this at length > in some posts to George and Tom. The identification between number and > machine is similar to the geometric identification of real numbers and > points once a coordinate system has been fixed. If you prefer I should > have said "associate" instead of "identifying". I do prefer! >In computer science, a > fixed universal machine plays the role of a coordinate system in > geometry. That's all. With Church Thesis, we don't even have to name > the particular universal machine, it could be a universal cellular > automaton (like the game of life), or Python, Robinson Aritmetic, > Matiyasevich Diophantine universal polynomial, Java, ... rational > complex unitary matrices, universal recursive group or ring, billiard > ball, whatever. Ye-e-es. But if all this is taking place in Platonia, the only thing it *can* be is a number. But *that* number can't be associated with a computaiton by *another* machine, or you get infinite regress. >Then just list the programs describable in the language > of that machine to get the Fi. The domain of the Fi gives the Wi which > can be shown to be the mechanically generable sets (of numbers, or > entities nameable, associable or identifiable with numbers in some > context like the partial recursive (computable) functions). > > > David Nyman wrote: > > > [Scene: Night-time. Fathers Ted and Dougal are in bed. > > > > Ted: "Dougal, that's a great idea! Can you tell me more?" > > Dougal: "Whoa, Ted - I want out! I can't take the pressure."] > > > I love them :) > > > Bruno > > PS For a while I will let Colin and David continue their discussion > before interfering. I have other comments but will regroup them for > making minimal the number of posts. Just try not to confuse > computability and provability (in a formal system or by a machine). > Computability is an absolute notion (with CT), but provability is a > relative (with respect to a machine) notion. Put in another way: > computations admits a universal dovetailer which generates and run all > computations, but there is no universal dovetailer for proofs. By > Godel's theorem for any proof system (with checkable proofs) you can > build a richer proof system. Without this all 'hypostases' would > collapse, for example, and the interview with a universal machine would > be ... infinitely boring, and probably unrelated to both quanta and > qualia. > Not also that the relative aspect of provability does not prevent the > finding of universal feature of provability (like obeying G and G* for > example). Note also that most provability systems (like RA and PA or > ZF) are universal computers, but still only relative (and different) > theorem provers. Seen as a universal machine, RA and PA and ZF can > simulate each others. As provability systems, ZF extends properly PA > which extends properly RA. > ZF and PA are lobian machines. RA isn't. > > > 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 firstname.lastname@example.org To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---