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. 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". 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. 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 [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 -~----------~----~----~----~------~----~------~--~---
