Le 16-mai-06, à 17:31, Tom Caylor a écrit :

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> > Bruno Marchal wrote: >> >> Now I think I should train you with diagonalization. I give you an >> exercise: write a program which, if executed, will stop on the biggest >> possible natural number. Fairy tale version: you meet a fairy who >> propose you a wish. You ask to be immortal but the fairy replies that >> she has only finite power. So she can make you living as long as you >> wish, but she asks precisely how long. It is up too you to describe >> precisely how long you want to live by writing a program naming that >> big (but finite) number. You have a limited amount of paper to write >> your answer, but the fairy is kind enough to give you a little more if >> you ask. >> You can ask the question to very little children. The cutest answer I >> got was "7 + 7 + 7 + 7 + 7" (by a six year old). Why seven? It was the >> age of his elder brother! >> >> Hint: try to generate an infinite set S of more and more growing and >> (computable) functions, and then try to diagonalize it. S can be >> {addition, multiplication, exponentiation, .... (?)....}. More hints >> and answers later. I let you think a little bit before. (Alas it looks >> I will be more busy in may than I thought because my (math) students >> want supplementary lessons this year ...). >> >> Hope this can help; feel free to make *any* comments. >> >> Remember that if all this is too technical, you can also just read >> Plotinus and the (neo)platonist which, accepting comp or weaker form >> of >> Pythagorism, do have a tremendous advance on most materialist of >> today >> ... I think it could even provide more light on the practical death >> issue. The role of G and G* is just to get the math correct for some >> notion of quantifying the 1-person probabilities. >> >> Bruno >> >> (*)SANE paper html: >> http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.htm >> SANE paper pdf: >> http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf >> >> http://iridia.ulb.ac.be/~marchal/ > > In keeping with the incremental interactive process, here is a first > guess. You simply start naming off the natural numbers in order. > After naming each number you say, "That's not the largest possible > natural number", or "That's not how long I want to live." This > statement seems to play the role of diagonalization. But it is not a finite process. The fairy asks you to give a well defined number, in a finite time. > The process I've > just described can be defined with a finite number of symbols (I just > did it). Thus, in a way you can say I've just "named" the largest > natural number. You have just given a procedure for building a bigger number from any number. The function which send n on n+1 does that trick. But the fairy asks you for a number, not a function. > > First question: Is this the same as Douglas Hoftstadter's supernatural > numbers (in his book Godel, Escher, Bach)? I have read that quite good book, but I don't have it under the hand, and I don't think the big number problem is related to its supernatural numbers. > It seems the only way to > really understand his book is to read it cover-to-cover (because of all > the acronyms and his defining ideas with stories, etc.). I wish I > would have read it cover-to-cover when I was young and had lots of time > on my hands (and lots of spare brain cells).... or may I can just start > reading it cover-to-cover now and simply ask the fairy for more > (quality) time as I need it. Hofstadter wrote a good book, yes, but on the pedagogical side it does not help so much by diluting the proof of Godel's theorem in many interesting themes (Bach, Escher, AI, etc.). > > Second question: When we switch over from natural numbers to "length > of life", it seems we need to specify "units of time" in order for the > specification of length of life to have any meaning. You are right. Let us take *years". > This crosses us > over into the realm of meaning. Length of life has no meaning apart > from an assignment of meaning or quality to the events that make up > life. There seems to be some kind of diagonalization going on here (or > perhaps transcendence, independent from any diagonalization argument). > What good is MWI "immortality" (or any kind of "immortality") if the > infinite sum of (units of time) * (quality or meaning) adds up to some > finite number? Is it really immortality? Life is more than existence. In the big number problem, immortality is not proposed by the fairy, what is proposed is just a long but finite life. Here too the quality is important. To stay a very long time awake in a coffin is not pleasant. Also, to stay alive for a very long period makes almost no sense if your brain is limited in space (bounded finite machine eventually cycle when running a long time. Do you see why?). The big number problem has been tackled by Archimedes. He got the number 10^63. This is remarkable if you recall the very bad notation for number used at that time. Today 10^63, although very big (the universe seems to exist since less than 10^17 sec and I think that number has shrinked recently) seems rather little. It is little than a Googol (10^100) and a Googolplex is far bigger: 10^[a googol]. But really we will end up with far bigger (but still finite) number once we will diagonalise on set of growing functions (soon). Meanwhile just a few questions to help me. They are hints for the problem too. Are you familiar with the following "recursive" program for computing the factorial function? fact(0) = 1 fact (n) = n * fact(n - 1) Could you compute "fact 5", from that program? Could you find a similar recursive definition (program) for multiplication (assuming your machine already know how to add)? Could you define exponentiation from multiplication in a similar way? Could you find a function which would grow more quickly than exponentiation and which would be defined from exponentiation like exponentiation is defined from multiplication? Could you generalize all this and define a sequence of more and more growing functions. Could you then diagonalise effectively (= writing a program who does the diagonalization) that sequence of growing functions so as to get a function which grows more quickly than any such one in the preceding sequence? I have no precise idea of your background, so ask if there is anything unclear. Anyone can ask. 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 everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---