Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Le 30-mai-06, à 19:13, Tom Caylor wrote : From what you've said about dovetailing before, you don't have to have just a single sequence in order to dovetail. You can jump among multiple sequences. I have yet to understand how you could dovetail on something that is not effective. That seems to be where you're going. I guess we have to get to non-effective diagonalization first. It is true that we can dovetail on multiple sequences, once we can generate their codes, like all the sequence of growing computable functions we have visited. But none of them are universal. Once I can name a sequence of growing function, and thus can program an enumeration of their codes, I will be able to diagonalize it, showing it was not universal. --- Quentin Anciaux wrote: I think dovetailing is possible because the dovetailer only complete sequences at infinity. So when you construct the matrice on which you will diagonalize, you are already diagonilizing it at the same time. Example: when you have the first number of the first growing function, you can also have the first number of the diagonalize function (by adding 1) and the first number of the diagonalize*diagonalize function and ... ad recursum. By dovetailing you execute in fact everything in parallel but all infinites sequences are only completed at infinity. Same answer as the one for Tom. You can diagonalize only on the transfinite sequences up to a nameable ordinal, and clearly this cannot be closed for diagonalization. Even in the limit, the transfinite construction will fail to name some computable growing function. --- Hal Finney wrote: The dovetailer I know does not seem relevant to this discussion about functions. It generates programs, not functions. So does our sequence of growing functions. They are given by the programmable generation of the code of the growing function. The same for the diagonalization. For example, it generates all 1 bit programs and runs each for one cycle; then generates all 2 bit programs and runs each for 2 cycles; then generates all 3 bit programs and runs each for 3 cycles; and so on indefinitely. (This assumes that the 3 bit programs include all 2- and 1-bit programs, etc.) In this way all programs get run with an arbitrary number of cycles. Close :) - Quentin Anciaux comments on Hal Finney: In fact it is relevant because of this : - Bruno showed us that it is not possible to write a program that will list sequentially all growing functions. ...that will list sequentially all computable growing functions. Right. - But the dovetailer will not do it too, but what it will do instead is generate all program that list all growing functions. Mmh... So the dovetailer will not list all the growing function but will generate (and execute in dovetailing) the infinite sequence of programs that generate all of them. The shadow of the truth, but what you describe would still be diagonalized out. (Or you are very unclear, to be sure, you will tell me later ... or you will not tell me later :) -- Jesse Mazer (on Hal Finney): I was being a little sloppy...it's true that a non-halting program would not be equivalent to a computable function, but I think we can at least say that the set of all computable functions is a *subset* of the set of all programs. Key remark. As for the programs taking input or not, if you look at the set of all programs operating on finite input strings, each one of these can be emulated by a program which has no input string (the input string is built into the design of the program). Actually this is a key remark too, but it will be needed only later (I recall that I am trying to explain a the missing link between computer science and Smullyan's heart of the Matter in FU, after a question by George Levy). This key remark is related to a simple but basic theorem in computer science known as the SMN theorem, S for substitution and M and N are parameter, and grosso modo the theorem says that you can programs can be mechanically parametrized by freezing some of their inputs. So for any computable function, there should be some member of the set of all halting programs being run by the dovetailer that gives the same output as the function, no? Yes. Do you see or know or guess the many consequences? --- George Levy wrote: To speak only for myself, I think I have a sufficient understanding of the thread. Essentially you have shown that one cannot form a set of all numbers/functions because given any set of numbers/functions it is always possible, using diagonalization, to generate new numbers/functions: the Plenitude is too large to be a set. This leads to a problem with the assumption of the
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Le Mercredi 31 Mai 2006 00:21, Hal Finney a écrit : The dovetailer I know does not seem relevant to this discussion about functions. It generates programs, not functions. For example, it generates all 1 bit programs and runs each for one cycle; then generates all 2 bit programs and runs each for 2 cycles; then generates all 3 bit programs and runs each for 3 cycles; and so on indefinitely. (This assumes that the 3 bit programs include all 2- and 1-bit programs, etc.) In this way all programs get run with an arbitrary number of cycles. In fact it is relevant because of this : - Bruno showed us that it is not possible to write a program that will list sequentially all growing functions. - But the dovetailer will not do it too, but what it will do instead is generate all program that list all growing functions. So it will first generate the programme that create the first sequence and also the pogram that create the sequence composed of diagonalisation of the first and so on... it can because program are countable because they are mapped to N. So the dovetailer will not list all the growing function but will generate (and execute in dovetailing) the infinite sequence of programs that generate all of them. --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Hal Finney wrote: Jesse Mazer writes: The dovetailer is only supposed to generate all *computable* functions though, correct? And the diagonalization of the (countable) set of all computable functions would not itself be computable. The dovetailer I know does not seem relevant to this discussion about functions. It generates programs, not functions. For example, it generates all 1 bit programs and runs each for one cycle; then generates all 2 bit programs and runs each for 2 cycles; then generates all 3 bit programs and runs each for 3 cycles; and so on indefinitely. (This assumes that the 3 bit programs include all 2- and 1-bit programs, etc.) In this way all programs get run with an arbitrary number of cycles. These programs differ from functions in two ways. First, programs may never halt and hence may produce no fixed output, while functions must have a well defined result. And second, these programs take no inputs, while functions should have at least one input variable. What do you understand a dovetailer to be, in the context of computable functions? Hal Finney I was being a little sloppy...it's true that a non-halting program would not be equivalent to a computable function, but I think we can at least say that the set of all computable functions is a *subset* of the set of all programs. As for the programs taking input or not, if you look at the set of all programs operating on finite input strings, each one of these can be emulated by a program which has no input string (the input string is built into the design of the program). So for any computable function, there should be some member of the set of all halting programs being run by the dovetailer that gives the same output as the function, no? Jesse --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Le 30-mai-06, à 03:14, Tom Caylor a écrit : OK. I see that so far (above) there's no problem. (See below for where I still have concern(s).) Here I was taking a fixed N, but G is defined as the diagonal, so my comparison is not valid, and so my proof that G is infinite for a fixed N is not valid. I was taking G's assignment of an ordinal of omega as being that it is in every way larger than all Fi's, but in fact G is larger than all Fi's only when comparing G(n) to Fn(n), not when comparing G(Nfixed) to Fi(N) for all i's. Right. OK. I think you are just throwing me off with your notation. Do you have to use transfinite ordinals (omega) to do this? Couldn't you just stay with successively combining and diagonalizing, like below, without using omegas? G(n) = Fn(n)+1 Gi(n) = G(...G(n)), G taken i times Then instead of using more and more additional letters, just add subscripts... H1(n) = Gn(n)+1 H1i(n) = H1(...H1(n)), H1 taken i times H2(n) = H1n(n)+1 H2i(n) = H2(...H2(n)), H2 taken i times Then the subscripts count the number of diagonalizations you've done, and every time you do the Ackermann thing, instead of adding an omega you add another subscript. Then it continues ad infinitum. You can do the Ackermann thing with the *number* of subscripts, i.e. do the Ackermann thing on the number of times you've done the Ackermann thing... etc. This may just be a technical point, but it doesn't seem precise to do very much arithmetic with ordinals, like doing omega [omega] omega, because you're just ordering things, and after a while you forget the computations that are being performed. I can see that it works for just proving that you can continue to diagonalize and grow, which is what you're doing. I just don't want to be caught off guard and suddenly realize you've slipped actual infinities in without me realizing it. I don't think you have. OK. And you are right, I could have done this without mentioning the constructive ordinal. But it is worth mentioning it, even at this early stages, because they will reappear again and again. Note that all those infinite but constructive ordinal are all countable (in bijection with N), and even constructively so. Note also, if you haven't already done so, that omega is just N, the set of natural numbers. I will soon give a more set-theoretical motivation for those ordinals. Actually there is a cute theorem about constructive ordinal. Indeed they are equivalent to the recursive (programmable) linear well-ordering on the natural numbers. Examples: An order of type omega: the usual order on N (0123456...) An order of type omega+1 : just decide that 0 is bigger than any non null natural numbers: 123456 0 It is recursive in the sense that you can write a program FORTRAN (say) capable of deciding it. For example such a program would stop on yes when asked if 48, and no if you ask 08, etc. An order of type omega+omega: just decide that all odd numbers are bigger than the even ones, and take the usual order in case the two numbers which are compared have the same parity: 0246810 . 13579... An order of type omega+omega+omega: just decide that multiple of 3 are bigger than the multiple of two, themselves bigger than the remaining numbers: 15711131417... 0246810... 3691215... Again it should be easy to write a Fortran program capable of deciding that order (that is to decide for any x and y if x y with that (unusual) order. Exercise: could you find an order of type omega*omega? (Hint: use the prime numbers). Those omega names are quite standard. OK. So we haven't left the finite behind yet. It makes intuitive sense to me that you can diagonalize till the cows come home, staying within countability, and still not be done. Otherwise infinity wouldn't be infinite. On the tricky question, it also makes intuitive sense that you can sequence effectively on all computable growing functions. This is because the larger the growing function gets, the more uncovered space (gaps) there are between the computable functions. Any scheme for generating growing functions will also leave behind every-growing uncomputed gaps. Very unmathematical of me to be so vague, but you've already given us the answer, and I know you will fill in the gaps. :) I will. Unfortunately this week is full of duty charges. Meanwhile, I would like to ask George and the others if they have a good understanding of the present thread, that is on the fact that growing functions has been well defined, that each sequence of such functions are well defined, and each diagonalisation defines quite well a precise programmable growing function (growing faster than the one in the sequence it comes from). Just a tiny effort, and I think we will have all we need to go into the heart of the matter, and to understand why comp makes our universe a godelized one in the Smullyan sense. I meant
Re: Smullyan Shmullyan, give me a real example
Bruno, It's been a long holiday weekend here in the US, Bruno, thank you for your reply, and your patience for my responce. Fromconventional math, everything you said was correct, put to me by a co-list friend as .. should I offer you a financial reimbursement for your answer: 1m$ that is: 0m$ :-) . Well, I'm not sending you 1m$, but I will continue commentary. Consider for a moment, the possibility that the entire used ediface of mathematics is an analog of Abbott's Flatland. That though we may think we are 'calculating' in a completely identified domain, that the 'environment' of mathematics is extensive in new ways, and that there are new/different operators needed to access the extended mathematics. Consider G.Cantor. Suppose I said that not only are Aleph0 regions of math calculations, but that addional functions make all of those infinities - calculation accessible. That 'normal math' still applies .. but if and only if .. notated as referencing each frame-of-reference Aleph n. That to segue (equationally transduce) from any Aleph to any Aleph requires additional notations marks, in order to keep separate what Aleph the immediate notation referneces, or, mores into or out of. You remarked that it is absurd to : From (-5)^2 = 5^2 you will not infer that 5 = (-5), right? Actually, what I suggest does -relate- to this question. We make such presumption about positive or absolute value numeration that when we do back-functioning we overlook relations and information that might be inconvenient or cumbersome to treat. Such as differentiating an already integrated operation. That pesky throw-away scalar transform value of (+C) is unceremoniously thrown out because we assume is to be a non-consequential shift- or spatial-translation factor that needn't be considered in mathematical generalization. When we take a square-root, we ignore the minus signs option. When we look at quantum equations, we keep the positive set and ignore the negative set .. which in and of itself is contrary to quantum-math philosophy .. where all variables are included, even if anti-thetical. [M and not-M are concurrent rather than computationally mutually exclusive.] A closing thought for this morning (possible discussion of particulars being left for another day): --from an off-list letter, same list-subject Dear __ , I am broaching a substantially new logic. 1m$ that is: 0m$ -is- a patent absurdity in current math. The version that I came up with essentially restructures the analysis of mathematics as comparisons of dimensions. I did one analysis around the pythagorean theorem that results in a statement b=b^2 for any and all numbers, b. [with the autonomous inclusion of new +/- markers that arrive everytime a dimension is added to or calculated to.] What is missing in math notation are markers that help a person to remember they may be co-navigating several different dimensional fields at the same time, where the left side of an equation is in 'm' dimensions and the right in 'other than m' dimensions, yet the equation is valid. The trouble persists if the notations presume that native dimensionality on both sides is identical. In -that- presumption, the numbers have to match conventional math concepts and no such thing as b=b^2 for any and all numbers, b is allowed or even sensible. It is like trying to have perfect translation among human languages. Not possible. It's only when we convert languages into the larger information network of memes, that 'equal translation' makes sense. That's what I'm doing. Identifying a core realm of 'information' (albeit, mathematical notions, concepts, information) that can transduce as real and valid 'equalities' across the equals sign. When the realm of dimensions is recognized as the larger realm of mathematical memes. If a person doesn't do that shift of consciousness/sensibilities, they'll never 'get it'. ... but I see a shining country of mathematics that no one else seems to recognize .. yet. Jamie Bruno, I know you are still going to treat this line of thought/conversation as sophomoric. A natural reaction. I can assure you it is 'of significance' however. Best Regards, Jamie Rose Bruno Marchal wrote: Le 26-mai-06, à 02:50, James N Rose a écrit : An example at the core of it is a most simplistic definition/equation. 1^1 = 1^0 [one to the exponent one equals one to the exponent zero] To all mathematicians, this is a toss-out absurdity, with no 'real meaning'. n^0 is a convenience tool at best ; n^0 = 1, because 1= (n^m)/(n^m) = n^(m-m) = n^0. Or better n^0 = the number of functions from the empty set (cardinal 0) to the set with cardinal n. This justifies also 0^0 = 1 (there is one (empty) function from the empty set to the empty set). along with 'n/0 is 'undefined''. We note the consistent/valid notation, but walk away from any
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Bruno Marchal wrote: OK. And you are right, I could have done this without mentioning the constructive ordinal. But it is worth mentioning it, even at this early stages, because they will reappear again and again. Note that all those infinite but constructive ordinal are all countable (in bijection with N), and even constructively so. Note also, if you haven't already done so, that omega is just N, the set of natural numbers. I will soon give a more set-theoretical motivation for those ordinals. OK. I feel power sets coming. Actually there is a cute theorem about constructive ordinal. Indeed they are equivalent to the recursive (programmable) linear well-ordering on the natural numbers. Examples: An order of type omega: the usual order on N (0123456...) An order of type omega+1 : just decide that 0 is bigger than any non null natural numbers: 123456 0 It is recursive in the sense that you can write a program FORTRAN (say) capable of deciding it. For example such a program would stop on yes when asked if 48, and no if you ask 08, etc. An order of type omega+omega: just decide that all odd numbers are bigger than the even ones, and take the usual order in case the two numbers which are compared have the same parity: 0246810 . 13579... An order of type omega+omega+omega: just decide that multiple of 3 are bigger than the multiple of two, themselves bigger than the remaining numbers: 15711131417... 0246810... 3691215... Again it should be easy to write a Fortran program capable of deciding that order (that is to decide for any x and y if x y with that (unusual) order. Exercise: could you find an order of type omega*omega? (Hint: use the prime numbers). You could use the Sieve of Eratosthenes (spelling?): 2*12*22*3... 3*13*33*5(all multiples of 3 that are not multiples of 2)... 5*15*55*7(all multiples of 5 that are not multiples of 3 or 2)... It sounds like the cute theorem says that you can keep dividing up the natural numbers like this forever. Those omega names are quite standard. OK. So we haven't left the finite behind yet. It makes intuitive sense to me that you can diagonalize till the cows come home, staying within countability, and still not be done. Otherwise infinity wouldn't be infinite. On the tricky question, it also makes intuitive sense that you can sequence effectively on all computable growing functions. This is because the larger the growing function gets, the more uncovered space (gaps) there are between the computable functions. Any scheme for generating growing functions will also leave behind every-growing uncomputed gaps. Very unmathematical of me to be so vague, but you've already given us the answer, and I know you will fill in the gaps. :) I will. Unfortunately this week is full of duty charges. Meanwhile, I would like to ask George and the others if they have a good understanding of the present thread, that is on the fact that growing functions has been well defined, that each sequence of such functions are well defined, and each diagonalisation defines quite well a precise programmable growing function (growing faster than the one in the sequence it comes from). Just a tiny effort, and I think we will have all we need to go into the heart of the matter, and to understand why comp makes our universe a godelized one in the Smullyan sense. I meant that it makes intuitive sense that you *cannot* sequence effectively on all computable growing functions. You are right. But would that mean we cannot dovetail on all growing computable functions? I let you ponder this not so easy question. Bruno PS About Parfit, I have already said some time ago in this list that I appreciate very much its Reasons and Persons book, but, in the middle of the book he makes the statement that we are token, where it follows---[easily? not really: you need the movie graph or some strong form of Occam]--- that comp makes us type, even abstract type. It just happens that from a first person point of view we cannot take ourselves as type because we just cannot distinguish between our many instances generated by the Universal Dovetailer. A similar phenomenon occur already with the quantum. But from the point of view of this thread, this is an anticipation. The things which look the more like token, with the comp hyp, are the numbers. This makes the second half part of Parfit's book rather inconsistent with comp, but, still, his analysis of personal identity remains largely genuine. (I don't like at all his use of the name reductionism in that context, also, it's quite misleading). http://iridia.ulb.ac.be/~marchal/ From what you've said about dovetailing before, you don't have to have just a single sequence in order to dovetail. You can jump among multiple sequences. I have yet to understand how you could dovetail on something that is not effective. That seems to be where
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Tom Caylor wrote: It sounds like the cute theorem says that you can keep dividing up the natural numbers like this forever. Oops. I slipped in an actual infinity when I said forever. Perhaps I should have said indefinitely ;) Tom --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Hi, From what you've said about dovetailing before, you don't have to have just a single sequence in order to dovetail. You can jump among multiple sequences. I have yet to understand how you could dovetail on something that is not effective. I think dovetailing is possible because the dovetailer only complete sequences at infinity. So when you construct the matrice on which you will diagonalize, you are already diagonilizing it at the same time. Example: when you have the first number of the first growing function, you can also have the first number of the diagonalize function (by adding 1) and the first number of the diagonalize*diagonalize function and ... ad recursum. By dovetailing you execute in fact everything in parallel but all infinites sequences are only completed at infinity. Quentin Anciaux --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Quentin Anciaux wrote: Hi, From what you've said about dovetailing before, you don't have to have just a single sequence in order to dovetail. You can jump among multiple sequences. I have yet to understand how you could dovetail on something that is not effective. I think dovetailing is possible because the dovetailer only complete sequences at infinity. So when you construct the matrice on which you will diagonalize, you are already diagonilizing it at the same time. Example: when you have the first number of the first growing function, you can also have the first number of the diagonalize function (by adding 1) and the first number of the diagonalize*diagonalize function and ... ad recursum. By dovetailing you execute in fact everything in parallel but all infinites sequences are only completed at infinity. Quentin Anciaux OK. Thanks. But so far we have done only effective diagonalization. I'll follow along as Bruno goes step by step. Also, it seems to me even with non-effective diagonalization there will be another problem to solve: When we dovetail, how do we know we are getting sufficient (which means indefinite) level of substitution in finite amount of computation? (Also, I am waiting for a good explanation of how Church Thesis comes into this.) Again, I'll wait for the step by step argument. Tom --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Bruno Marchal wrote: Meanwhile, I would like to ask George and the others if they have a good understanding of the present thread, that is on the fact that growing functions has been well defined, that each sequence of such functions are well defined, and each diagonalisation defines quite well a precise programmable growing function (growing faster than the one in the sequence it comes from). Just a tiny effort, and I think we will have all we need to go into the heart of the matter, and to understand why comp makes our universe a godelized one in the Smullyan sense. To speak only for myself, I think I have a sufficient understanding of the thread. Essentially you have shown that one cannot form a set of all numbers/functions because given any set of numbers/functions it is always possible, using diagonalization, to generate new numbers/functions: the Plenitude is too large to be a set. This leads to a problem with the assumption of the existence of a Universal Dovetailer whose purpose is to generate all functions. I hope this summary is accurate. George --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
George Levy wrote: Bruno Marchal wrote: Meanwhile, I would like to ask George and the others if they have a good understanding of the present thread, that is on the fact that growing functions has been well defined, that each sequence of such functions are well defined, and each diagonalisation defines quite well a precise programmable growing function (growing faster than the one in the sequence it comes from). Just a tiny effort, and I think we will have all we need to go into the heart of the matter, and to understand why comp makes our universe a godelized one in the Smullyan sense. To speak only for myself, I think I have a sufficient understanding of the thread. Essentially you have shown that one cannot form a set of all numbers/functions because given any set of numbers/functions it is always possible, using diagonalization, to generate new numbers/functions: the Plenitude is too large to be a set. This leads to a problem with the assumption of the existence of a Universal Dovetailer whose purpose is to generate all functions. I hope this summary is accurate. George The dovetailer is only supposed to generate all *computable* functions though, correct? And the diagonalization of the (countable) set of all computable functions would not itself be computable. Jesse --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Jesse Mazer writes: The dovetailer is only supposed to generate all *computable* functions though, correct? And the diagonalization of the (countable) set of all computable functions would not itself be computable. The dovetailer I know does not seem relevant to this discussion about functions. It generates programs, not functions. For example, it generates all 1 bit programs and runs each for one cycle; then generates all 2 bit programs and runs each for 2 cycles; then generates all 3 bit programs and runs each for 3 cycles; and so on indefinitely. (This assumes that the 3 bit programs include all 2- and 1-bit programs, etc.) In this way all programs get run with an arbitrary number of cycles. These programs differ from functions in two ways. First, programs may never halt and hence may produce no fixed output, while functions must have a well defined result. And second, these programs take no inputs, while functions should have at least one input variable. What do you understand a dovetailer to be, in the context of computable functions? Hal Finney --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Rich, are you familiar with the work of R.D. Laing? He was the illustrious founder of the anti-psychiatry movement in the 60s. One never hears of him these days. He had all the other thinkers on the hop for quite a while. Your thoughts represent no interruption whatsoever. Kim On 29/05/2006, at 1:09 PM, Rich Winkel wrote: At the risk of wasting more bandwidth than I alread have I'd like to apologize for any discomfort I've caused on the list. Sometimes I feel like a jewish person arguing the reality of the holocaust to doubters. Such is the hidden record of psychiatry and the power of its PR machine. Please excuse the interruption. Rich --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Bruno Marchal wrote:
Le 26-mai-06, à 19:35, Tom Caylor a écrit :
Bruno,
You are starting to perturb me! I guess that comes with the territory
where you're leading us.
You should not worry too much. I confess I am putting your mind in the
state of mathematicians before the Babbage Post Markov Turing Church
discovery. Everything here will be transparently clear.
But of course being perturbed doesn't
necessarily imply being correct. I will summarize my perturbation
below. But for now, specifically, you're bringing in transfinite
cardinals/ordinals.
Only transfinite ordinal which are all countable, and even nameable,
for example by name of growing computable functions as I am
illustrating.
Be sure you understand why G is a well defined computable growing
function, and why it grows faster than each initial Fi. If you know a
computer programming language, write the program!
This is where things get perverse and perhaps
inconsistent. For instance, couldn't I argue that G is also infinite?
In which sense? All functions are infinite mathematical object.
Factorial is defined by its infiinite set of inputs outputs: {(0,1)
(1,1)(2,2) (3,6) (4,24) (5,120) ...}.
Take n = some fixed N1. Then F1(N) 1, F2(N) 2, F3(N) 3, ...
and Fn(N) n, for all n. So each member of the whole sequence F1, F2,
F3 ... G is greater than the corresponding member of the sequence 1, 2,
3, ... aleph_0 (countable infinity). Thus, G (=) countable infinity,
even for a fixed n=N1.
You are right but G is a function. Actually it just does what it has
been programmed to. I don't see any problem here.
OK. I see that so far (above) there's no problem. (See below for
where I still have concern(s).) Here I was taking a fixed N, but G is
defined as the diagonal, so my comparison is not valid, and so my proof
that G is infinite for a fixed N is not valid. I was taking G's
assignment of an ordinal of omega as being that it is in every way
larger than all Fi's, but in fact G is larger than all Fi's only when
comparing G(n) to Fn(n), not when comparing G(Nfixed) to Fi(N) for all
i's.
Oh Oh Oh Oh Oh A new pattern emerge (the Ackerman Caylor one, at
a
higher level).
F_omega,
F_omega + omega
F_omega * omega
F_omega ^ omega
F_omega [4] omega (omega tetrated to omega, actually this ordinal got
famous and is named Epsilon Zéro, will say some words on it later)
F_omega [5] omega
F_omega [6] omega
F_omega [7] omega
F_omega [8] omega
F_omega [9] omega
F_omega [10] omega
F_omega [11] omega
...
In this case they are all obtained by successive diagonalzations, but
nothing prevent us to diagonalise on it again to get
F_omega [omega] omega
OK, I think the following finite number is big enough:
F_omega [omega] omega (F_omega [omega] omega (9 [9] 9))
Next, we will meet a less constructivist fairy, and take some new kind
of big leap.
Be sure to be convinced that, despite the transfinite character of the
F_alpha sequence, we did really defined at all steps precise
computable growing functions ... (if not: ask question please).
It seems to me that you are on very shaky ground if you are citing
transfinite numbers in your journey to showing us your ultimate
argument.
Please Tom, I did stay in the realm of the finitary. Even intutionist
can accept and prove correct the way I named what are just big finite
number. I have not until now transgressed the constructive field, I
have not begin to use Platonism! There is nothing controversial here,
even finitist mathematician can accept this. (Not ultra-finitist,
though, but those reject already 10^100)
I also think that if you could keep your arguments totally
in the finite arena it would less risky.
I have. You must (re)analyse the construction carefully and realize I
have not go out of the finite arena. Ordinals are just been used as a
way to put order on the successive effective diagonalizations. Those
are defined on perfectly well defined and generable sequence of well
defined functions. I have really just written a program (a little bit
sketchily, but you should be able to add the details once you should a
programming language).
OK. I think you are just throwing me off with your notation. Do you
have to use transfinite ordinals (omega) to do this? Couldn't you just
stay with successively combining and diagonalizing, like below, without
using omegas?
G(n) = Fn(n)+1
Gi(n) = G(...G(n)), G taken i times
Then instead of using more and more additional letters, just add
subscripts...
H1(n) = Gn(n)+1
H1i(n) = H1(...H1(n)), H1 taken i times
H2(n) = H1n(n)+1
H2i(n) = H2(...H2(n)), H2 taken i times
Then the subscripts count the number of diagonalizations you've done,
and every time you do the Ackermann thing, instead of adding an omega
you add another subscript. Then it continues ad infinitum. You can
do the Ackermann thing with the *number* of subscripts,
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
I meant that it makes intuitive sense that you *cannot* sequence effectively on all computable growing functions. Tom --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Thanks for that, Jesse. History-by-Hollywood has been my downfall before...scriptwriter Akiva Goldsman should perhaps get six cuts of the school cane for using such a high degree of creative licence with the facts. A Best Movie vote seems to screw up one's scholarly instincts to check the truth of the matter. Goldsman has been mercifully truer to the original with his job on The Da Vinci Code ;) Nevertheless, do we see any interest in Nash's flitting in and out of sanity and the corresponding strengthening and weakening of his maths genius? As a reasonable (= non-schizo) machine he seemed capable of a form of almost transcendental mathematical perception - possibly almost as fabulous as Bruno's :) - during his extensive schizophrenic interludes, a different form of perception seemed to hold sway. Was he tripping between parallel versions of himself - or what? The relevance to the discussion (for me) lies in his apparent inabilty to choose his mental orientation. In both (opposing) mental states he was being true to himself. He was surely a plural-self. Maybe the imposition of a certain chemistry is the key to a multiversal (ie non-linear) perception of self. Don't experiment with this thought at home, though. Kim On 28/05/2006, at 11:18 AM, Jesse Mazer wrote: Kim Jones wrote: Well, in the case of schizoid mathematician John Nash, his psychotic behaviour was also clearly linked to his maths ability. After imbibing anti-psychotic medication, not only did his unreal friends disappear, but his mathematical perception as well. I don't think that's true, my understanding is that once he became schizophrenic he no longer did any useful mathematical work, just mystical numerology. In discussing the movie, the wikipedia entry at http://en.wikipedia.org/wiki/A_Beautiful_Mind says: The movie also misrepresents the effect Nash's mental illness had on his work. The movie depicts Nash as already suffering from schizophrenia when he wrote his doctoral thesis. In reality, Nash's schizophrenia did not appear until years later and once it did his mathematical work ceased until he was able to bring it under control. And the page at http://www.pnas.org/misc/classics5.shtml says that he once again started doing useful work after his recovery: In 1970, Nash moved back to Princeton, where he took to shuffling through the halls of the mathematics building, occasionally scribbling enigmatic numerological messages on the walls. Students referred to him as the Phantom of Fine Hall. Gradually, however, Nash's mental condition began to improve. Schizophrenia rarely disappears completely, but by the 1990s Nash appeared to have made a remarkable recovery, and he had turned once again to mathematical research. The wikipedia article elaborates on what his recent work has been about: The 1990s brought a return of his genius, and Nash has taken care to manage the symptoms of his mental illness. He is still hoping to score substantial scientific results. His recent work involves ventures in advanced game theory including partial agency which show that, as in his early career, he prefers to select his own path and problems (though he continues to work in a communal setting to assist in managing his illness). Jesse --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
RE: Smullyan Shmullyan, give me a real example
John M writes: Stathis: 1. to Kim's question to Bruno (and your reply): I call reasonable the items matching OUR (human) logic, even if we call it a machine. There is no norm in the existence for 'reasonable', as Cohen and Stewart showed in their chef d'oeuvre on Chaos in the imaginary Zarathustrans. We, with our 100 years ahead thinking and Bruno with his 200 should be above such narrowminded limitations. 2.to your 'delusion': it is correctG. Yes, but even with a normative definition of rational(if that is what you mean) there are those who are either unable (due to illness, or because they are infants, for example) or unwilling (due to expediency, or laziness, or whatever) to be rational. Moreover, this is very common, not just an exception to the rule. )...The single best test is to treat someone with antipsychotic medication and see if the delusion goes away.) is this to implant new delusions and see how the poor fellow reacts? We had some intelligent dicussions about 'everybody is crazy' (George at al.) and so crazy is 'normal' and the norm may be crazy. Are the psych-professionals exceptions? Let me give you a real example. A patient claims that in the last month, whenever he is away, or asleep, someone comes into his house and does annoying things, like shifting personal items from one place to another, putting holes in his socks, opening cupboards that were closed or closing cupboards that were open, and so on. The culprit must know a lot about electronic surveillance, because the cameras the poor victim has installed do not show evidence of intrusion, and the neighbours might also be involved because they all say they have seen nothing unusual, even though it is happening almost every day. The patient stops going to work or socializing, and drinks endless cups of coffee in order to stay up as long as possible and catch the intruder, but it's no use: the incidents become more frequent and more brazen. Finally, when the patient starts talking about homicidal and suicidal impulses, his family contact the local psychiatric crisis team, who make a diagnosis of schizophreniform psychosis and persuade the patient to start taking aripiprazole 15mg mane, which they supervise by visiting daily. The patient only agrees to this because the alternative would be involuntary hospital admission: he can explain, in great detail, why he is convinced that the intruder is real, and to be fair, there is no way the psychiatric team can prove him wrong, or be completely certain that he is wrong. Two weeks later, the alleged intrusions have almost completely stopped, and our patient is thinking about going back to work again. He argues it has nothing to do with the medication: maybe the culprit was scared off by the psychiatric team, or maybe he just got sick of annoying him. Two years later, the patient has gone through a cycle of stopping and starting the medication half a dozen times. Within a few weeks of stopping, the intrusions start up again, and within a few weeks of recommencing treatment, they stop. He still isn't convinced that he has ever been paranoid, but to stop his family and psychiatric services from nagging him, he agrees to stay on the medication indefinitely, and he remains well. Given this story, are you still prepared to say that the patient's reasoning that strangers are putting holes in his socks is just as valid as that of the psychiatric team or his concerned family? It's worth noting that his *deductive* reasoning remained intact throughout, so if you could use this to explain logically where his *inductive* reasoning went wrong, you might have saved the state the cost of an expensive antipsychotic. 3. You wrote: An unreasonable machine would look like a brain. The minds of living organisms, such as they are, evolved ... Because we know so little about the ways a brain works and assume too much based on our present ignorance to explain everything still unknown. There is the terror of physicists forcing their primitive model on the world, especially on domains where SOME features can be measured in established 'phisics-invented' concepts by the so fa physics-invented instruments and read in physics-invented units, although the conclusions come from 'non-physics-related' activities (mentality, ideation, feelings, delusions, etc.,) all having parallel and physically measurable phenomena in the neurological sciences. we use the 'brain' as a tool and have no idea how it works and for what. In your quoted fragment I feel an equating of brain and mind, which I find at least premature. I don't know what a mind may be. I know(?) it must be both atemporal and aspatial, while the material of the brain is imagined (physically) to be space and time related. The mind is not the same as the brain, of course, but the two are connected. We know that whenever certain complex physical processes which we call brain activity occur, certain other mysterious
Re: Re: Smullyan Shmullyan, give me a real example
According to Stathis Papaioannou: On the other hand, about 1/3 of people who present with psychotic symptoms will have either a complete or a near-complete response to medication with minimal side-effects, and it would be tragic if they missed out due to anti-psychiatry prejudice or (more commonly) because they don't believe they have an illness. Please don't bait me or the other unfortunates who have good reasons for anti-psychiatry prejudice. If psychiatry had any insight into its own ignorance and destructiveness there'd be no reason for such prejudice. Tweaking brain chemicals or running house current through brains in pursuit of behavior modification and perception management is like trying to program a computer with a soldering iron. Selling such quackery as medicine against nonconsenting children while in the pay of their parents or other conflicted adults is an atrocity. Shrinks have forgotten the principle of first no harm. There is a theory that the genes predisposing to schizophrenia have survived because they also give rise to original thinkers, conferring an advantage to society which outweighs the disadvantage of having floridly psychotic people around. You're jumping between individual and meta-social fitness here. Where is the feedback loop? Back to the shadows :) Rich --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Re: Smullyan Shmullyan, give me a real example
At the risk of wasting more bandwidth than I alread have I'd like to apologize for any discomfort I've caused on the list. Sometimes I feel like a jewish person arguing the reality of the holocaust to doubters. Such is the hidden record of psychiatry and the power of its PR machine. Please excuse the interruption. Rich --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
RE: Smullyan Shmullyan, give me a real example
Kim Jones writes: Bruno, what would an unreasonable machine be like? You seem to be implying they exist, also that they can prove things about their possible neighborhoods and or histories. (?) Kim An unreasonable machine would look like a brain. The minds of living organisms, such as they are, evolved to promote survival and reproduction, and apparently being rational is only a minor advantage towards this end. I am sure that even logicians, at least when they are off duty, pluck axioms out of the air according to whim or fashion, hold contradictory beliefs simultaneously or sequentially, decide that the correct course of action is x and then do ~x anyway, and so on. It is interesting that in psychiatry, it is impossible to give a reliable method for recognizing a delusion. The usual definition is that a delusion is a fixed, false belief which is not in keeping with the patient's cultural background. If you think about it, why should cultural background have any bearing on whether a person's reasoning is faulty? And even including this criterion, it is often difficult to tell without looking at associated factors such as change in personality, mood disturbance, etc. The single best test is to treat someone with antipsychotic medication and see if the delusion goes away. This means that in theory there might be two people with exactly the same belief, justified in exactly the same way, but one is demonstrably psychotic while the other is not! Crazy thinking is so common that, by itself, it is generally not enough reason to diagnose someone as being crazy. Stathis Papaioannou --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Le 26-mai-06, à 19:35, Tom Caylor a écrit :
Bruno Marchal wrote:
Hi,
OK, let us try to name the biggest natural (finite) number we can, and
let us do that transfinite ascension on the growing functions from N
to
N.
We have already build some well defined sequence of description (code)
of growing functions.
Let us choose the Hall Finney sequence to begin with (but the one by
Tom Caylor can be use instead).
F1 F2 F3 F4 F5 ...
With F1(n) = factorial(n), F2(n) = factorial(factorial n), etc.
Note this: Hal gave us a trick for getting from a growing function f,
a
new function growing faster, actually the iteration of the function.
That is, Hal gave us a notion of successor for the growing function.
Now the diagonalization of the sequence F1 F2 F3 F4 ..., which is
given
by the new growing function defined by
G(n) = Fn(n) + 1
gives us a growing function which grows faster than any Fi from Hal's
initial sequence. Precisely, G will grow faster than any Fi on *almost
all* number (it could be that some Fi will grow faster than G on some
initial part of N, but for some finite value (which one?) G will keep
growing faster. Technically we must remember to apply our growing
function on sufficiently big input' if we want to benefit of the
growing phenomenon. We will make a rough evaluation on that input
later, but let us not being distract by technical point like that.
The diagonalization gives an effective way to take the limit of the
sequence F1, F2, F3, ...
G grows faster than any Fi. Mathematician will say that the order type
of g, in our our new sequence F1 F2 F3 ... G, is omega (the greek
letter).
Bruno,
You are starting to perturb me! I guess that comes with the territory
where you're leading us.
You should not worry too much. I confess I am putting your mind in the
state of mathematicians before the Babbage Post Markov Turing Church
discovery. Everything here will be transparently clear.
But of course being perturbed doesn't
necessarily imply being correct. I will summarize my perturbation
below. But for now, specifically, you're bringing in transfinite
cardinals/ordinals.
Only transfinite ordinal which are all countable, and even nameable,
for example by name of growing computable functions as I am
illustrating.
Be sure you understand why G is a well defined computable growing
function, and why it grows faster than each initial Fi. If you know a
computer programming language, write the program!
This is where things get perverse and perhaps
inconsistent. For instance, couldn't I argue that G is also infinite?
In which sense? All functions are infinite mathematical object.
Factorial is defined by its infiinite set of inputs outputs: {(0,1)
(1,1)(2,2) (3,6) (4,24) (5,120) ...}.
Take n = some fixed N1. Then F1(N) 1, F2(N) 2, F3(N) 3, ...
and Fn(N) n, for all n. So each member of the whole sequence F1, F2,
F3 ... G is greater than the corresponding member of the sequence 1, 2,
3, ... aleph_0 (countable infinity). Thus, G (=) countable infinity,
even for a fixed n=N1.
You are right but G is a function. Actually it just does what it has
been programmed to. I don't see any problem here.
But G is just a well defined computable growing function and we can
use
Hall Finney successor again to get the still faster function, namely
G(G(n)).
The order type of G(G(n)) is, well, the successor of omega: omega+1
And, as Hall initially, we can build the new sequance of growing
functions (all of which grows more than the preceding sequence):
G(n) G(G(n)) G(G(G(n))) G(G(G(G(n etc.
which are of order type omega, omega+1, omega+2, omega+3, omega+4,
etc.
Now we have obtained a new well defined infinite sequence of growing
function, and, writing it as:
G1, G2, G3, G4, G5, G6, ... or better, as
F_omega, F_omega+1, F_omega+2, F_omega+3
just showing such a sequence can be generated so that we can again
diagonalise it, getting
H(n) = Gn(n) + 1, or better
H(n) = F_omega+n (n) + 1
Getting a function of order type omega+omega: we can write H =
F_omega+omega
And of course, we can apply Hall's successor again, getting
F_omega+omega+1
which is just H(H(n), and so we get a new sequence:
F_omega+omega+1, F_omega+omega+2, F_omega+omega+3, ...
Which can be diagonalise again, so we get
F_omega+omega+omega,
and then by Hal again, and again ...:
F_omega+omega+omega+1, F_omega+omega+omega+2, F_omega+omega+omega+3
...
Oh Oh! a new pattern emerges, a new type of sequence of well defined
growing functions appears:
F_omega, F_omega+omega, F_omega+omega+omega,
F_omega+omega+omega+omega,
And we can generated it computationnaly, so we can diagonalise again
to
get:
F_omega * times omega,
and of course we can apply Hal's successor (or caylor one of course)
again, and again
Oh Oh Oh Oh Oh A new pattern emerge (the Ackerman Caylor one, at
a
higher
Re: Smullyan Shmullyan, give me a real example
Stathis: 1. to Kim's question to Bruno (and your reply): I call reasonable the items matching OUR (human) logic, even if we call it a machine. There is no norm in the existence for 'reasonable', as Cohen and Stewart showed in their chef d'oeuvre on Chaos in the imaginary Zarathustrans. We, with our 100 years ahead thinking and Bruno with his 200 should be above such narrowminded limitations. 2.to your 'delusion': it is correctG. )...The single best test is to treat someone with antipsychotic medication and see if the delusion goes away.) is this to implant new delusions and see how the poor fellow reacts? We had some intelligent dicussions about 'everybody is crazy' (George at al.) and so crazy is 'normal' and the norm may be crazy. Are the psych-professionals exceptions? 3. You wrote: An unreasonable machine would look like a brain. The minds of living organisms, such as they are, evolved ... Because we know so little about the ways a brain works and assume too much based on our present ignorance to explain everything still unknown. There is the terror of physicists forcing their primitive model on the world, especially on domains where SOME features can be measured in established 'phisics-invented' concepts by the so fa physics-invented instruments and read in physics-invented units, although the conclusions come from 'non-physics-related' activities (mentality, ideation, feelings, delusions, etc.,) all having parallel and physically measurable phenomena in the neurological sciences. we use the 'brain' as a tool and have no idea how it works and for what. In your quoted fragment I feel an equating of brain and mind, which I find at least premature. I don't know what a mind may be. I know(?) it must be both atemporal and aspatial, while the material of the brain is imagined (physically) to be space and time related. John M - Original Message - From: Stathis Papaioannou [EMAIL PROTECTED] To: everything-list@googlegroups.com Cc: [EMAIL PROTECTED] Sent: Saturday, May 27, 2006 8:25 AM Subject: RE: Smullyan Shmullyan, give me a real example Kim Jones writes: Bruno, what would an unreasonable machine be like? You seem to be implying they exist, also that they can prove things about their possible neighborhoods and or histories. (?) Kim An unreasonable machine would look like a brain. The minds of living organisms, such as they are, evolved to promote survival and reproduction, and apparently being rational is only a minor advantage towards this end. I am sure that even logicians, at least when they are off duty, pluck axioms out of the air according to whim or fashion, hold contradictory beliefs simultaneously or sequentially, decide that the correct course of action is x and then do ~x anyway, and so on. It is interesting that in psychiatry, it is impossible to give a reliable method for recognizing a delusion. The usual definition is that a delusion is a fixed, false belief which is not in keeping with the patient's cultural background. If you think about it, why should cultural background have any bearing on whether a person's reasoning is faulty? And even including this criterion, it is often difficult to tell without looking at associated factors such as change in personality, mood disturbance, etc. The single best test is to treat someone with antipsychotic medication and see if the delusion goes away. This means that in theory there might be two people with exactly the same belief, justified in exactly the same way, but one is demonstrably psychotic while the other is not! Crazy thinking is so common that, by itself, it is generally not enough reason to diagnose someone as being crazy. Stathis Papaioannou --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 26-mai-06, à 02:50, James N Rose a écrit : Bruno, You struck a personal nerve in me with your following remarks: Bruno Marchal wrote: They are degrees. The worst unreasonableness of a (platonist or classical or even intuitionist) machine is when she believes some plain falsity (like p ~p, or 0 = 1). The false implies all propositions, so that such machine believes everything, including everything about their maximal consistent extensions or histories (which does not exist). Those machines are just inconsistent. particularly , some plain falsity (like p ~p, or 0 = 1). Rather than treat these as 'blatantly false' I have been exploring the notion for several years .. 'what conditions, situations, criteria or states would allow such statements to be 'true', and what would it mean in how we define and manipulate and operate the rest of mathematics?'. I have discovered that an unprecedentedly un-appreciated realm of mathematical relations has existed right before our minds. The lack, having kept us trying to cope with 'anomalies' and math issues without the full toolkit of mathematical instruments. An example at the core of it is a most simplistic definition/equation. 1^1 = 1^0 [one to the exponent one equals one to the exponent zero] To all mathematicians, this is a toss-out absurdity, with no 'real meaning'. n^0 is a convenience tool at best ; n^0 = 1, because 1= (n^m)/(n^m) = n^(m-m) = n^0. Or better n^0 = the number of functions from the empty set (cardinal 0) to the set with cardinal n. This justifies also 0^0 = 1 (there is one (empty) function from the empty set to the empty set). along with 'n/0 is 'undefined''. We note the consistent/valid notation, but walk away from any active utility or application. My thesis is that doing so was a missed opportunity. To be hyper-consistent, the equation set-up 1^1 = 1^0 indicates that there -must- be some valid states/conditions (not just 'interpretation') when 0 and 1 are 'equal' in some real meaning/use of the word equal. Why? It is usual that a function (like y = 1^x) can have the same value for different argument. From (-5)^2 = 5^2 you will not infer that 5 = (-5), right? From sinus(x) = sinus(pi - x) you will not deduce that x = pi - x, right? If they can be substituted in the above equation, without changing a resultant of calculations (they are embedded in), then they must somewhere somehow in fact be identical in some way or condition. You talk like if all functions are bijections (one to one function). The entire ediface of physics is hamstrung because of this, because mathematical definitions and language compounded the error by applying - actually DIS-applying - a related concept .. the notion of 'extent' .. also known as 'dimension'. Physics and mathematics transform and wholly open up when we throw away the old concept of 'dimensionless' and instead reformulate -everything- as 'dimensional'. Including zero; including numbers unassociated with variables. As musch as you are brilliant and mathematically inventive, your statement some plain falsity (like p ~p, or 0 = 1) shows you haven't quite awoken to everything yet. I hope I'm in the process of stirring you from your slumber. I am using the name 0, 1, ... for the usual numbers. 1 is different from 0 for the same reason that 1 cup of coffee is different from 0 cup of coffee, or that 1 joke is different from 0 joke ... 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 -~--~~~~--~~--~--~---
Re: Ascension (was Re: Smullyan Shmullyan, give me a real example)
Bruno Marchal wrote: Hi, OK, let us try to name the biggest natural (finite) number we can, and let us do that transfinite ascension on the growing functions from N to N. We have already build some well defined sequence of description (code) of growing functions. Let us choose the Hall Finney sequence to begin with (but the one by Tom Caylor can be use instead). F1 F2 F3 F4 F5 ... With F1(n) = factorial(n), F2(n) = factorial(factorial n), etc. Note this: Hal gave us a trick for getting from a growing function f, a new function growing faster, actually the iteration of the function. That is, Hal gave us a notion of successor for the growing function. Now the diagonalization of the sequence F1 F2 F3 F4 ..., which is given by the new growing function defined by G(n) = Fn(n) + 1 gives us a growing function which grows faster than any Fi from Hal's initial sequence. Precisely, G will grow faster than any Fi on *almost all* number (it could be that some Fi will grow faster than G on some initial part of N, but for some finite value (which one?) G will keep growing faster. Technically we must remember to apply our growing function on sufficiently big input' if we want to benefit of the growing phenomenon. We will make a rough evaluation on that input later, but let us not being distract by technical point like that. The diagonalization gives an effective way to take the limit of the sequence F1, F2, F3, ... G grows faster than any Fi. Mathematician will say that the order type of g, in our our new sequence F1 F2 F3 ... G, is omega (the greek letter). Bruno, You are starting to perturb me! I guess that comes with the territory where you're leading us. But of course being perturbed doesn't necessarily imply being correct. I will summarize my perturbation below. But for now, specifically, you're bringing in transfinite cardinals/ordinals. This is where things get perverse and perhaps inconsistent. For instance, couldn't I argue that G is also infinite? Take n = some fixed N1. Then F1(N) 1, F2(N) 2, F3(N) 3, ... and Fn(N) n, for all n. So each member of the whole sequence F1, F2, F3 ... G is greater than the corresponding member of the sequence 1, 2, 3, ... aleph_0 (countable infinity). Thus, G (=) countable infinity, even for a fixed n=N1. But G is just a well defined computable growing function and we can use Hall Finney successor again to get the still faster function, namely G(G(n)). The order type of G(G(n)) is, well, the successor of omega: omega+1 And, as Hall initially, we can build the new sequance of growing functions (all of which grows more than the preceding sequence): G(n) G(G(n)) G(G(G(n))) G(G(G(G(n etc. which are of order type omega, omega+1, omega+2, omega+3, omega+4, etc. Now we have obtained a new well defined infinite sequence of growing function, and, writing it as: G1, G2, G3, G4, G5, G6, ... or better, as F_omega, F_omega+1, F_omega+2, F_omega+3 just showing such a sequence can be generated so that we can again diagonalise it, getting H(n) = Gn(n) + 1, or better H(n) = F_omega+n (n) + 1 Getting a function of order type omega+omega: we can write H = F_omega+omega And of course, we can apply Hall's successor again, getting F_omega+omega+1 which is just H(H(n), and so we get a new sequence: F_omega+omega+1, F_omega+omega+2, F_omega+omega+3, ... Which can be diagonalise again, so we get F_omega+omega+omega, and then by Hal again, and again ...: F_omega+omega+omega+1, F_omega+omega+omega+2, F_omega+omega+omega+3 ... Oh Oh! a new pattern emerges, a new type of sequence of well defined growing functions appears: F_omega, F_omega+omega, F_omega+omega+omega, F_omega+omega+omega+omega, And we can generated it computationnaly, so we can diagonalise again to get: F_omega * times omega, and of course we can apply Hal's successor (or caylor one of course) again, and again Oh Oh Oh Oh Oh A new pattern emerge (the Ackerman Caylor one, at a higher level). F_omega, F_omega + omega F_omega * omega F_omega ^ omega F_omega [4] omega (omega tetrated to omega, actually this ordinal got famous and is named Epsilon Zéro, will say some words on it later) F_omega [5] omega F_omega [6] omega F_omega [7] omega F_omega [8] omega F_omega [9] omega F_omega [10] omega F_omega [11] omega ... In this case they are all obtained by successive diagonalzations, but nothing prevent us to diagonalise on it again to get F_omega [omega] omega OK, I think the following finite number is big enough: F_omega [omega] omega (F_omega [omega] omega (9 [9] 9)) Next, we will meet a less constructivist fairy, and take some new kind of big leap. Be sure to be convinced that, despite the transfinite character of the F_alpha sequence, we did really defined at all steps precise computable growing functions ... (if not: ask question please). It seems to me that you are on very
Re: Smullyan Shmullyan, give me a real example
Bruno, what would an unreasonable machine be like? You seem to be implying they exist, also that they can prove things about their possible neighborhoods and or histories. (?) Kim On 23/05/2006, at 8:25 PM, Bruno Marchal wrote: Is it not utterly obvious that, IF we are (hopefully reasonable) machine, THEN we will learn something genuine by studying what (reasonable) machines can prove about themselves and about their possible neighborhoods and or histories? --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 25-mai-06, à 09:04, Kim Jones a écrit : what would an unreasonable machine be like? You seem to be implying they exist, also that they can prove things about their possible neighborhoods and or histories. (?) They are degrees. The worst unreasonableness of a (platonist or classical or even intuitionist) machine is when she believes some plain falsity (like p ~p, or 0 = 1). The false implies all propositions, so that such machine believes everything, including everything about their maximal consistent extensions or histories (which does not exist). Those machines are just inconsistent. Then you have machines which, although they are consistent, are not self-referentially correct. They are unsound, and does also believe some falsity, but here the falsity is irrefutable. Like consistent machine asserting they are inconsistent, or, curiously enough (it is a consequence of the second incompleteness theorem), consistent machine asserting they are consistent. Although it is true that they are consistent they cannot assert it without becoming either inconsistent, or, if they assert it in some special cautious way, they become different machine (and in that case they remain consistent and also get some new provability power). But again here we anticipate. I hope I will make clear that with Church thesis the notion of computability will appear as absolute and universal, and then we will see that the notion of provability is relative and never universal, although some universal pattern can appear there too (like the logic G and G*, etc.). I guess we will come back on this, but we have to do a transfinite ascension before :) 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 24-mai-06, à 18:30, Tom Caylor a écrit : Exercises: 0) Could you evaluate roughly the number of digit of 4 [4] 4 ? What about the number of digit of fact(fact(fact(fact 4 1) is the diagonal g function a growing function? Could g belong to the initial sequence, does g grows more quickly than any function in the initial sequence, and in what sense precisely. 2) Could you find a function, and even a new sequence of functions more and more growing, and growing more than the function g? 3) Do you see why it is said that g is build by diagonalization? Where is the diagonal? 4) Is there a universal sequence of growing functions, i. e. containing all computable growing functions? Must already go. Sorry for this quick piece. Solution tomorrow. Hope things are clear. Ask any elementary question (even about notation) before missing the real start ... Any comments , critics or suggestions are welcome ... Bruno http://iridia.ulb.ac.be/~marchal/ I don't have time right now for detailed computations, but I'll give a few quick answers and questions. g is the same as my f(n,n,n)+1, and I already commented that f(n,m,n) is a growing function, since f(N,m,n) is growing for fixed N. So clearly g is growing. As I said about f(n,m,n), the degree or -ation of g grows as n (or x) grows. I recognize that adding 1 to make g is the classical diagonalization move. It makes g different from any Fi in the sequence Fi, i=1,2,3,... And in fact, since we add 1, rather than subtract 1, g is larger than any Fi. I'm having a problem with accepting g, or even my original f(n,n,n), as a function in the same sense as with a fixed degree or -ation of operation. This is because the definition of the function changes depending on the value taken in the domain of the function. Is this valid? It is. Actually the definition of your f does not change, given that you are using a parameter to capture that change. It is valid because you did build a well defined computable function. Actually Ackermann invented his function for showing the existence of a computable function (from N to N) which does not belong to the already very large (but not universal) set of so called primitive recursive function. However, if we just ignore this problem, throwing caution to the wind, then the next logical iteration of diagonalization is to do the -ation thing on g and then diagonalize. Let Gi(x) = g(x) [i] g(x), then let h(x) = Gx(x) + 1. Good idea. And there is no reason to stop there in case the fairy give you some more paper. See my next post. 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 -~--~~~~--~~--~--~---
Ascension (was Re: Smullyan Shmullyan, give me a real example)
Hi, OK, let us try to name the biggest natural (finite) number we can, and let us do that transfinite ascension on the growing functions from N to N. We have already build some well defined sequence of description (code) of growing functions. Let us choose the Hall Finney sequence to begin with (but the one by Tom Caylor can be use instead). F1 F2 F3 F4 F5 ... With F1(n) = factorial(n), F2(n) = factorial(factorial n), etc. Note this: Hal gave us a trick for getting from a growing function f, a new function growing faster, actually the iteration of the function. That is, Hal gave us a notion of successor for the growing function. Now the diagonalization of the sequence F1 F2 F3 F4 ..., which is given by the new growing function defined by G(n) = Fn(n) + 1 gives us a growing function which grows faster than any Fi from Hal's initial sequence. Precisely, G will grow faster than any Fi on *almost all* number (it could be that some Fi will grow faster than G on some initial part of N, but for some finite value (which one?) G will keep growing faster. Technically we must remember to apply our growing function on sufficiently big input' if we want to benefit of the growing phenomenon. We will make a rough evaluation on that input later, but let us not being distract by technical point like that. The diagonalization gives an effective way to take the limit of the sequence F1, F2, F3, ... G grows faster than any Fi. Mathematician will say that the order type of g, in our our new sequence F1 F2 F3 ... G, is omega (the greek letter). But G is just a well defined computable growing function and we can use Hall Finney successor again to get the still faster function, namely G(G(n)). The order type of G(G(n)) is, well, the successor of omega: omega+1 And, as Hall initially, we can build the new sequance of growing functions (all of which grows more than the preceding sequence): G(n) G(G(n)) G(G(G(n))) G(G(G(G(n etc. which are of order type omega, omega+1, omega+2, omega+3, omega+4, etc. Now we have obtained a new well defined infinite sequence of growing function, and, writing it as: G1, G2, G3, G4, G5, G6, ... or better, as F_omega, F_omega+1, F_omega+2, F_omega+3 just showing such a sequence can be generated so that we can again diagonalise it, getting H(n) = Gn(n) + 1, or better H(n) = F_omega+n (n) + 1 Getting a function of order type omega+omega: we can write H = F_omega+omega And of course, we can apply Hall's successor again, getting F_omega+omega+1 which is just H(H(n), and so we get a new sequence: F_omega+omega+1, F_omega+omega+2, F_omega+omega+3, ... Which can be diagonalise again, so we get F_omega+omega+omega, and then by Hal again, and again ...: F_omega+omega+omega+1, F_omega+omega+omega+2, F_omega+omega+omega+3 ... Oh Oh! a new pattern emerges, a new type of sequence of well defined growing functions appears: F_omega, F_omega+omega, F_omega+omega+omega, F_omega+omega+omega+omega, And we can generated it computationnaly, so we can diagonalise again to get: F_omega * times omega, and of course we can apply Hal's successor (or caylor one of course) again, and again Oh Oh Oh Oh Oh A new pattern emerge (the Ackerman Caylor one, at a higher level). F_omega, F_omega + omega F_omega * omega F_omega ^ omega F_omega [4] omega (omega tetrated to omega, actually this ordinal got famous and is named Epsilon Zéro, will say some words on it later) F_omega [5] omega F_omega [6] omega F_omega [7] omega F_omega [8] omega F_omega [9] omega F_omega [10] omega F_omega [11] omega ... In this case they are all obtained by successive diagonalzations, but nothing prevent us to diagonalise on it again to get F_omega [omega] omega OK, I think the following finite number is big enough: F_omega [omega] omega (F_omega [omega] omega (9 [9] 9)) Next, we will meet a less constructivist fairy, and take some new kind of big leap. Be sure to be convinced that, despite the transfinite character of the F_alpha sequence, we did really defined at all steps precise computable growing functions ... (if not: ask question please). Tricky Problem: is there a sequence in which all growing computable functions belong? Is it possible to dovetail on all computable growing functions, ... I let you think, 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Bruno, You struck a personal nerve in me with your following remarks: Bruno Marchal wrote: They are degrees. The worst unreasonableness of a (platonist or classical or even intuitionist) machine is when she believes some plain falsity (like p ~p, or 0 = 1). The false implies all propositions, so that such machine believes everything, including everything about their maximal consistent extensions or histories (which does not exist). Those machines are just inconsistent. particularly , some plain falsity (like p ~p, or 0 = 1). Rather than treat these as 'blatantly false' I have been exploring the notion for several years .. 'what conditions, situations, criteria or states would allow such statements to be 'true', and what would it mean in how we define and manipulate and operate the rest of mathematics?'. I have discovered that an unprecedentedly un-appreciated realm of mathematical relations has existed right before our minds. The lack, having kept us trying to cope with 'anomalies' and math issues without the full toolkit of mathematical instruments. An example at the core of it is a most simplistic definition/equation. 1^1 = 1^0 [one to the exponent one equals one to the exponent zero] To all mathematicians, this is a toss-out absurdity, with no 'real meaning'. n^0 is a convenience tool at best ; along with 'n/0 is 'undefined''. We note the consistent/valid notation, but walk away from any active utility or application. My thesis is that doing so was a missed opportunity. To be hyper-consistent, the equation set-up 1^1 = 1^0 indicates that there -must- be some valid states/conditions (not just 'interpretation') when 0 and 1 are 'equal' in some real meaning/use of the word equal. If they can be substituted in the above equation, without changing a resultant of calculations (they are embedded in), then they must somewhere somehow in fact be identical in some way or condition. The entire ediface of physics is hamstrung because of this, because mathematical definitions and language compounded the error by applying - actually DIS-applying - a related concept .. the notion of 'extent' .. also known as 'dimension'. Physics and mathematics transform and wholly open up when we throw away the old concept of 'dimensionless' and instead reformulate -everything- as 'dimensional'. Including zero; including numbers unassociated with variables. As musch as you are brilliant and mathematically inventive, your statement some plain falsity (like p ~p, or 0 = 1) shows you haven't quite awoken to everything yet. I hope I'm in the process of stirring you from your slumber. Jamie Rose --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Hi Russell, You wrote (24 may): On Tue, May 23, 2006 at 12:25:35PM +0200, Bruno Marchal wrote: In a sense, you are obviously right. That is why I said some knowledge of comp science or even just in math will make the existence of the UD, and of the Universal Machine astonishing. Precisely it is the knowledge of diagonalization. Godel will miss the universal machine and Church thesis, and will describe those things as a sort of miracle. More later. I will comment again with much more detail the rest of your post much later. If I comment it here now I will introduce confusion. It is preferable people get much more familiarity with the effective and not effective daigonalisations procedures before, I think. I guess by this you mean that whilst it is impossible enumerate all descriptions (the books in the infinite version of the Library of Babel such as I take as my starting point), nor all true mathematical facts, or even all programs (not sure on this one, obviously one can enumerate all halting programs), it is however possible to execute all possible programs. Yes, put that way, I suppose it is astonishing. You put your finger on the difficulty. If we can enumerate all halting programs then we can diagonalize it and extract an halting program not belonging to the list. Actually when you say in your preceding post (21 May): That one can dovetail on all possible programs must be pretty obvious once one realises that these can be enumerated. You are pointing on the main difficulty. Once we can enumerate a list of functions from N to N, then we can diagonalize that list, and by this we can show the list being not complete. Now, it is not so hard for a computer programmer to single out a solution to this difficulty, but without a good understanding of diagonalization, it is easy to miss what is going on, and to miss in this way the tremendous impact of Church thesis. I will show that Church thesis will literally rehabilitate Pythagorus doctrine: all is number, despite irrational or transcendent numbers. Let me proceed further with the others because we are far ahead in the thread, Best regards, 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Hi George, Tom, Hal, and others, OK. I hope it is clear for everybody that, exactly like we have a natural infinite sequence of positive integer or natural numbers: 0, 1, 2, 3, 4, etc. We have a natural sequence of growing functions, (also called operations): ADDITION MULTIPLICATION EXPONENTIATION TETRATION PENTATION HEXATION HEPTATION OCTATION ENNEATION DECATION 11-ATION 12-ATION TRISKAIDEKATION 14-ATION 15-ATION 16-ATION 17-ATION ... (I remember the greek name of 13 thanks to the disease triskadekaphobia : the fear of the number 13 :) We can use the notation [n] for any n-ation, so that for example: 4 [1] 3 = 7, 4 [2] 3 = 12, 4 [3] 3 = 64, 4 [4] 3 = 134078079299425970995740249982058461274793658205923933777235614437217640 300735469768018742981669034276900318581864860508537538828119465699464336 49006084096, 4 [4] 4 = 4 ^ the preceding number [out-of-range of most computer without additional work!] etc. Let us write Fi(x) = x [i] x ; Indeed it will be more easy to illustrate diagonalization on one variable function: Thus F1(x) = x + x; F2(x) = x * x, F3(x) = x ^ x, F4(x) = x [4] x, F5(x) = x [5] x, F6(x) = x [6] x, etc. This gives us an infinite list of one variable growing functions F0 F1, F2, F3, F4, F5, F6, F7, ... Please note that I could have taken Hal Finney list, H0 H1 H2 H3 H4 H5 H6 H7 H8 H9 ...where H0(x) = factorial(x), H1(x) = factorial(factorial 2), H2(x) = factorial(factorial (factorial (x))), ... Mmmh... I am realizing it will even be easier to diagonalize transfinitely with Hal Finney's functions than with the traditional one, because with Hal Finney's one we will not been obliged of doing some back and forth between one and two variable functions. Anyway, let F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 ... be your favorite sequence of one-variable more and more growing function. (I recall all function here are function defined on N and with value in N; where N = the set of natural numbers : 0, 1, 2, 3, ... Here is a growing function, build from that class from diagonalization: g(x) = Fx(x) + 1 (in english: to compute g(x), search the xth function in your sequence, and apply it to x and then add 1. For example g(3) = F3(3) + 1, g(245) = F245(245) + 1, etc. Exercises: 0) Could you evaluate roughly the number of digit of 4 [4] 4 ? What about the number of digit of fact(fact(fact(fact 4 1) is the diagonal g function a growing function? Could g belong to the initial sequence, does g grows more quickly than any function in the initial sequence, and in what sense precisely. 2) Could you find a function, and even a new sequence of functions more and more growing, and growing more than the function g? 3) Do you see why it is said that g is build by diagonalization? Where is the diagonal? 4) Is there a universal sequence of growing functions, i. e. containing all computable growing functions? Must already go. Sorry for this quick piece. Solution tomorrow. Hope things are clear. Ask any elementary question (even about notation) before missing the real start ... Any comments , critics or suggestions are welcome ... 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Bruno Marchal wrote: Hi George, Tom, Hal, and others, OK. I hope it is clear for everybody that, exactly like we have a natural infinite sequence of positive integer or natural numbers: 0, 1, 2, 3, 4, etc. We have a natural sequence of growing functions, (also called operations): ADDITION MULTIPLICATION EXPONENTIATION TETRATION PENTATION HEXATION HEPTATION OCTATION ENNEATION DECATION 11-ATION 12-ATION TRISKAIDEKATION 14-ATION 15-ATION 16-ATION 17-ATION ... (I remember the greek name of 13 thanks to the disease triskadekaphobia : the fear of the number 13 :) We can use the notation [n] for any n-ation, so that for example: 4 [1] 3 = 7, 4 [2] 3 = 12, 4 [3] 3 = 64, 4 [4] 3 = 134078079299425970995740249982058461274793658205923933777235614437217640 300735469768018742981669034276900318581864860508537538828119465699464336 49006084096, 4 [4] 4 = 4 ^ the preceding number [out-of-range of most computer without additional work!] etc. Let us write Fi(x) = x [i] x ; Indeed it will be more easy to illustrate diagonalization on one variable function: Thus F1(x) = x + x; F2(x) = x * x, F3(x) = x ^ x, F4(x) = x [4] x, F5(x) = x [5] x, F6(x) = x [6] x, etc. This gives us an infinite list of one variable growing functions F0 F1, F2, F3, F4, F5, F6, F7, ... Please note that I could have taken Hal Finney list, H0 H1 H2 H3 H4 H5 H6 H7 H8 H9 ...where H0(x) = factorial(x), H1(x) = factorial(factorial 2), H2(x) = factorial(factorial (factorial (x))), ... Mmmh... I am realizing it will even be easier to diagonalize transfinitely with Hal Finney's functions than with the traditional one, because with Hal Finney's one we will not been obliged of doing some back and forth between one and two variable functions. Anyway, let F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 ... be your favorite sequence of one-variable more and more growing function. (I recall all function here are function defined on N and with value in N; where N = the set of natural numbers : 0, 1, 2, 3, ... Here is a growing function, build from that class from diagonalization: g(x) = Fx(x) + 1 (in english: to compute g(x), search the xth function in your sequence, and apply it to x and then add 1. For example g(3) = F3(3) + 1, g(245) = F245(245) + 1, etc. Exercises: 0) Could you evaluate roughly the number of digit of 4 [4] 4 ? What about the number of digit of fact(fact(fact(fact 4 1) is the diagonal g function a growing function? Could g belong to the initial sequence, does g grows more quickly than any function in the initial sequence, and in what sense precisely. 2) Could you find a function, and even a new sequence of functions more and more growing, and growing more than the function g? 3) Do you see why it is said that g is build by diagonalization? Where is the diagonal? 4) Is there a universal sequence of growing functions, i. e. containing all computable growing functions? Must already go. Sorry for this quick piece. Solution tomorrow. Hope things are clear. Ask any elementary question (even about notation) before missing the real start ... Any comments , critics or suggestions are welcome ... Bruno http://iridia.ulb.ac.be/~marchal/ I don't have time right now for detailed computations, but I'll give a few quick answers and questions. g is the same as my f(n,n,n)+1, and I already commented that f(n,m,n) is a growing function, since f(N,m,n) is growing for fixed N. So clearly g is growing. As I said about f(n,m,n), the degree or -ation of g grows as n (or x) grows. I recognize that adding 1 to make g is the classical diagonalization move. It makes g different from any Fi in the sequence Fi, i=1,2,3,... And in fact, since we add 1, rather than subtract 1, g is larger than any Fi. I'm having a problem with accepting g, or even my original f(n,n,n), as a function in the same sense as with a fixed degree or -ation of operation. This is because the definition of the function changes depending on the value taken in the domain of the function. Is this valid. However, if we just ignore this problem, throwing caution to the wind, then the next logical iteration of diagonalization is to do the -ation thing on g and then diagonalize. Let Gi(x) = g(x) [i] g(x), then let h(x) = Gx(x) + 1. Tom --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 22-mai-06, à 18:20, Tom Caylor a écrit : Bruno Marchal wrote: ... I give, for all, one last exercise before introducing diagonalization: define recursively in an explicit way the operation [i+1] from the preceding operation [i]. If you know a computer language (Fortran, Lisp, Prolog, c++, Java, whatever ...) write the program. If you don't know any such language, read my combinators posts and program those function with S and K, (if you have the time). Well, just be sure you follow the idea. Must leave now. Bruno I would have thought that my previous result captures this: Generalizing this, given the function in the sequence corresponding to the operation of degree N. f(N,m,n) = f(N-1,m,...f(N-1,m,n)...) (f(N-1) taken n times) Well, if the representation is explicit, you should avoid the But I think I see what you mean. If we express my f(i,m,n) as your m [i] n, then this would be m [i] n = m [i-1] (m [i-1] (...(m [i-1] n)...) ( [i-1] taken n times ) Same remark. Or if we just look at m [i] m to keep it simpler as you suggest, m [i] m = m [i-1] (m [i-1] (...(m [i-1] m)...) ( [i-1] taken n times ) I guess you mean m times (not n). In terms of a program, in a sort of pseudocode, to compute m [i] n, initialize result to (m [i-1] n) do the following n-1 times set result to (m [i-1] result) end do All right, this is explicit. Personally I prefer recursive coding. This is allowed in most modern language. But let us not take such implementation issue too seriously. The best would consist in implementing it by yourself on some real machine, in case you would doubt your code. The input is (m [i-1] n), the end result is m [i] n. If we simply want m [i] m, then set the input to (m [i-1] m). Of course in a real computer language you would have to worry about numerical representation and storage. Many high level language make it possible not to worry about such representations, and everything I will develop does not depend on those issues. More latter. Bruno PS I got your three last emails in double (in each post) I don't know why. 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 21-mai-06, à 10:53, Russell Standish a écrit : On Thu, May 18, 2006 at 11:38:24AM +0200, Bruno Marchal wrote: Also the universal dovetailer idea is also one of those that is fairly obvious, and might have been discovered a number of times independently. I'm not sure it is so easy, and in the present case I have never heard about some other papers. Frankly I am not sure you got it right. I guess it is subtle: there is a need of some amount in computer science to be astosnished that such a thing is logically possible. I will not develop this here because I intend to make this clear in my reply (or sequence of replies) to Tom and George. I'm not sure why a knowledge of computer science would make the UD astonishing. If anything, I would have thought the opposite. In a sense, you are obviously right. That is why I said some knowledge of comp science or even just in math will make the existence of the UD, and of the Universal Machine astonishing. Precisely it is the knowledge of diagonalization. Godel will miss the universal machine and Church thesis, and will describe those things as a sort of miracle. More later. I will comment again with much more detail the rest of your post much later. If I comment it here now I will introduce confusion. It is preferable people get much more familiarity with the effective and not effective daigonalisations procedures before, I think. I'm interested to read your post to Tom and George. Thanks for telling, The notion of dovetailing is really the theory behind timesharing, so simple dovetailing must be pretty obvious, at least since the early seventies. That one can dovetail on all possible programs must be pretty obvious once one realises that these can be enumerated. Of course the philosophial consequences of being able to do this is not so obvious, and as far as I know, you are the first person to have thought about that. Without the philosophical consequences, one would just think so what? So it is perhaps not surprising noone mentioned the UD before you. Then I am showing that the appearances of persons and realities are due to the incompleteness phenomena. I guess this is also a fairly simple idea in the air, but, like the UD, I have not seen it develop elsewhere, and it still gives me an hard and long time to make it clear as this very list can illustrate. And of course I can also be wrong, also. My work mainly consists in making that idea testable (and *partially* tested). I sympathise, but I'm still having trouble getting the connection too. Nevertheless, I find it intriguing. Which connection? Is it not utterly obvious that, IF we are (hopefully reasonable) machine, THEN we will learn something genuine by studying what (reasonable) machines can prove about themselves and about their possible neighborhoods and or histories? What is not obvious, is that computer science put strong constraints on the nature of possible machine realities, and with comp a case is made that all constraints comes from number theoretical relations (intensional and extensional(*)) and associated measures. And then the comp hyp is used just for making things easy. The only fundamental assumption which is needed for the reversal physics/numbers is the hypothesis of correct self-reference. But I don't want to anticipate at this stage. Bruno (*) extensional: number represents themselves; intensional: number can be used as code. Grosso modo: extensional number theory = number theory; intensional number theory = computer science, information science, provability logic. 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 23-mai-06, à 06:57, George Levy a écrit : One can create faster and faster rising functions and larger and larger number until one is blue in the face. The point is that no matter how large a finite number n one defines, I can stand on the shoulder of giants and do better by citing n+1 using simple addition. Now if somehow one came up with a finite number n so large that I am not allowed to say n+1 as if I was up against an overflow limitation similar to that found in computers, then there would be no physical way for me to invent or cite a larger number. So it seems that if we are to define a largest finite number we must define it in conjunction with the number b of bits that we are allowed to use to express this number. For a given number of bits b the largest number would be n(b). If we use the Ackerman series of functions we need 1 bit for addition, 2 bits for multiplication, 3 bits for exponentiation, 4 bits for tetration etc... These bits are required in addition to the bits for the input parameter(s) of the function. What is the largest number of bits which are available to me to define an Ackerman function or some other fast rising function? Possibly the number of particles in the universe? I don't know if the fairy would be satisfied or if I could personally herd all those bits. The fairy gives you some amount of papers. She is fair enough to provide more if you ask politely ;) The goal is to name a finite but as huge as possible number. It does not (obviously) consist in naming the biggest number (which does not exist as your little reasoning above shows clearly), nor does it consist in writing the best possible solution with respect to an available number of bits, (although we *will* arrive at this (much less simple) problem later). Is she expecting me to hand in a piece of paper with the number written on it? Maybe then the answer would be the number generated by the largest Ackerman function that I can write with a very fine pen on this piece of paper. Actually this can be considered as a good answer in the sense that the Ackermann number are already unimaginably gigantic, but that's nothing compared to the number which we will obtain by diagonalizations. Remember that my goal is to explain diagonalization. Actually, the goal of this thread is to explain Smullyan's heart of the matter in his FU book. For this we need not only to understand diagonalisation, but we will need to understand varieties of effective (programmable) and non effective diagonalizations before. I'm a bit sorry for the work I'm asking you ... 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
of addition, which is the preceding function in the sequence of functions. f(2,m,n) = f(2,m,n-1) + m = f(1,m,f(2,m,n-1)) = f(1,m,f(2,m,n-2)+m) = f(1,m,f(1,m,f(2,m,n-2))) = f(1,m,f(1,m,f(2,n-3,m)+m)) = f(1,m,f(1,m,f(1,m,f(2,m,n-3 = f(1,m,...f(2,m,n-n)+m)...) (f(1) taken n-1 times, f(2) taken 1 time) = f(1,m,...f(2,m,n-n)) (f(1) taken n times, f(2) taken 1 time) = f(1,m,...f(2,m,0)) (f(1) taken n times, f(2) taken 1 time) = f(1,m,...f(1,m,0)) (f(1) taken n times) = m * n The above is a formal way of saying that multiplication of m and n is adding n m's together. We knew that. Generalizing this, given the function in the sequence corresponding to the operation of degree N. f(N,m,n) = f(N-1,m,...f(N-1,m,n)) (f(N-1) taken n times) The above is a formal way of saying that f(N)ing of m and n together is the same as f(N-1)ing n m's together. The sequence of ever growing functions is defined as f(N,m,n) for N= 1,2,3,... Given a function of degree N, I take the growth of f(N,m,n) as defined as its magnitude as n approaches infinity. So here's a thought toward finding a function that's bigger than any function in this sequence. Define the following function. d(m,n) = f(1,m,...f(n,m,n)) Note that n has been placed not only as the counting operation, but also as the degree! So now as n approaches infinity, the degree approaches infinity. (!) So here is a single function that has a degree of operation that is higher than any function of a given degree of operation. Am I on the right track? Tom http://iridia.ulb.ac.be/~marchal/ X-Google-Language: ENGLISH,ASCII Received: by 10.54.160.18 with SMTP id i18mr191684wre; Mon, 22 May 2006 05:29:35 -0700 (PDT) Return-Path: [EMAIL PROTECTED] Received: from bonito.ulb.ac.be (bonito.ulb.ac.be [164.15.59.220]) by mx.googlegroups.com with ESMTP id v23si963617cwb.2006.05.22.05.29.34; Mon, 22 May 2006 05:29:35 -0700 (PDT) Received-SPF: pass (googlegroups.com: best guess record for domain of [EMAIL PROTECTED] designates 164.15.59.220 as permitted sender) Received: from mach.vub.ac.be (maxi.ulb.ac.be [164.15.128.8]) by bonito.ulb.ac.be (Postfix) with ESMTP id 9EDD72F for everything-list@googlegroups.com; Mon, 22 May 2006 14:29:33 +0200 (CEST) Received: by mach.vub.ac.be (Postfix, from userid 21099) id 7E9C78D27; Mon, 22 May 2006 14:29:33 +0200 (CEST) Received: from [164.15.10.83] (post.ulb.ac.be [164.15.10.83]) by mach.vub.ac.be (Postfix) with ESMTP id 1AE668D02 for everything-list@googlegroups.com; Mon, 22 May 2006 14:29:32 +0200 (CEST) Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable In-Reply-To: [EMAIL PROTECTED] References: [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] Message-Id: [EMAIL PROTECTED] From: Bruno Marchal [EMAIL PROTECTED] Subject: Re: Smullyan Shmullyan, give me a real example Date: Mon, 22 May 2006 14:29:26 +0200 To: everything-list@googlegroups.com X-Mailer: Apple Mail (2.623) X-Spam-Checker-Version: maxi.ulb.ac.be SA 2.63 (2004-01-11) X-Spam-Status: No, hits=0.0, level Hi Tom, Apparently you have (re)discover Ackermann function which indeed provide formally a sequence of more and more growing functions, similar to the sequence I was pointing too. I will present it in a easier way for the benefit of the others, but also for using a presentation which will facilitate the future successive diagonalizations. In your second recent post you got the first diagonalization right (or almost right) but the diag is quite hidden and it would be hard to continue the process. (Note that you could have define f(0, m, n) as the successor function) So your sequence of functions are, in your Ackerman like parametrized presentation, the functions f(1, m, n)is m + n (or m [1] n, writting + as [1] ; for first function) f(2, m, n)is m * n (or m [2] n) f(3, m, n)is m ^ n (or m [3] n) f(4, m, n)is m [4] n f(5, m, n)is m [5] n etc It will be easier to diagonalize functions of one argument, that is x + x, x * x, x ^ x, x [4] x, x[5]x, etc. Let us see their values on 10: 1) 10 + 10 = 20 2) 10 * 10 = 10+10+10+10+10+10+10+10+10 = 100 (ten sums) 3) 10 ^ 10 = 10*10*10*10*10*10*10*10*10 = 100 (ten products) 4) 10 [4] 10 = 10 ^ (10 ^ (10 ^ (10 ^ (10 ^ (10 ^ (10 ^ (10 ^ (10 ^ 10) (ten exponentiations) Note the parenthesis: schoolboys/girls know that a ^ (b ^ c) is in general different and bigger than (a ^ b) ^ c . Note also that 10 [4] 10 is already so big that the known observed part
Re: Smullyan Shmullyan, give me a real example
Bruno Marchal wrote: ... I give, for all, one last exercise before introducing diagonalization: define recursively in an explicit way the operation [i+1] from the preceding operation [i]. If you know a computer language (Fortran, Lisp, Prolog, c++, Java, whatever ...) write the program. If you don't know any such language, read my combinators posts and program those function with S and K, (if you have the time). Well, just be sure you follow the idea. Must leave now. Bruno I would have thought that my previous result captures this: Generalizing this, given the function in the sequence corresponding to the operation of degree N. f(N,m,n) = f(N-1,m,...f(N-1,m,n)...) (f(N-1) taken n times) If we express my f(i,m,n) as your m [i] n, then this would be m [i] n = m [i-1] (m [i-1] (...(m [i-1] n)...) ( [i-1] taken n times ) Or if we just look at m [i] m to keep it simpler as you suggest, m [i] m = m [i-1] (m [i-1] (...(m [i-1] m)...) ( [i-1] taken n times ) In terms of a program, in a sort of pseudocode, to compute m [i] n, initialize result to (m [i-1] n) do the following n-1 times set result to (m [i-1] result) end do The input is (m [i-1] n), the end result is m [i] n. If we simply want m [i] m, then set the input to (m [i-1] m). Of course in a real computer language you would have to worry about numerical representation and storage. Tom X-Google-Language: ENGLISH,ASCII-7-bit Received: by 10.11.53.63 with SMTP id b63mr128186cwa; Mon, 22 May 2006 09:20:25 -0700 (PDT) X-Google-Token: _qvdtQwAAACtJ1G4Hlrpr8kdXHeLayAm Received: from 199.46.245.234 by j73g2000cwa.googlegroups.com with HTTP; Mon, 22 May 2006 16:20:25 + (UTC) From: Tom Caylor [EMAIL PROTECTED] To: Everything List everything-list@googlegroups.com Subject: Re: Smullyan Shmullyan, give me a real example Date: Mon, 22 May 2006 09:20:25 -0700 Message-ID: [EMAIL PROTECTED] References: [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] User-Agent: G2/0.2 X-HTTP-UserAgent: Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.0),gzip(gfe),gzip(gfe) X-HTTP-Via: 1.0 tus-gate5.raytheon.com:8080 (Squid/2.4.STABLE7) Mime-Version: 1.0 Content-Type: text/plain Bruno Marchal wrote: ... I give, for all, one last exercise before introducing diagonalization: define recursively in an explicit way the operation [i+1] from the preceding operation [i]. If you know a computer language (Fortran, Lisp, Prolog, c++, Java, whatever ...) write the program. If you don't know any such language, read my combinators posts and program those function with S and K, (if you have the time). Well, just be sure you follow the idea. Must leave now. Bruno I would have thought that my previous result captures this: Generalizing this, given the function in the sequence corresponding to the operation of degree N. f(N,m,n) = f(N-1,m,...f(N-1,m,n)...) (f(N-1) taken n times) If we express my f(i,m,n) as your m [i] n, then this would be m [i] n = m [i-1] (m [i-1] (...(m [i-1] n)...) ( [i-1] taken n times ) Or if we just look at m [i] m to keep it simpler as you suggest, m [i] m = m [i-1] (m [i-1] (...(m [i-1] m)...) ( [i-1] taken n times ) In terms of a program, in a sort of pseudocode, to compute m [i] n, initialize result to (m [i-1] n) do the following n-1 times set result to (m [i-1] result) end do The input is (m [i-1] n), the end result is m [i] n. If we simply want m [i] m, then set the input to (m [i-1] m). Of course in a real computer language you would have to worry about numerical representation and storage. Tom --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
On Thu, May 18, 2006 at 11:38:24AM +0200, Bruno Marchal wrote: Also the universal dovetailer idea is also one of those that is fairly obvious, and might have been discovered a number of times independently. I'm not sure it is so easy, and in the present case I have never heard about some other papers. Frankly I am not sure you got it right. I guess it is subtle: there is a need of some amount in computer science to be astosnished that such a thing is logically possible. I will not develop this here because I intend to make this clear in my reply (or sequence of replies) to Tom and George. I'm not sure why a knowledge of computer science would make the UD astonishing. If anything, I would have thought the opposite. I'm interested to read your post to Tom and George. The notion of dovetailing is really the theory behind timesharing, so simple dovetailing must be pretty obvious, at least since the early seventies. That one can dovetail on all possible programs must be pretty obvious once one realises that these can be enumerated. Of course the philosophial consequences of being able to do this is not so obvious, and as far as I know, you are the first person to have thought about that. Without the philosophical consequences, one would just think so what? So it is perhaps not surprising noone mentioned the UD before you. Then I am showing that the appearances of persons and realities are due to the incompleteness phenomena. I guess this is also a fairly simple idea in the air, but, like the UD, I have not seen it develop elsewhere, and it still gives me an hard and long time to make it clear as this very list can illustrate. And of course I can also be wrong, also. My work mainly consists in making that idea testable (and *partially* tested). I sympathise, but I'm still having trouble getting the connection too. Nevertheless, I find it intriguing. Bruno http://iridia.ulb.ac.be/~marchal/ -- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
One can create faster and faster rising functions and larger and larger number until one is blue in the face. The point is that no matter how large a finite number n one defines, I can stand on the shoulder of giants and do better by citing n+1 using simple addition. Now if somehow one came up with a finite number n so large that I am not allowed to say n+1 as if I was up against an overflow limitation similar to that found in computers, then there would be no physical way for me to invent or cite a larger number. So it seems that if we are to define a largest finite number we must define it in conjunction with the number b of bits that we are allowed to use to express this number. For a given number of bits b the largest number would be n(b). If we use the Ackerman series of functions we need 1 bit for addition, 2 bits for multiplication, 3 bits for exponentiation, 4 bits for tetration etc... These bits are required in addition to the bits for the input parameter(s) of the function. What is the largest number of bits which are available to me to define an Ackerman function or some other fast rising function? Possibly the number of particles in the universe? I don't know if the fairy would be satisfied or if I could personally herd all those bits. Is she expecting me to hand in a piece of paper with the number written on it? Maybe then the answer would be the number generated by the largest Ackerman function that I can write with a very fine pen on this piece of paper. George --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
I've been working on this off and on when I get a chance, even before my first guess. My version of this defines an operation as a recursive function f(N,m,n), where N is the degree of the operation. m is one of the operands. n is the other operand, which is the counting operand. n is the number iterations that the recursive function is evaluated. I'll call addition the operation of degree 1. So addition can be defined as follows. Initial value: f(1,m,0) = m Recursion rule: f(1,m,k) = f(1,m,k-1) + 1 So for a given n, f(1,m,n) = f(1,...f(1,m,0) + 1,... + 1) (f(1) taken n times) = f(1,m,0) + 1 + ... + 1(1 added n times) = f(1,m,0) + n = m + n Note that counting can be separately defined as the operation of degree 0, but this didn't add to the current argument. Counting is also equivalent to adding with m=0. Multiplication is defined in a similar manner, as the operation of degree 2. Initial value: f(2,m,0) = 0 Recursion rule: f(2,m,k) = f(2,m,k-1) + m So for a given n, f(2,m,n) = f(2,...f(2,m,0) + m,... + m) (f(2) taken n times) = f(2,m,0) + m + ... + m(m added n times) = f(2,m,0) + m * n = m * n Exponentiation is the operation of degree 3. f(3,m,0) = 1 f(3,m,k) = f(3,m,k-1) * m f(3,m,n) = f(3,...f(3,m,0) * m,...* m) (f(3) taken n times) = f(3,m,0) * m * ... * m(m multiplied n times) = f(3,m,0) * m ^ n = m ^ n Just for kicks, I tried to define hyper-nentiation as the operation of degree 4, with operator symbol @. I was interested in what the initial value of this would be: the mth root of m. Any further than this gets too weird for me. f(4,m,0) = m^(1/m) f(4,m,1) = m f(4,m,k) = f(4,m,k-1) ^ m f(4,m,n) = f(4,...f(4,m,0) ^ m,...^ m) (f(4) taken n times) = m ^ m ^ ... ^ m(m exponentiated n times) = m @ n Looking closer specifically at multiplication, we see that it is defined in terms of addition, which is the preceding function in the sequence of functions. f(2,m,n) = f(2,m,n-1) + m = f(1,m,f(2,m,n-1)) = f(1,m,f(2,m,n-2)+m) = f(1,m,f(1,m,f(2,m,n-2))) = f(1,m,f(1,m,f(2,n-3,m)+m)) = f(1,m,f(1,m,f(1,m,f(2,m,n-3 = f(1,m,...f(2,m,n-n)+m)...) (f(1) taken n-1 times, f(2) taken 1 time) = f(1,m,...f(2,m,n-n)) (f(1) taken n times, f(2) taken 1 time) = f(1,m,...f(2,m,0)) (f(1) taken n times, f(2) taken 1 time) = f(1,m,...f(1,m,0)) (f(1) taken n times) = m * n The above is a formal way of saying that multiplication of m and n is adding n m's together. We knew that. Generalizing this, given the function in the sequence corresponding to the operation of degree N. f(N,m,n) = f(N-1,m,...f(N-1,m,n)) (f(N-1) taken n times) The above is a formal way of saying that f(N)ing of m and n together is the same as f(N-1)ing n m's together. The sequence of ever growing functions is defined as f(N,m,n) for N= 1,2,3,... Given a function of degree N, I take the growth of f(N,m,n) as defined as its magnitude as n approaches infinity. So here's a thought toward finding a function that's bigger than any function in this sequence. Define the following function. d(m,n) = f(1,m,...f(n,m,n)) Note that n has been placed not only as the counting operation, but also as the degree! So now as n approaches infinity, the degree approaches infinity. (!) So here is a single function that has a degree of operation that is higher than any function of a given degree of operation. Am I on the right track? Tom X-Google-Language: ENGLISH,ASCII-7-bit Received: by 10.11.53.63 with SMTP id b63mr96488cwa; Sun, 21 May 2006 00:08:25 -0700 (PDT) X-Google-Token: MV9CDAwAAABW1UuTDcJqFpeal26hqLve Received: from 207.200.116.67 by y43g2000cwc.googlegroups.com with HTTP; Sun, 21 May 2006 07:08:25 + (UTC) From: Tom Caylor [EMAIL PROTECTED] To: Everything List everything-list@googlegroups.com Subject: Re: Smullyan Shmullyan, give me a real example Date: Sun, 21 May 2006 00:08:25 -0700 Message-ID: [EMAIL PROTECTED] References: [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] User-Agent: G2/0.2 X-HTTP-UserAgent: Mozilla/4.0 (compatible; MSIE 6.0; AOL 9.0; Windows NT 5.1; Q312461; SV1; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) X-HTTP-Via: HTTP/1.1 (Velocity/1.3.32 [uScMs f p eN:t cCMp s ]), HTTP/1.1 Turboweb [ntc-tc091 8.4.0], HTTP/1.0 cache-ntc-ab03.proxy.aol.com[CFC87443] (Traffic-Server/6.1.0 [uScM]) Mime-Version: 1.0 Content-Type: text/plain I've been working on this off and on when I get a chance, even before my first guess. My version of this defines an operation as a recursive function f(N,m,n), where N
Re: Smullyan Shmullyan, give me a real example
To be slightly more clear d(m,n) = f(1,m,f(2,m,f(3,m,f(4,m,...f(n,m,n)...) Note that the it's only the innermost function that has degree n. To simplify things, I suppose we could just consider f(n,m,n) by itself. This has the same property that as n approaches infinity, the degree of operation approaches infinity. This gives a larger growth (as n approaches infinity) than fixing the degree at any finite number. And then, instead of substituting n into the degree, we could substitute things like f(n,m,n) into the degree to get f(f(n,m,n),m,n). Tom X-Google-Language: ENGLISH,ASCII-7-bit Received: by 10.11.88.14 with SMTP id l14mr36988cwb; Sun, 21 May 2006 19:49:11 -0700 (PDT) X-Google-Token: xMTf1AwAAADf4R2x5ktCHDVWo87JexXS Received: from 207.200.116.67 by u72g2000cwu.googlegroups.com with HTTP; Mon, 22 May 2006 02:49:11 + (UTC) From: Tom Caylor [EMAIL PROTECTED] To: Everything List everything-list@googlegroups.com Subject: Re: Smullyan Shmullyan, give me a real example Date: Sun, 21 May 2006 19:49:11 -0700 Message-ID: [EMAIL PROTECTED] In-Reply-To: [EMAIL PROTECTED] References: [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] [EMAIL PROTECTED] User-Agent: G2/0.2 X-HTTP-UserAgent: Mozilla/4.0 (compatible; MSIE 6.0; AOL 9.0; Windows NT 5.1; Q312461; SV1; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) X-HTTP-Via: HTTP/1.1 (Velocity/1.3.32 [uScMs f p eN:t cCMp s ]), HTTP/1.1 Turboweb [ntc-td054 8.4.0], HTTP/1.0 cache-ntc-ab03.proxy.aol.com[CFC87443] (Traffic-Server/6.1.0 [uScM]) Mime-Version: 1.0 Content-Type: text/plain To be slightly more clear d(m,n) = f(1,m,f(2,m,f(3,m,f(4,m,...f(n,m,n)...) Note that the it's only the innermost function that has degree n. To simplify things, I suppose we could just consider f(n,m,n) by itself. This has the same property that as n approaches infinity, the degree of operation approaches infinity. This gives a larger growth (as n approaches infinity) than fixing the degree at any finite number. And then, instead of substituting n into the degree, we could substitute things like f(n,m,n) into the degree to get f(f(n,m,n),m,n). Tom --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 19-mai-06, à 23:46, George Levy a écrit :
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 ...).
Any potentially largest finite number n that I could name could be incremented by 1 so this finite number could not be the largest. The trick is not to name a particular number but to specify a method to reach the unreachable.
Well, if *you* try to give the biggest natural number, *you* will never stop, and you will not succeed in specifying a large number, and the fairy will not make your wish coming true, and you will gain nothing. It would be absurd not saying 10^100 under the pretext that you would have prefer to live (10^100)+1 years.
By trying to reach the unreachable, well, you are anticipating another fairy which I will present to you once you will be able to diagonalize without even thinking ...
Method 1) Use the fairy power against her.
I see you like to live dangerously. Or to die dangerously should I say
She says she has finite power. Ask for precisely the largest number of days she can provide with her finite power.
*any* FINITE number !
It is up to you to choose one in particular. And to succeed in describing which one.
Recall that the set of all finite things (or numbers) is infinite.
This method is similar to the robber's response when the victim asks him how much money do you want?: All the money in your pocket.
Method 2) Use the concept of limits Ask for as many days it would take to obtain a sum of 2 as terms in the series 1+1/2 + 1/4 + 1/8 + 1/16. If the fairies knows any math she may argue that the series never reaches 2.
And she is right!
On the other hand I may argue that in the limit it does reach 2.
Yes but to reach that limit you need an infinity of additions. The serie is convergent, and you can go as close as you want to 2 in finite steps, but 2 itself requires an infinity of steps, and the fairy asks you to specify a precise number, actually the biggest you can specify precisely (through some algorithm, program, description) using a reasonable amount of paper. The better is to describe a growing function applied to some number. Although it looks a little bit ridiculous, the child solution: 9 + 9 contains the basic good idea: applying a growing function you know (or can program), like + on a big number that you know, like 9. Of course 9 * 9 is better, and 9^9 is still better, ... (?)
Method 3) Come up with a unprovably non-halting problem:
Again, you are anticipating on the next sort of fairy I will present later. The current fairy is somehow constructivist, and ask you to specify a number as big as you can describe, but it must be a precisely well defined number.
For example ask for as many days as required digits in PI to prove that PI has a single repetition of a form such that digits 1 to n match digits n+1 to 2n. For example 2^0.5 = 1.4142135... has a single repetition (1 4 match 1 4) in which digits 1 to 2 match digits 3 to 4. Similarly 79^0.5=8.8881944 and 147^0.5= 12.12435565. Note that the repetition must include all numbers 1 to n from the beginning and match all number n+1 to 2n The problem with this approach is I don't know for sure if PI is repeatable or non-repeatable (according to above requirements.) I don't even know if this problem is unprovable. All I know is that the probability for any irrational to have a single repeat is about 0.. For PI the probability is much lower since I already know PI to a large number of digits and as far as I can see it does not repeat. However, with this approach I could be taking chances.
Indeed.
Diagonalization clearly allows you to specify a number outside any given set of number, but I have not been able to weave it into this argument.
I will say more in my reply to Hal Finney who gives a good start.
Re: Smullyan Shmullyan, give me a real example
Le 20-mai-06, à 01:17, Hal Finney a écrit : Bruno writes: Meanwhile just a few questions to help me. They are hints for the=20 problem too. Are you familiar with the following recursive program=20 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=20 recursive definition (program) for multiplication (assuming your=20 machine already know how to add)? Could you define exponentiation from multiplication in a similar way? =20 Could you find a function which would grow more quickly than=20 exponentiation and which would be defined from exponentiation like=20 exponentiation is defined from multiplication? Could you generalize all=20 this and define a sequence of more and more growing functions. Could=20 you then diagonalise effectively (=3D writing a program who does the=20 diagonalization) that sequence of growing functions so as to get a=20 function which grows more quickly than any such one in the preceding=20 sequence? Here's what I think you are getting at with the fairy problem. The point is not to write down the last natural number, because of course there is no such number. Right! Rather, you want to write a program which represents (i.e. would compute) some specific large number, and you want to come up with the best program you can for this, i.e. the program that produces the largest number from among all the programs you can think of. ... and write on a reasonable amount of paper provided by the fairy. Of course it must be a finite description. If we start with factorial, we could define a function func0 as: func0(n) = fact(n) Now this gets big pretty fast. func0(100) is already enormous, it's like a 150 digit number. However we can stack this function by calling it on itself. func0(func0(100)) is beyond comprehension. And we can generalize, to call it on itself as many times as we want, like n times: func1(n) = func0(func0(func0( ... (n))) ... ))) where we have nested calls of func0 on itself n times. All right. You provide a sequence of growing functions: func0(n), func0(func0(n), etc. And then you get func1(n) by an n-iteration of func0. This really gets bigger fast, much faster than func0. Then we can nest func1: func2(n) = func1(func1(func1( ... (n))) ... ))) where again we have nested calls of func1 on itself n times. We know that func1(n) gets bigger so fast, func1(func1(n)) will get bigger amazingly faster, and of course with n of them it is that much faster yet. This clearly generalizes to func3, func4, Now we can step up a level and define hfunc1(n) = funcn(n), the nth function along the path from func1, func2, func3, Wow, imagine how fast that gets bigger. hfunc is for hyperfunc. Then we can stack the hfuncs, and go to an ifunc, a jfunc, etc. Well, my terminology is getting in the way since I used letters instead of numbers here. But if I were more careful I think it would be possible to continue this process more or less indefinitely. You'd have program P1 which continues this process of stacking and generalizing, stacking and generalizing. Then you could define program P2 which runs P1 through n stack-and-generalize sequences. Then we stack-and-generalize P2, etc. It never ends. But it's not clear to me how to describe the process formally. All right. I will do this a little bit more formally, but you got the right idea. Eventually we will see that there is no systematic way to get the biggest number by such a procedure, so that it is not so important which one to choose, except it is better to choose one which is such that we can easily describe a big part of that sequence of sequences of sequences ... of growing functions. Your stack and generalize seems to correspond to diagonalizations. So we have this ongoing process where we define a series of functions that get big faster and faster than the ones before. I'm not sure how we use it. Maybe at some point we just tell the fairy, okay, let me live P1000(1000) years. Yes. Well, if you have still some remaining place you can write P1000(P1000(1000)). That's a number so big that from our perspective it seems like it's practically infinite. Absolutely! We should ask the fairy if she provides the growing brain needed for living a so long time. If the brain is not growing, given that its dimension are very small compare to such big number, it will cycle! But of course from the infinite perspective it seems like it's practically zero. Right, but then this will be the case with any fairy capable of providing only finite (but long) life. In a sense, all numbers are big, and even *very big*, and very very big, etc ... Except for a finite number of them. Computer scientist say almost all number have property P when all numbers have
Re: Smullyan Shmullyan, give me a real example
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 ...).
Any potentially largest finite number n that I could name could be
incremented by 1 so this finite number could not be the largest. The
trick is not to name a particular number but to specify a method to
reach the unreachable.
Method 1) Use the fairy power against her. She says she has "finite
power". Ask for precisely the largest number of days she can provide
with her "finite power." This method is similar to the robber's
response when the victim asks him "how much money do you want?": "All
the money in your pocket."
Method 2) Use the concept of "limits" Ask for as many days it would
take to obtain a sum of 2 as terms in the series 1+1/2 + 1/4 + 1/8 +
1/16. If the fairies knows any math she may argue that the series
never reaches 2. On the other hand I may argue that "in the limit" it
does reach 2.
Method 3) Come up with a unprovably non-halting problem: For
example ask for as many days as required digits in PI to prove that PI
has a single repetition of a form such that digits 1 to n match
digits n+1 to 2n. For example 2^0.5 = 1.4142135... has a single
repetition (1 4 match 1 4) in which digits 1 to 2 match digits 3 to 4.
Similarly 79^0.5=8.8881944 and 147^0.5= 12.12435565. Note that the
repetition must include all numbers 1 to n from the beginning and match
all number n+1 to 2n The problem with this approach is I don't know for
sure if PI is repeatable or non-repeatable (according to above
requirements.) I don't even know if this problem is unprovable. All I
know is that the probability for any irrational to have a single repeat
is about 0.. For PI the probability is much lower since I already
know PI to a large number of digits and as far as I can see it does not
repeat. However, with this approach I could be taking chances.
Diagonalization clearly allows you to specify a number outside any
given set of number, but I have not been able to weave it into this
argument.
George
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Re: Smullyan Shmullyan, give me a real example
Bruno writes: Meanwhile just a few questions to help me. They are hints for the=20 problem too. Are you familiar with the following recursive program=20 for computing the factorial function? fact(0) =3D 1 fact (n) =3D n * fact(n - 1) Could you compute fact 5, from that program? Could you find a similar=20 recursive definition (program) for multiplication (assuming your=20 machine already know how to add)? Could you define exponentiation from multiplication in a similar way? =20 Could you find a function which would grow more quickly than=20 exponentiation and which would be defined from exponentiation like=20 exponentiation is defined from multiplication? Could you generalize all=20 this and define a sequence of more and more growing functions. Could=20 you then diagonalise effectively (=3D writing a program who does the=20 diagonalization) that sequence of growing functions so as to get a=20 function which grows more quickly than any such one in the preceding=20 sequence? Here's what I think you are getting at with the fairy problem. The point is not to write down the last natural number, because of course there is no such number. Rather, you want to write a program which represents (i.e. would compute) some specific large number, and you want to come up with the best program you can for this, i.e. the program that produces the largest number from among all the programs you can think of. If we start with factorial, we could define a function func0 as: func0(n) = fact(n) Now this gets big pretty fast. func0(100) is already enormous, it's like a 150 digit number. However we can stack this function by calling it on itself. func0(func0(100)) is beyond comprehension. And we can generalize, to call it on itself as many times as we want, like n times: func1(n) = func0(func0(func0( ... (n))) ... ))) where we have nested calls of func0 on itself n times. This really gets bigger fast, much faster than func0. Then we can nest func1: func2(n) = func1(func1(func1( ... (n))) ... ))) where again we have nested calls of func1 on itself n times. We know that func1(n) gets bigger so fast, func1(func1(n)) will get bigger amazingly faster, and of course with n of them it is that much faster yet. This clearly generalizes to func3, func4, Now we can step up a level and define hfunc1(n) = funcn(n), the nth function along the path from func1, func2, func3, Wow, imagine how fast that gets bigger. hfunc is for hyperfunc. Then we can stack the hfuncs, and go to an ifunc, a jfunc, etc. Well, my terminology is getting in the way since I used letters instead of numbers here. But if I were more careful I think it would be possible to continue this process more or less indefinitely. You'd have program P1 which continues this process of stacking and generalizing, stacking and generalizing. Then you could define program P2 which runs P1 through n stack-and-generalize sequences. Then we stack-and-generalize P2, etc. It never ends. But it's not clear to me how to describe the process formally. So we have this ongoing process where we define a series of functions that get big faster and faster than the ones before. I'm not sure how we use it. Maybe at some point we just tell the fairy, okay, let me live P1000(1000) years. That's a number so big that from our perspective it seems like it's practically infinite. But of course from the infinite perspective it seems like it's practically zero. Hal Finney --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 11-mai-06, à 13:38, Russell Standish a écrit : On Thu, May 11, 2006 at 01:00:31PM +0200, Bruno Marchal wrote: I think Schroedinger used the cat for explaining a paradoxical feature of QM, and I have not see suggestions by him that comp leads to either many world or quantum immortality (as Everett and Deutsch will do for the many-world, but not the immortality question though. I think that in the priority matter (a boring subject but then friends said that I must defend myself a little bit more) the criteria is the date of the publication. It is one thing to get an idea and a different thing to publish it. You need to fçind the idea but also to James Higgo found a 1986 publication by Euan Sqires that mentions the immortality argument. Perhaps that's not too much earlier for you to claim independent discovery in your 1988 paper. Still the point is, its one of those ideas that's floating around anyway - in the ether, so to speak. Sure. Also the universal dovetailer idea is also one of those that is fairly obvious, and might have been discovered a number of times independently. I'm not sure it is so easy, and in the present case I have never heard about some other papers. Frankly I am not sure you got it right. I guess it is subtle: there is a need of some amount in computer science to be astosnished that such a thing is logically possible. I will not develop this here because I intend to make this clear in my reply (or sequence of replies) to Tom and George. In some ways, these ideas are too simple for the issue of priority to be taken seriously. Perhaps, but the fame game is fickle indeed. Famous people are often not famous for their most important work. My most cited paper according to Google Scholar On complexity and emergence doesn't contain any original ideas at all! (Its a digestion of what I've read on the topics) On the other hand your COMP ontological reversal idea is truly unique. Hopefully you are right, and it goes down in history as your greatest contribution to human knowledge. Well thanks for that. In my opinion the UD, the UDA, the Universal Machine and Church Thesis are all deeply linked together (once in the TOE context). About the priority, I don't care so much, my point consisted mainly in the fact that the quantum immortality is a sub-case of comp immortality, which, by the way, can even be considered as a sub-case of the usual Pythagorico-Platonist-Plotino-Cartesian argument for the immortality of the soul which has been proposed by the intellectual greeks for about a millennium in Occident (more or less -500 to +500 JC era). Then I am showing that the appearances of persons and realities are due to the incompleteness phenomena. I guess this is also a fairly simple idea in the air, but, like the UD, I have not seen it develop elsewhere, and it still gives me an hard and long time to make it clear as this very list can illustrate. And of course I can also be wrong, also. My work mainly consists in making that idea testable (and *partially* tested). 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 16-mai-06, à 02:22, Russell Standish a écrit : An observer attaches a meaning to the data e observes. The set of all such meanings is semantic space or meaning space. I believe this is necessarily a discrete set (but not necessarily finite). If you have the time to define formally your meaning space I could be interested. I don't founf it in your writings. The details, of course are in my paper Why Occams Razor. To summarise, an observer induces a map O(x) from the space of descriptions, which is equivalent AFAIK to the output of your UD, ? The UD has neither inputs nor outputs. (like any universe or everything, note) Perhaps I'm being a little casual in my terminology. What I'm referring to is UD*. But UD* is not even a program. It is the trace (at some level) of the entire (infinite) execution of the UD. Why does constructibility, or otherwise have anything to do with the 1/3 person distinction? It is the logic of the self-extending self. It is akin to Brouwer's theory of consciousness, which is a root of its intuitionist philosophy. It is among the confirmed point through the fact that the theaetetical variant Bp p (the soul hypostase), although not constructive per se, does lead to an arithmetical interpretation of intuitionism, like the intelligible and sensible matter hypostases lead to a form of arithmetical quantization. I am willing to concede that there is possibly more to the WR problem, but I have yet to see it expressed in a manner I can understand :). It is the whole point of the UDA to help making this clear. Perhaps I am wrong but I think you underestimate the fact that the first person are not aware of the delays (numbers of steps) make by the UD to reach computational continuing states. Think about a highly discontinuous function with continuous derivatives: we can only be conscious of the derivative because we does not feel either the splitting or bifurcation, nor the discontinuous jumps. Everyone is free to download my last presentation of the UDA (my SANE paper), and tell me at which point they believed the argument does not proceed. A step could be wrong or not well supported perhaps, but until now, honest scientists who take time to verify the argument does not see anything wrong, and most abandon comp in a way or another because they cannot really swallow the platonist reversal ... Well, in the worst case I will come back next millennium ;-) 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 16-mai-06, à 17:31, Tom Caylor a écrit :
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
Re: Smullyan Shmullyan, give me a real example
On Mon, May 15, 2006 at 03:51:56PM +0200, Bruno Marchal wrote:
Le 15-mai-06, à 13:59, Russell Standish a écrit :
OK, why not taking that difference [description/computation] into
account. I think it is a
crucial point.
I do :). However, its makes no difference as far as I can tell to the
Occam's razor issue.
You do? See below.
given a reference Turing machine U. This appears
to be a 3rd person description, but it need not be so.
I am not sure I understand.
Do you mean you don't think it is a 3rd person description, or do you
mean you don't think it can be anything else?
I think it is a third person description.
That's what I suspect most people think. My point is that it needn't
be, and it is in fact inherently first person. I make this point in
many different papers, as well as my book.
In the fairness of scientific discussion, I am willing to be shown
wrong, of course :)
The details, of course are in my paper Why Occams Razor. To
summarise, an observer induces a map O(x) from the space of
descriptions, which is equivalent AFAIK to the output of your UD,
? The UD has neither inputs nor outputs. (like any universe or
everything, note)
Perhaps I'm being a little casual in my terminology. What I'm
referring to is UD*.
to
the space of meanings.
Which space is it? What do you mean (here) by meanings?
An observer attaches a meaning to the data e observes. The set of all
such meanings is semantic space or meaning space. I believe this is
necessarily a discrete set (but not necessarily finite).
If it is a
mathematical semantics then which one, if not, I don't understand. I
already ask you similar question after my first reading of your Occam).
For any given meaning y, let omega(y,l) be the
number of equivalent descriptions of length l mapping to y (for
infinite length description we need the length l prefixes). So
omega(y,l) = |{x: O(x)=y len(x)=l}|
Now P(y) = lim_{l-\infty} omega(y,l)/2^l is a probability
distribution, related to the Solomonoff-Levin universal
distribution.
C(y)=-log_2 P(y)
is a complexity measure related to Kolmogorov Complexity.
Note that this approach is non constructive (and thus cannot be first
person, at least as I use it and modelize it). I have already argued
that it can be refined through the notion of depth (a la Bennett),
which takes a notion of long computation into account; but it is
still incomplete relatively to the first person indeterminacy problem
(pertaining on the set of *all* (relative) computations, and not at all
on the set of descriptions).
The non-constructibility is a problem here, given the goal of deducing
physical laws or principles without physics.
And now I don't understand you. Why does constructibility, or
otherwise have anything to do with the 1/3 person distinction?
If you have succeed in eliminating all the many person pov - white
rabbits, then publish!
Well, I have! One thing you can't accuse me of is not publishing my
ideas.
Frankly it seems to me you don't really address the first person issue
(and thus the mind/body issue).
Yes - you've said that before, and its a point I've never understood.
For example, what is your theory of
mind? In particular, do you say yes to the comp doctor?
Pretty much everything thing I've done summarises the theory of the
mind by the function O(x). It maps descriptions (aka bitstrings) to
meaning. I do make use of a robustness property, which essentially is
that O^{-1}(y) is not of measure zero, but that is about it.
In particular, none of my results depend on whether I would say yes to
the comp doctor or not.
I think that eventually, we have to limit ourself to the discourses
that a self-referentially correct machine (or entity, or growing
entities of such lobian kind) can have about herself and her
possibilities.
And I think you could be right, or even approximately right. At this
stage, we need to explore.
I am not saying your argument is wrong, just that is incomplete (and
unclear, but this could be my incompetence).
Bruno
Well, of course it is incomplete if you're looking for a TOE. For the
White Rabbit issue, the argument is quite simple. I have conceived of
the White Rabbit problem in a certain way: the unreasonable
effectiveness of mathematics, the (non-)failure of induction. It
certainly appears to me that the argument addresses this conclusively,
from a first person point of view, however, there is always room for
doubt that I have overlooked some nuance.
I am willing to concede that there is possibly more to the WR problem,
but I have yet to see it expressed in a manner I can understand :).
Where I suspect most people might come unstuck is justifying why
formula (1) from On Complexity and Emergence should be called
complexity. The reason comes down its connection with Kolmogorov
complexity - it is the
Re: Smullyan Shmullyan, give me a real example
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. 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.
First question: Is this the same as Douglas Hoftstadter's supernatural
numbers (in his book Godel, Escher, Bach)? 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.
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. 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.
Tom
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Re: Smullyan Shmullyan, give me a real example
Le 15-mai-06, à 02:04, Russell Standish a écrit : I guess it is a delicate point, a key point though, which overlaps the ASSA/RSSA distinction (that is: the Absolute Self Sampling Assumption versus the Relative Self Sampling Assumption). If you identify a conscious first person history with a third person describable computation, it can be argued that an explanation for physics can be given by Bayesian sort of anthropic reasoning based on some universal probability distribution like Hall Finney's Kolmogorovian UDist. Note tat this approach relies also on Church Thesis. Here somehow the TOE will be a winning little program. I agree that this would hunt away the third person white rabbits. I disagree. The UDist comes from looking at the measure induced on a set of descriptions OK. (or computations if your prefer, It is not the same. It changes the whole problem, especially from the Relative SSA (Self-sampling assumption). although the two are not equivalent), OK, why not taking that difference into account. I think it is a crucial point. given a reference Turing machine U. This appears to be a 3rd person description, but it need not be so. I am not sure I understand. As I have pointed out (but suspect it hasn't really sunk in yet), U can be taken to be the observer erself. I could agree, but U cannot *know* e is U. Need some bet or act of faith. In general if U describes the observer, he is a big number in need of an explanation. I mean, the existence of big stable U is what we try to explain. When done this way, there is a 1st person universal distribution, with a corresponding 1st person Occam razor theorem. And this implies the absence of 1st person white rabbits. I really don't understand. 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
On Mon, May 15, 2006 at 11:17:35AM +0200, Bruno Marchal wrote:
Le 15-mai-06, à 02:04, Russell Standish a écrit :
I guess it is a delicate point, a key point though, which overlaps the
ASSA/RSSA distinction (that is: the Absolute Self Sampling Assumption
versus the Relative Self Sampling Assumption).
If you identify a conscious first person history with a third
person
describable computation, it can be argued that an explanation for
physics can be given by Bayesian sort of anthropic reasoning based on
some universal probability distribution like Hall Finney's
Kolmogorovian UDist. Note tat this approach relies also on Church
Thesis. Here somehow the TOE will be a winning little program. I agree
that this would hunt away the third person white rabbits.
I disagree. The UDist comes from looking at the measure induced on a
set of descriptions
OK.
(or computations if your prefer,
It is not the same. It changes the whole problem, especially from the
Relative SSA (Self-sampling assumption).
although the two
are not equivalent),
OK, why not taking that difference into account. I think it is a
crucial point.
I do :). However, its makes no difference as far as I can tell to the
Occam's razor issue.
given a reference Turing machine U. This appears
to be a 3rd person description, but it need not be so.
I am not sure I understand.
Do you mean you don't think it is a 3rd person description, or do you
mean you don't think it can be anything else?
As I have
pointed out (but suspect it hasn't really sunk in yet), U can be
taken to be the observer erself.
I could agree, but U cannot *know* e is U. Need some bet or act of
faith.
In general if U describes the observer, he is a big number in need of
an explanation. I mean, the existence of big stable U is what we try
to explain.
Sure. In fact the argument does not even hinge upon the observer being a
UTM. However, if not, then the distribution is not exactly universal
(hence my quotes below).
When done this way, there is a 1st
person universal distribution, with a corresponding 1st person Occam
razor theorem. And this implies the absence of 1st person white
rabbits.
I really don't understand.
Bruno
The details, of course are in my paper Why Occams Razor. To
summarise, an observer induces a map O(x) from the space of
descriptions, which is equivalent AFAIK to the output of your UD, to
the space of meanings. For any given meaning y, let omega(y,l) be the
number of equivalent descriptions of length l mapping to y (for
infinite length description we need the length l prefixes). So
omega(y,l) = |{x: O(x)=y len(x)=l}|
Now P(y) = lim_{l-\infty} omega(y,l)/2^l is a probability
distribution, related to the Solomonoff-Levin universal
distribution.
C(y)=-log_2 P(y)
is a complexity measure related to Kolmogorov Complexity.
Basically this is an Occams Razor theorem - the probability of
observing something decreases dramatically with its observed
complexity. And this is a pure 1st person result. It doesn't get rid
of all white rabbits, but the remaining ones are dealt with the
Malcolm-Standish argument.
http://iridia.ulb.ac.be/~marchal/
--
A/Prof Russell Standish Phone 8308 3119 (mobile)
Mathematics0425 253119 ()
UNSW SYDNEY 2052 [EMAIL PROTECTED]
Australiahttp://parallel.hpc.unsw.edu.au/rks
International prefix +612, Interstate prefix 02
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Re: Smullyan Shmullyan, give me a real example
Le 15-mai-06, à 13:59, Russell Standish a écrit :
OK, why not taking that difference [description/computation] into
account. I think it is a
crucial point.
I do :). However, its makes no difference as far as I can tell to the
Occam's razor issue.
You do? See below.
given a reference Turing machine U. This appears
to be a 3rd person description, but it need not be so.
I am not sure I understand.
Do you mean you don't think it is a 3rd person description, or do you
mean you don't think it can be anything else?
I think it is a third person description.
snip
I really don't understand.
Bruno
The details, of course are in my paper Why Occams Razor. To
summarise, an observer induces a map O(x) from the space of
descriptions, which is equivalent AFAIK to the output of your UD,
? The UD has neither inputs nor outputs. (like any universe or
everything, note)
to
the space of meanings.
Which space is it? What do you mean (here) by meanings? If it is a
mathematical semantics then which one, if not, I don't understand. I
already ask you similar question after my first reading of your Occam).
For any given meaning y, let omega(y,l) be the
number of equivalent descriptions of length l mapping to y (for
infinite length description we need the length l prefixes). So
omega(y,l) = |{x: O(x)=y len(x)=l}|
Now P(y) = lim_{l-\infty} omega(y,l)/2^l is a probability
distribution, related to the Solomonoff-Levin universal
distribution.
C(y)=-log_2 P(y)
is a complexity measure related to Kolmogorov Complexity.
Note that this approach is non constructive (and thus cannot be first
person, at least as I use it and modelize it). I have already argued
that it can be refined through the notion of depth (a la Bennett),
which takes a notion of long computation into account; but it is
still incomplete relatively to the first person indeterminacy problem
(pertaining on the set of *all* (relative) computations, and not at all
on the set of descriptions).
The non-constructibility is a problem here, given the goal of deducing
physical laws or principles without physics.
Basically this is an Occams Razor theorem - the probability of
observing something decreases dramatically with its observed
complexity. And this is a pure 1st person result.
?
It doesn't get rid
of all white rabbits, but the remaining ones are dealt with the
Malcolm-Standish argument.
If you have succeed in eliminating all the many person pov - white
rabbits, then publish!
Frankly it seems to me you don't really address the first person issue
(and thus the mind/body issue). For example, what is your theory of
mind? In particular, do you say yes to the comp doctor?
I think that eventually, we have to limit ourself to the discourses
that a self-referentially correct machine (or entity, or growing
entities of such lobian kind) can have about herself and her
possibilities.
I am not saying your argument is wrong, just that is incomplete (and
unclear, but this could be my incompetence).
Bruno
http://iridia.ulb.ac.be/~marchal/
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Re: Smullyan Shmullyan, give me a real example
On Thu, May 11, 2006 at 01:00:31PM +0200, Bruno Marchal wrote: I think Schroedinger used the cat for explaining a paradoxical feature of QM, and I have not see suggestions by him that comp leads to either many world or quantum immortality (as Everett and Deutsch will do for the many-world, but not the immortality question though. I think that in the priority matter (a boring subject but then friends said that I must defend myself a little bit more) the criteria is the date of the publication. It is one thing to get an idea and a different thing to publish it. You need to fçind the idea but also to James Higgo found a 1986 publication by Euan Sqires that mentions the immortality argument. Perhaps that's not too much earlier for you to claim independent discovery in your 1988 paper. Still the point is, its one of those ideas that's floating around anyway - in the ether, so to speak. Also the universal dovetailer idea is also one of those that is fairly obvious, and might have been discovered a number of times independently. In some ways, these ideas are too simple for the issue of priority to be taken seriously. Perhaps, but the fame game is fickle indeed. Famous people are often not famous for their most important work. My most cited paper according to Google Scholar On complexity and emergence doesn't contain any original ideas at all! (Its a digestion of what I've read on the topics) On the other hand your COMP ontological reversal idea is truly unique. Hopefully you are right, and it goes down in history as your greatest contribution to human knowledge. Cheers -- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
On Sat, May 13, 2006 at 05:28:31PM +0200, Bruno Marchal wrote: Le 12-mai-06, à 09:41, Kim Jones a écrit : Bruno, I almost understand this. Just expand a little Kim On 11/05/2006, at 9:00 PM, Bruno Marchal wrote: Schmidhuber did leave the list by refusing explicitly the first-third person distinction (which explain why his great programmer does not need to dovetail). I guess it is a delicate point, a key point though, which overlaps the ASSA/RSSA distinction (that is: the Absolute Self Sampling Assumption versus the Relative Self Sampling Assumption). If you identify a conscious first person history with a third person describable computation, it can be argued that an explanation for physics can be given by Bayesian sort of anthropic reasoning based on some universal probability distribution like Hall Finney's Kolmogorovian UDist. Note tat this approach relies also on Church Thesis. Here somehow the TOE will be a winning little program. I agree that this would hunt away the third person white rabbits. I disagree. The UDist comes from looking at the measure induced on a set of descriptions (or computations if your prefer, although the two are not equivalent), given a reference Turing machine U. This appears to be a 3rd person description, but it need not be so. As I have pointed out (but suspect it hasn't really sunk in yet), U can be taken to be the observer erself. When done this way, there is a 1st person universal distribution, with a corresponding 1st person Occam razor theorem. And this implies the absence of 1st person white rabbits. Cheers -- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 12-mai-06, à 09:41, Kim Jones a écrit : Bruno, I almost understand this. Just expand a little Kim On 11/05/2006, at 9:00 PM, Bruno Marchal wrote: Schmidhuber did leave the list by refusing explicitly the first-third person distinction (which explain why his great programmer does not need to dovetail). I guess it is a delicate point, a key point though, which overlaps the ASSA/RSSA distinction (that is: the Absolute Self Sampling Assumption versus the Relative Self Sampling Assumption). If you identify a conscious first person history with a third person describable computation, it can be argued that an explanation for physics can be given by Bayesian sort of anthropic reasoning based on some universal probability distribution like Hall Finney's Kolmogorovian UDist. Note tat this approach relies also on Church Thesis. Here somehow the TOE will be a winning little program. I agree that this would hunt away the third person white rabbits. Despite the obvious appeal for such an approach, once we take into account the fact that we cannot know in which computations we belong, and that we are not aware of the delay of a universal dovetailer to rich the computationally accessible computational states, then we realize that we need to take into account the fact that almost all programs which generate us are *big*. Our consciousness is somehow distributed in the whole of the comp-platonia (a non comp structure!). Here somehow the TOE could still be given by a little program, but it needs a justification how it can win an infinite battle with the big programs, and eliminate a vaster collection of first person white rabbits. (BTW we are very close to Descartes fifth meditation if you know. His malin génie generates first person hallucinations). All this follows from the UDA (Universal Dovetailer Argument). From empiry it could be that the winning little program describes some quantum universal dovetailer, or an universal unitary transformation, modular functor (topological quantum computer), etc. but all what I try to explain is that such little program must be justified as being invariant for some notion of first person (plural) observable taking into account the infinities of infinite computations (once we make explicit the comp (or weaker) assumption. By identifying first and third person experience we need only one successful computation as an explanation. By being aware of the 1-3 distinction we have to dovetail on all computations and (re)defined reality as a relative measure on the possible ways of glueing consistent first person experience; if not, I'm afraid the mind body problem remains under the rug. Hope that help a little bit. Don't hesitate to ask more explanations. Just be patient if I don't answer so quickly. Some more technical points will be made clearer through the deepening of diagonalization, perhaps. Critics from ASSA people are welcome! 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Bruno, I almost understand this. Just expand a little Kim On 11/05/2006, at 9:00 PM, Bruno Marchal wrote: Schmidhuber did leave the list by refusing explicitly the first-third person distinction (which explain why his great programmer does not need to dovetail). --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
On Fri, 12 May 2006, Saibal Mitra wrote: Einstein seems to have believed in ''immortal observer moments''. In a BBC documentary about time it was mentioned that Einstein consoled a friend whose son had died in a tragic accident by saying that relativity suggests that the past and the future are as real as the present. I'm sure Einstein would turn in his grave at your quoted expression. An immortal moment is a contradiction in terms, unless it implies a second time which passes as we contemplate first time embedded in 4-D space-time. Unfortunately a lot of popular discussion of space-time implicitly invokes this spurious second time, because it is hard to decouple the language of existence from the language of existence *in time*. To believe, with Einstein, that all points in space-time are equally real (because the relativity of simultaneity means that there is no unique now) is quite the opposite of the nutty idea that all events exist now --- not even wrong, from Einstein's point of view. Einstein actually expressed this view in a letter of condolence to the widow of his old friend Michele Besso. His words are worth quoting accurately: In quitting this strange world he has once again preceded me by just a little. That doesn't mean anything. For we convinced physicists the distinction between past, present, and future is only an illusion, however persistent. Later physicists, in particular John Bell, have pointed out that relativity doesn't *prove* that now is an illusion, it just makes it impossible to identify any objective now. Not that any of this has anything to do with the sort of immortality contemplated by Everett, which is not at all enticing: like the Sibyl in classical myth, his immortality would not be accompanied by eternal youth... a rather horrible fate. --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
From: Patrick Leahy [EMAIL PROTECTED] To: everything-list@googlegroups.com Sent: Friday, May 12, 2006 12:56 PM Subject: Re: Smullyan Shmullyan, give me a real example On Fri, 12 May 2006, Saibal Mitra wrote: Einstein seems to have believed in ''immortal observer moments''. In a BBC documentary about time it was mentioned that Einstein consoled a friend whose son had died in a tragic accident by saying that relativity suggests that the past and the future are as real as the present. I'm sure Einstein would turn in his grave at your quoted expression. An immortal moment is a contradiction in terms, unless it implies a second time which passes as we contemplate first time embedded in 4-D space-time. Unfortunately a lot of popular discussion of space-time implicitly invokes this spurious second time, because it is hard to decouple the language of existence from the language of existence *in time*. To believe, with Einstein, that all points in space-time are equally real (because the relativity of simultaneity means that there is no unique now) is quite the opposite of the nutty idea that all events exist now --- not even wrong, from Einstein's point of view. Einstein actually expressed this view in a letter of condolence to the widow of his old friend Michele Besso. His words are worth quoting accurately: In quitting this strange world he has once again preceded me by just a little. That doesn't mean anything. For we convinced physicists the distinction between past, present, and future is only an illusion, however persistent. Later physicists, in particular John Bell, have pointed out that relativity doesn't *prove* that now is an illusion, it just makes it impossible to identify any objective now. Not that any of this has anything to do with the sort of immortality contemplated by Everett, which is not at all enticing: like the Sibyl in classical myth, his immortality would not be accompanied by eternal youth... a rather horrible fate. Thanks for the correction and the exact quote. I only vaguely remembered what was said in the program. Of course, ''immortal observer moment'' is indeed contradictory. The point is, of course, that ''now'' implies a localization in time just like ''here'' implies localization in space. Just like things that don't exist here but do exist elsewhere are ''real'' so should things that don't exist now anymore but did exist at some time in the past. Saibal --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 11-mai-06, à 01:07, Russell Standish a écrit : (Sadly, Everett's daughter Liz, in her later suicide note, said she was going to a parallel universe to be with her father...) Sadly, because this is based on a total misunderstanding of QTI, I guess. I guess and/or hope it was just a poetical way to express herself on her suicide. 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Hi George, Bruno, Thank you for still working on my post. I am working on the reply, in particular designing the set of function or number that can be diagonalized to generate a large number. I shall be busy this weekend with family matters but I will reply to you in detail. Take it easy we have all the time. In case of trouble don't hesitate to ask for hints or supplementary explanations. This is said for everyone. I agree that the idea of quantum suicide did not originate with Tegmark, even though he is the one who popularized it. Sure. He wrote a beautiful paper. Note that he is unaware of the more general and more simple (?) comp immortality. The idea also came to me independently in the early 1990's as I was pondering the Scroedinger cat experiment. What if I was the cat? How would I feel? What if I was the scientist conducting the experiment and I was inside a larger box enclosing the whole experiment? Would I feel the superposition? These are very obvious questions to ask. This Scroedinger cat experiment approximately dates to the 1920-1930's (?) and it is very well possible that others have had the same thought. I think Schroedinger used the cat for explaining a paradoxical feature of QM, and I have not see suggestions by him that comp leads to either many world or quantum immortality (as Everett and Deutsch will do for the many-world, but not the immortality question though. I think that in the priority matter (a boring subject but then friends said that I must defend myself a little bit more) the criteria is the date of the publication. It is one thing to get an idea and a different thing to publish it. You need to fçind the idea but also to get the nerves to make it public. I have not publish so much easy readable (original) thing not to insist a little bit on this, especially given that I still somehow paying a hard price for having dare to work on such questions in the seventies. I suspect a little bit Russell (notably in his book) to dismiss how much both the universal dovetailer and comp-immortality was (and still is actually) original in the TOE framework. Russell makes often (in posts and in his preprint book) the confusion between the notion of Universal Machine, Schmidhuber great programmer (which does not dovetail) and the Universal Dovetailer. Those notions are related but are not at all equivalent in the search for a TOE, neither extensionally, nor intensionally (different programs and different functions). I developed and defended those ideas very early. This explains in part why I have been confronted with an obvious natural skepticism, and this is why I have provided the logical analysis (well to be true I got this one simultaneously as I explained in the 1988 paper: it is even the reason why I have chose to do math and not physics). Actually the heart of the matter explanations will consist in showing how much universal dovetailing and Church thesis are non trivial notions. Paradoxically enough, the widespread use of computer hide the complexity. You need training in diagonalization to doubt Church's thesis! The same with the comp first person indeterminacy. Probably my main easy (for you in this quite open-minded list) discovery. Remember that Schmidhuber did leave the list by refusing explicitly the first-third person distinction (which explain why his great programmer does not need to dovetail). It is not just a question of priority, it is a question of getting the notions right before. Another point. james Higgo told us explicitly that, despite quantum suicide, Tegmark did not believe in the quantum immortality consequence of the quantum hyp, showing the big nuance between the immortality and suicide points, often confuse in posts or elsewhere. Oops I must leave ... Best, 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Einstein seems to have believed in ''immortal observer moments''. In a BBC documentary about time it was mentioned that Einstein consoled a friend whose son had died in a tragic accident by saying that relativity suggests that the past and the future are as real as the present. Saibal From: Russell Standish [EMAIL PROTECTED] To: everything-list@googlegroups.com Sent: Thursday, May 11, 2006 01:07 AM Subject: Re: Smullyan Shmullyan, give me a real example On Wed, May 10, 2006 at 11:13:27PM +0100, Patrick Leahy wrote: On who invented quantum suicide, the following is from the biography of Hugh Everett by Eugene B. Shikhovtsev and Kenneth W. Ford, at http://space.mit.edu/home/tegmark/everett/ Atheist or not, Everett firmly believed that his many-worlds theory guaranteed him immortality: His consciousness, he argued, is bound at each branching to follow whatever path does not lead to death --- and so on ad infinitum. (Sadly, Everett's daughter Liz, in her later suicide note, said she was going to a parallel universe to be with her father...) Sadly, because this is based on a total misunderstanding of QTI, I guess. The reference is to Everett's views in 1979-80, but there is no reason to suppose that Everett had only just thought of it at the time. On a personal note, some time in the '80s I met one of Everett's co-workers who told me that Everett used to justify his very unhealthy lifestyle on exactly these grounds. In our world, Everett died of a heart attack aged 52. I have always assumed that John Bell was thinking along these lines when he commented on Everett's theory: But if such a theory was taken seriously it would hardly be possible to take anything else seriously. (1981, reprinted in _Speakable Unspeakable in Quantum Mechanics). These dates all mesh with Don Page's anecdote about Ed Teller : immortality consequences widely known, but rarely talked about by the early '80s. For that matter, this idea is implicit in Borges' story The Garden of Forking Paths (written before 1941), which provides the epigraph to the DeWitt Graham anthology on The Many Worlds Interpretation. == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618 Very interesting. Its a shame my manuscript is already at the printers, I would have loved this for my background info on QTI. -- -- -- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 -- -- --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 10-mai-06, à 04:19, Russell Standish a écrit : James Higgo published a web page describing the history of quantum suicide aka comp suicide. The notion obvious predates both Tegmark and Marchal - and there is some anecdotal evidence that Edward Teller knew about the argument in the early eighties. It appears to have been a dirty little secret, which has only really been considered acceptable talk in polite scientific circles in the last 10 years or so. I explain quantum suicide, and I use it to explain the comp immortality in: Marchal B., Informatique théorique et philosophie de l’esprit. Actes du 3ème colloque international de l’ARC, Toulouse 1988. I have presented orally the paper at Toulouse in 1987. The paper contains the movie graph argument, and a much earlier version of that paper contains the RE paradox, one of many version of the UDA. That earlier paper has been published in two parts later under the forms: Marchal B., Mechanism and Personal Identity, proceedings of WOCFAI 91, M. De Glas D. Gabbay (Eds), Angkor, Paris, 1991. Marchal B., 1992, Amoeba, Planaria, and Dreaming Machines, in Bourgine Varela (Eds), Artificial Life, towards a practice of autonomous systems, ECAL 91, MIT press. Look, you can see my work as the given of a purely arithmetical (more generally lobian) reconstruction of Lucas-Penrose type of argument against mechanism. Only, such argument does not show that we are not machine but only that *in case* we are machine *then* we cannot know which machine we are, nor can we know which computational paths support us, and there is already an indeterminacy there. Then I illustrate that we (I mean the (hopefully) lobian machines) can reflect that indeterminacy. You can see it as a generalization of Everett's embedding of the physicists in the physical world; where instead I embed the mathematician (actually some arithmetican) in the mathematical (arithmetical) world. In both case this makes sense only when we distinguished first person and third person discourse. But now, my preceding point was just that the existence of the discourse about quantum suicide or quantum immortality, which appears from empirical reasons, confirms the general statement that comp implies that any machine looking at herself below its substitution level should discover empirically the indetermination about which computations which support her, from which the comp immortality follows. Obviously (?) I am suspecting a big part of the physical emerges already from the impossible statistics on number relations once you mix addition and multiplication. The advantage of the self-referential approach (just made easier by comp, but it works on many type of non-machine or generalized infinite machines) is that it provides at its roots a difference between the truth and the true discourses on those questions (got through G* \ G and its intensional variants), the arithmetical Hypostases as I am tempted to call them since I read Plotinus. You can see what I am mainly trying to say as: oh look we can *already* interview a universal machine about fundamental questions. I illustrate this by interviewing a lobian machine on the logics of the communicable, knowable and bettable (by Universal Machines) pertaining on verifiable propositions (here verifiable = accessible by the Universal Dovetailer. The goal: extract the whole measure on the relative continuations. Not just the logic of certainty. The problem at this stage is mathematical and concerns the existence of not of some Hopf algebra of trees capable of explaining how to renormalize in front of the arithmetical white rabbits. 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Bruno, Thank you for still working on my post. I am working on the reply, in particular designing the set of function or number that can be diagonalized to generate a large number. I shall be busy this weekend with family matters but I will reply to you in detail. I agree that the idea of quantum suicide did not originate with Tegmark, even though he is the one who popularized it. The idea also came to me independently in the early 1990's as I was pondering the Scroedinger cat experiment. What if I was the cat? How would I feel? What if I was the scientist conducting the experiment and I was inside a larger box enclosing the whole experiment? Would I feel the superposition? These are very obvious questions to ask. This Scroedinger cat experiment approximately dates to the 1920-1930's (?) and it is very well possible that others have had the same thought. George --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
On who invented quantum suicide, the following is from the biography of Hugh Everett by Eugene B. Shikhovtsev and Kenneth W. Ford, at http://space.mit.edu/home/tegmark/everett/ Atheist or not, Everett firmly believed that his many-worlds theory guaranteed him immortality: His consciousness, he argued, is bound at each branching to follow whatever path does not lead to death --- and so on ad infinitum. (Sadly, Everett's daughter Liz, in her later suicide note, said she was going to a parallel universe to be with her father...) The reference is to Everett's views in 1979-80, but there is no reason to suppose that Everett had only just thought of it at the time. On a personal note, some time in the '80s I met one of Everett's co-workers who told me that Everett used to justify his very unhealthy lifestyle on exactly these grounds. In our world, Everett died of a heart attack aged 52. I have always assumed that John Bell was thinking along these lines when he commented on Everett's theory: But if such a theory was taken seriously it would hardly be possible to take anything else seriously. (1981, reprinted in _Speakable Unspeakable in Quantum Mechanics). For that matter, this idea is implicit in Borges' story The Garden of Forking Paths (written before 1941), which provides the epigraph to the DeWitt Graham anthology on The Many Worlds Interpretation. == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618 --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
On Wed, May 10, 2006 at 11:13:27PM +0100, Patrick Leahy wrote: On who invented quantum suicide, the following is from the biography of Hugh Everett by Eugene B. Shikhovtsev and Kenneth W. Ford, at http://space.mit.edu/home/tegmark/everett/ Atheist or not, Everett firmly believed that his many-worlds theory guaranteed him immortality: His consciousness, he argued, is bound at each branching to follow whatever path does not lead to death --- and so on ad infinitum. (Sadly, Everett's daughter Liz, in her later suicide note, said she was going to a parallel universe to be with her father...) Sadly, because this is based on a total misunderstanding of QTI, I guess. The reference is to Everett's views in 1979-80, but there is no reason to suppose that Everett had only just thought of it at the time. On a personal note, some time in the '80s I met one of Everett's co-workers who told me that Everett used to justify his very unhealthy lifestyle on exactly these grounds. In our world, Everett died of a heart attack aged 52. I have always assumed that John Bell was thinking along these lines when he commented on Everett's theory: But if such a theory was taken seriously it would hardly be possible to take anything else seriously. (1981, reprinted in _Speakable Unspeakable in Quantum Mechanics). These dates all mesh with Don Page's anecdote about Ed Teller : immortality consequences widely known, but rarely talked about by the early '80s. For that matter, this idea is implicit in Borges' story The Garden of Forking Paths (written before 1941), which provides the epigraph to the DeWitt Graham anthology on The Many Worlds Interpretation. == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618 Very interesting. Its a shame my manuscript is already at the printers, I would have loved this for my background info on QTI. -- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Tom wrote My beside myself statement was a punny reference to self-reference. I meant that I am looking forward to your post(s) with positive eagerness. Thanks. Also, I will follow your suggestion to force me writing little post. I will first answer an older post by George (not to confuse with Georges). Le 25-mars-06, à 00:51, George Levy a écrit : I have read or rather tried to read Smullyan's book. His examples are totally fabricated. I will never meet the white knight in the island of liars and truthtellers. I need examples which are relevant to life, at least the way I understand it in the context of the many-worlds. The role of the knight-knave island is just to give an easy way to produce self-referential sentences? This is explain on page 48 in Forever Undecided (hereafter FU). Einstein (or maybe someone writing about relativity) came up with the paradox of the travelling aging twin. Was it not Langevin? Schroedinger came up with his cat's paradox. Tegmark came up with the quantum suicide experiment. I came up first with the comp suicide, and much later after with the quantum suicide and with the kill the user sort of quantum computation, well before Tegmark, and I am not sure this has helped to make my work more acceptable or comprehensible. For me quantum suicide was a confirmation of the fact, easily derivable from comp, that even for purely empirical reason we can doubt mortality, or doubt that the mortality issue is simple, like so many materialist tend to think. Granted, I will never travel near the speed of light; I will never put a cat in a box equipped with a random and automatized killing device; and I will not attempt suicide; my wife would just kill me. However, these examples fired up my imagination: travelling near the speed of light, existing in a superposition of state, surviving a nuclear bomb under your chair. Smullyan's white knigth had the mission to teach me about the logic of G and G*. Sorry, he failed. The white knight does not fire up my imagination. I don't care about his island and about his questions. It is exactly with the diagonal principle (FU page 211) that the logical role of the Knight-knave Island is eventually eliminated. Now his diagonal principle arises in the context of his Godelized Universe for which Smullyan don't provide motivation (it still look like a fairy tale). What makes Universe *Godelized* is really Church Thesis, and that is really the missing key in FU (and actually in the whole work of Smullyan). I make those steps more transparent in my SANE paper. have you print it? It should help. However I do care about life, death and immortality. The many-world does seem to guarantee a form of immortality, at least according to some interpretations. Yes, we have discuss this a lot. I think most people agree on this in the list, both with the quantum MWI, or with some all computations exist. I think the most serious involved people in this list just disagree on how to quantify the (quantum or comp) indeterminacy. Of course progress have been made on the quantum part of that problem, but hardly on the comp part, which is actually presupposed in the quantum MWI (cf Everett, Deutsch, ...). I consider this issue to be very relevant since sooner or later each one of us will be facing the issue of death or of non-death. Be careful with such motivation because it could lead t wishful thinking. I am not sure you can appreciate the comp lesson which shows above all the abyssal googelplexity or our ignorance. But then such an ignorance appears to have a mathematical shape capable of providing information, but this leads today just to hard mathematical questions. I would like someone to come up with an extreme adventure story like the travelling twin, Schroedinger's cat, or Tegmark's suicide experiment to illustrate G and G*. I think the whole UDA (Universal Dovetailer Argument) and the movie graph paradox (or argument), and/or Maudlin's Olympia device are going in that direction. G and G* are just the tools for making this technical enough so that it leads to testable propositions. UDA is really the intuitive (and rigorous although informal) path for an understanding of the reversal between physics and numbers, including the showing of how hard the comp (im)mortality puzzle is. For example this story would describe a close brush with death.. It would create a paradox by juxtaposing 1) classical or common sense logic assuming a single world, It seems to me that the UDA just does that. Do you see that classical physics is a priori untenable with comp once we take the 1-3 person distinction into account? I mean Classical physics is just epistemologically incompatible with common sense, once comp is assumed. 2) classical or common sense logic assuming the many-world, and See just above. 3) G/G* logic assuming the many-world.
Re: Smullyan Shmullyan, give me a real example
Le 27-mars-06, à 06:09, George Levy a écrit : I am looking forward to being diagonalized. I hope it won't hurt too much. Asap. Meanwhile you could already medidate on my first diagonalization post here. You can ask (out or online) any question including about notations or definitions: http://www.mail-archive.com/everything-list@eskimo.com/msg01561.html If you find that unreadable, tell me and I will think about other ways to present it, or links ... Also: did you grasp in FU the notions of: reasoner of type 1 reasoner of type 1* reasoner of type 2 reasoner of type 3 reasoner of type 4 and reasoner of type G ? Bruno x-tad-bigger /x-tad-biggerhttp://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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Bruno Marchal wrote: <> Le 25-mars-06, 00:51, George Levy a crit : Smullyan's white knigth had the mission to teach me about the logic of G and G*. Sorry, he failed. All right, but this is just because he miss Church Thesis and Comp. His purpose actually is just to introduce you to Godel and Lob theorems, not to computer science. The heart of the matter is that mathematical systems (machines, angels, whatever) cannot escape the diagonalisation lemma, and so life for them is like the life of those reasoners travelling on fairy knight Knave island with curious self-referential question. With comp *we* cannot escape those diagonal propositions. I am looking forward to examples involving people being diagonalized...hmmm Hilbert did come up with a thought experiment with an infinite number of people lodged in a hotel actually we want to go further than that and assume an infinite number of selves in the many-worldOnce upon many times (Ils etaient des fois...), there were several princesses...they looked into self referential magic mirrorsand they lived ever after. I would like someone to come up with an extreme adventure story like the travelling twin, Schroedinger's cat, or Tegmark's suicide experiment to illustrate G and G*. For example this story would describe a close brush with death.. It would create a paradox by juxtaposing 1) classical or common sense logic assuming a single world, I think you miss the diagonalization notion. I will work on that. I am looking forward to being diagonalized. I hope it won't hurt too much. I will give you "real examples", but don't throw out FU to quickly. \ OK. George --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
Le 25-mars-06, à 00:51, George Levy a écrit : Dear members of the list, Bruno and those who understand G. I have read or rather tried to read Smullyan's book. His examples are totally fabricated. I will never meet the white knight in the island of liars and truthtellers. Nor will any Lobian machines. The knight/knave Island is just a trick for having simple self-referential statements. I think you miss the heart of the matter section and the godelized universe. Not your fault and despite my love of Smullyan I am quasi willing to say Smullyan miss it too. The missing piece is Church thesis. And then with comp we can understand that lobian machines live in a Godelized Universe. I will come back on this. I need examples which are relevant to life, at least the way I understand it in the context of the many-worlds. OK, OK, I work on this since many years. Modal logics and Solovay's theorems provides a tools for progressing, but this need some understanding of computer science and mathematical logic. Einstein (or maybe someone writing about relativity) came up with the paradox of the travelling aging twin. Schroedinger came up with his cat's paradox. Tegmark came up with the quantum suicide experiment. Actually I came up before but this is anecdotical. But I have elaborated it in the comp frame. It is the UDA. You have acknowledge understanding it years ago. The interview of the lobian machine just illustrate how we can already interview of universal machine on the UDA question, and extract the logic of the physical propositions. Granted, I will never travel near the speed of light; I will never put a cat in a box equipped with a random and automatized killing device; and I will not attempt suicide; my wife would just kill me. However, these examples fired up my imagination: travelling near the speed of light, existing in a superposition of state, surviving a nuclear bomb under your chair. Smullyan's white knigth had the mission to teach me about the logic of G and G*. Sorry, he failed. All right, but this is just because he miss Church Thesis and Comp. His purpose actually is just to introduce you to Godel and Lob theorems, not to computer science. The heart of the matter is that mathematical systems (machines, angels, whatever) cannot escape the diagonalisation lemma, and so life for them is like the life of those reasoners travelling on fairy knight Knave island with curious self-referential question. With comp *we* cannot escape those diagonal propositions. The white knight does not fire up my imagination. I don't care about his island and about his questions. However I do care about life, death and immortality. The many-world does seem to guarantee a form of immortality, at least according to some interpretations. I consider this issue to be very relevant since sooner or later each one of us will be facing the issue of death or of non-death. I thought you did understand that comp entails different forms of immortality. The interveiw of the lobian machine makes it possible to get more precise consequences, including testable one (some already tested). I would like someone to come up with an extreme adventure story like the travelling twin, Schroedinger's cat, or Tegmark's suicide experiment to illustrate G and G*. For example this story would describe a close brush with death.. It would create a paradox by juxtaposing 1) classical or common sense logic assuming a single world, UDA shows rather directly the impossibility of single world. 2) classical or common sense logic assuming the many-world, ? and 3) G/G* logic assuming the many-world. Assuming comp, the third person worlds are the computational histories. An history is a computations as seen by from some internal point of view. the fact that correct self-referential propositions obeys G and G* makes it possible to describe those histories What would the white knight do if he were living in the many-world? What kind of situations would highlight his talent to think in G. Would his behavior appear to be paradoxical from our logical point of view? The white knight, (well actually any Knight on the Knight Knave Island!) are not even reasoners. None types of reasoner applies including G. The intuitive explanation why physics emerges from numbers and numbers' dream is already given in the UDA. Smullyan just introduce the logics of self-reference (the provable one, G, and the true one, G*). The relation with our field is the content of one half of my posts (the other half being UDA itself). I think you miss the diagonalization notion. I will work on that. I will give you real examples, but don't throw out FU to quickly. He makes something hard easy, but indeed don't give to much motivations, except some allusions to AI here and there. Bruno PS. I will answer other posts asap. http://iridia.ulb.ac.be/~marchal/
Smullyan Shmullyan, give me a real example
Dear members of the list, Bruno and those who understand G. I have read or rather tried to read Smullyan's book. His examples are totally fabricated. I will never meet the white knight in the island of liars and truthtellers. I need examples which are relevant to life, at least the way I understand it in the context of the many-worlds. Einstein (or maybe someone writing about relativity) came up with the paradox of the travelling aging twin. Schroedinger came up with his cat's paradox. Tegmark came up with the quantum suicide experiment. Granted, I will never travel near the speed of light; I will never put a cat in a box equipped with a random and automatized killing device; and I will not attempt suicide; my wife would just kill me. However, these examples fired up my imagination: travelling near the speed of light, existing in a superposition of state, surviving a nuclear bomb under your chair. Smullyan's white knigth had the mission to teach me about the logic of G and G*. Sorry, he failed. The white knight does not fire up my imagination. I don't care about his island and about his questions. However I do care about life, death and immortality. The many-world does seem to guarantee a form of immortality, at least according to some interpretations. I consider this issue to be very relevant since sooner or later each one of us will be facing the issue of death or of non-death. I would like someone to come up with an extreme adventure story like the travelling twin, Schroedinger's cat, or Tegmark's suicide experiment to illustrate G and G*. For example this story would describe a close brush with death.. It would create a paradox by juxtaposing 1) classical or common sense logic assuming a single world, 2) classical or common sense logic assuming the many-world, and 3) G/G* logic assuming the many-world. What would the white knight do if he were living in the many-world? What kind of situations would highlight his talent to think in G. Would his behavior appear to be paradoxical from our logical point of view? George Levy --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
