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 ^ [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
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 ^ [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 -~--~~~~--~~--~--~---
platinum-eaters and alien abductees
Some thoughts on the idea of longevity, which has come up in the recent "Smullyan Shmullyan" thread: Firstly, although at present I think I would like to live forever, I don't actually need to live forever to be happy with my lifespan. Rather, I only need to live until such time as I no longer mind dying. I could refine this last statement further if I want: I only need to live until such time as (a) I no longer mind dying; (b) I either don't expect that or don't care if in future I will mind dying; (c) I have reached this conclusion in the absence of depression or despair; and (d) whatever other state of mind I care to name that has a non-zero probability of occurring. I figure the requisite state of mind for a happy death might occur in as little as a few hundred years, and almost certainly within a few hundred thousand years of continuous cognition. Secondly, although a wish to die or indifference to one's survival is usually seen as evidence of mental illness, it need not logically occur in the setting of other symptoms of mental illness, such as depression, delusions and hallucinations (even though in practice it usually does). A person who wishes to die might be going against the "prime directive" of every naturally evolved organism, but he is not as a result of his death wish committing an error of logic or of empirical fact, in the way a person who is paranoid is. Evolution throws up organisms which want to live and reproduce, organisms which want to live and reproduce but whose metabolism is dependent on some very rare element, and organisms which don't want to live and reproduce. The first of these thrives, while the other two die out. If we are interested in who is being rational, the suicide has more in common with the platinum-eaters than with people who think they have been abducted by aliens. Finally, the very notion of continuity of personal identity, which is necessary if "survival" is to have any meaning, is just as much a product of evolutionary expedience. That is, it is no more logically necessary that an organism is the "same" individual from one moment to the next than it is logically necessary that an organism will strive to survive from one moment to the next. Those organisms which run away when a predator approaches because they believe they will be the same individual in the next moment will thrive, while those which believe that the organism with their approximate shape, memories, position etc. in the next moment is a completely different individual, and don't care if that other individual gets eaten, will die out. Such considerations do not apply to most of the devices that humans produce, which "replicate" on the basis of usefulness rather than a desire to survive and have progeny. A car does not care if it is wrecked for spare parts for use in another car, or a modern sculpture, or whatever, while even a non-sentient organism such as a bacterium is essentially a machine with no purpose other than maintaining its structural integrity from moment to moment and producing exact copies of itself. Stathis Papaioannou _ Express yourself instantly with MSN Messenger! Download today it's FREE! http://messenger.msn.click-url.com/go/onm00200471ave/direct/01/ --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
