SUMMARY (was: OM = SIGMA_1)
Title: SUMMARY (was: OM = SIGMA_1) I send to David Nyman (the 06 Nov 2007) a little planning: 1) Cantor's diagonal 2) Does the universal digital machine exist? 3) Lobian machines, who and what are they? 4) The 1-person and the 3- machine. 5) Lobian machines' theology 6) Lobian machines' physics 7) Lobian machines' ethics Let me summarize what has been done and what remains to be done. 1) Cantor's diagonal I tend to consider that the point 1) is finished. Cantor's argument is that if there is a bijection between natural numbers, that is: 0, 1, 2, 3, 4, ..., and sequences of numbers, that is a bijection like 0 --- 45, 7, 8976, 4, 32, ... 1 --- 0, 0, 67, 78, 0, ... 2 --- 27, 1, 24, 24, 23, ... 3 --- 1, 1, 1, 345, 7, ... ... then the antidiagonal sequence 46, 1, 25, 346, ... cannot be in the list, because by construction it differs from each sequence in the list. See below how to make explicit the contradiction. The reasoning does not depend on the particular sequences exhibited, and it shows that no enumerable set of sequences can be put in 1-1 correspondence with the natural numbers. The conclusion is that the set of all sequences of natural numbers is innumerable (not enumerable, not countable, uncountable, etc. Important concept have many synonym in math). Let me recall the same proof, but with usual mathematical notation. A sequence of numbers, like f_0 = 56, 7897876, 67, 89, 1, 1, 45, ... is really just a function from N to N: f_0(0), f_0(1) f_0(2), f_0(3), f_0(4), ... with here: f_0(0) = 56, f_0(1) = 7897876, f_0(2) = 67, f_0(3) = 89, f_0(4) = 1, etc. So the bijection above becomes: 0 --- f_0 = f_0(0), f_0(1) f_0(2), f_0(3), f_0(4), ... 1 --- f_1 = f_1(0), f_1(1) f_1(2), f_1(3), f_1(4), ... 2 --- f_2 = f_2(0), f_2(1) f_2(2), f_2(3), f_2(4), ... 3 --- f_3 = f_3(0), f_3(1) f_3(2), f_3(3), f_4(4), ... ... You can see that the diagonal sequence can be described by: f_0(0), f_1(1), f_2(2), f_n(n), ... Then the antidiagonal sequence (function) g is given by f_0(0)+1, f_1(1)+1, f_2(2)+1, f_n(n)+1, ... That is: g(n) = f_n(n)+1 (definition of g) Now we can make the contradiction explicit. Suppose that g is in the list f_i. Then it exists a number k such that g = f_k. This means of course that for all numbers n we have g(n) = f_k(n). In particular g(k) = f_k(k). But by the definition of g: g applied on k = g(k) = f_k(k)+1. Thus (by Leibniz identity rule): f_k(k) = f_k(k)+1 Now, all f_i are functions from N to N, so they are defined on all natural numbers, so f_k(k) is a number. We have seen in high school that identical numbers can be subtract on both sides of an equation leading to 0 = 1. (contradiction). Thus the f_i cannot enumerate all functions from N to N. We say: N^N is innumerable. This was point 1). Hope it is ok for every one. Please be sure you get the point before proceeding. 2) Does the universal digital machine exist? I recall the informal notion of what is an (intuitively) computable function (from N to N). Def: A function f from N to N is computable if we can describe in some formal language L, in a finite way, how to compute, in a finite time, its value f(n) on each natural number n. Def. I will call code of f such a description of how to compute f. Def. A language L is said universal if all computable functions can be described in the language. Def. A machine is universal if she understands a universal language, (and thus can indeed compute all computable functions from N to N, at least in Platonia, where Platonia is defined by a place where you can always ask and get more time and more space/memory: we don't put deadline to the (universal) machine. Church thesis is the statement that a universal language (and machine) exists, and indeed that in particular lambda-calculus provides such a universal language. Church's thesis is not obvious. Indeed, when Church defined the computable functions by those capable of being computed by a lambda-expression (a symbolic expression or code written in the lambda-calculus), Stephen Cole Kleene thought at first that a reasoning similar to Cantor's proof of the non enumerability of N^N (see above) could be made against Church's pretension. Kleene's reasoning is the following, and works for any pretension that there is a universal language (so we have not to even define what lambda-calculus). Indeed, suppose that there is a universal machine and thus a universal language in which all computable functions from N to N can be given a code. Now the set of codes in the language L is enumerable, being a subset of all possible expression written in the language (which we have seen to be enumerable). Thus there is an enumeration of all computable functions from N to N f_1, f_2, f_3, f_4, f_5, f_6, f_7, f_8, f_9, ... but then the antidiagonal function g defined by g(n) = f_n(n) + 1 is computable,
Re: Joining Post
Brent Meeker wrote: Actually it collapses before, see quant-ph/0402146 v1. It is shown that in a Young's slit experiment with C70 buckyballs, the interference fringes disappear when the buckyballs are sufficiently heated to radiate some IR photons. No observer is needed, only the interaction with the environment. Decoherence doesn't defeat quantum uncertainty, it partially hides the multiplicity of other worlds due to thermal connections of the environment. It results inevitably in tracing any single history. Decoherence is why Schrödinger placed the cat in a box, to isolate the experiment from the external observer. The colleague walking in after I have opened the box to observe the cat is also disconnected from the experiment in my description. For them the room is the box. Decoherence still applies to each history. It helps to remember that where Schrödinger's cat paradox shows how the uncertainty of a single electron can be amplified to produce widely diverse timelines, normally the uncertainty of trillions upon trillions of microscopic events entangle to construct a path of history. Decoherence is like placing a mirror in with the cat, the cat doesn't see its own phase space, each branching time line observes a near classical history, while the global superposition of worlds exists beyond the realm of measurement. Puppet implies you are pulling the strings. So can you bend the universe to your will? I had previously implied a hand puppet, and I was considering the implications of sampling the whole set of many worlds, as if they all exist simultaneously, and in each proceeding moment we find ourselves in one particular universe. This places in question the individuality and will of observed others apart from the probabilistic selection of the experienced world. The hand in the puppet is the universe itself. I am undecided on if the observer can bend reality. I don't rule such things out based on skepticism. An individual's will would largely be a product of their personal history, and thus physical events or states, so I do expect a considerable measure of entanglement between the mind/brain and the environment. I will be un-subscribing from this list. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: What has Schmidhuber been up to
Very funny! For anyone who doesn't read German, the error message says something along the lines of The film cannot be opened. In the film, an invalid public Movie-Atom was found. Cheers On Mon, Jan 07, 2008 at 07:47:33PM +0100, Tim Boykett wrote: Moviomium, a new element, has been recently created at the CERN digital physics lab deep in the heart of the alps in Switzerland. Rumours suggest that renegade Theorist of Everything J. Schmidhuber has applied one of his Goedel machines to the creation of new types of matter. Obviously some of this matter escaped an infected the new version of OSX. (Apologies for the German error message) -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Quantum Interference and the Plentitude
On Mon, Jan 07, 2008 at 02:48:35PM +0100, Mirek Dobsicek wrote: If quantum mechanics was done using a real-valued Hilbert space, you simply don't get wavelike interference patterns. To my knowledge, you don't get interference patterns for *positive* real-valued Hilbert space, but for real-valued Hilbert space you do. Check http://mina4-49.mc2.chalmers.se/~dobsicek/PhDThesis.pdf on page 39 for a quick review and references. Sincerely, Mirek Thanks for the info - I will take a look, when I'm on top of a few things. However, I'm not sure what you mean by positive real Hilbert space, as the positive real numbers do not form a field. I can only guess you mean some kind of non-Hilbert space generalisation, a bit like my non-Hilbert space non-commutative division ring gadgets I've alluded to in the past. Cheers -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: dark energy
Hi George: I see no feeling of anything in a Something. There is only an absence of the information needed to answer meaningful questions that are asked and must is be answered. Hal Ruhl At 11:13 PM 1/17/2008, you wrote: Hal, Allright. You are saying that incompleteness is the (only) motivator of the members. In other words the members feel motivated by incompleteness. They do have the feeling of being incomplete that motivates their behavior. Is this correct? George Hal Ruhl wrote: Hi George: I see no motivator to any dynamics within the Everything other than the incompleteness of some of its members and the unavoidable necessity to progressively resolve this incompleteness. Hal Ruhl At 12:29 AM 1/17/2008, you wrote: Hal Ruhl wrote: This is an automatic process like a mass has to answer to the forces [meaningful questions] applied to it. What in the psyche of the mass makes it answer to the forces? George --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
