I am putting this into a new topic. I tried the other day, but it did not work. This topic has nothing to do with AI singularity as I can see. I continue below
On 1 Mar 2018, at 17:22, Lawrence Crowell <goldenfield...@gmail.com> wrote: I am shifting this over to a new topic. On Thursday, March 1, 2018 at 4:53:04 AM UTC-6, Bruno Marchal wrote: On 28 Feb 2018, at 20:38, Lawrence Crowell <goldenfield...@gmail.com> wrote: On Wednesday, February 28, 2018 at 2:08:43 AM UTC-6, Bruno Marchal wrote: On 26 Feb 2018, at 18:02, Lawrence Crowell <goldenfield...@gmail.com> wrote: On Monday, February 26, 2018 at 5:53:05 AM UTC-6, Bruno Marchal wrote: On 24 Feb 2018, at 00:36, Lawrence Crowell <goldenfield...@gmail.com> wrote: On 23 Feb 2018, at 17:15, Lawrence Crowell <goldenfield...@gmail.com> wrote: >>>> >>>> The MH spacetime in the case of the Kerr metric does permit an observer >>>> in principle to witness an infinite stream of bits or qubits up to the >>>> inner horizon r_- that is continuous with I^+ in the exterior spacetime. >>>> This means due to spacetime effects one could witness the diagonalization >>>> in a Zeno machine context. For instance a switch that is switched one in >>>> one second, off the next half second, on in the next quarter second and so >>>> forth will presumably have a final state. However, what does prevent this >>>> in a fundamental way is that a switch flipped in this chirped frequency >>>> will diverge in energy and become a black hole before returning a result. >>>> We could of course avoid the black hole with a ball that bounces, but of >>>> course one does not get an infinite number of little bounces at the end. >>>> Because of this an observer could in principle witness a universal Turing >>>> machine emulate all possible Turing machines. Thinking according to TMs is >>>> for me a bit simpler, but this does illustrate one could get around Godel. >>>> >>>> Good project. It is partially done—in some sense, even from one special Diophantine equation, which is Turing-Universal, and QM and GR should (with mechanism) be “theory independent”. If you can derive them from the self-reference implicit in a Diophantine equation, you can derive them from any first order specification of any Turing-complete theory or Turing universal machinery. Three quantum logics appear already where expected, and the whole “many-worlds” aspect of Nature appears formally, and intuitively (through the many computations or the universal dovetailing implicit in all Turing universal system. The big advantage, compared to a physical “conventional” approach is that the Gödelian division between proof and truth, inherited in the “material” variants of the provability logics provides a mean to distinguish the sharable quanta from the private first person qualia. The quanta appears as particular case of first person plural qualia, which justifies the “multiplication of population” in arithmetic, which is confirmed (retrospectively) the “contagion” of the superposition, and thus the linearity at the bottom of the physical reality. Formally we get also the symmetry (which is quite astonishing formally, given that the “1p” variant of provability/believability are typically anti-symmetrical). Bruno Godel proved his original theorem as an incompleteness with Diophantine equations. Gödel’s use the full logic of arithmetic (actually, even of a richer typed set theory, but its arithmetical part plays the key role). The unsolvability (and Turing universality) of Diophantine equation took more than 50 years of intense research begun with Davis and Putnam (around 1950), developed significantly by Julia Robinson (sixties, seventies) , and transformed eventually by Yuri Matiazevich in the eighties (in Russian). That solved the Hilbert 10th problem in the negative: there are no mechanical procedure to solve Diophantine equations (assuming the Church-Turing thesis of course). Diophantine equations are also a number theoretic model for quantum eigen-values. This could connect with spacetime physics and MH spacetimes with Schild's ladder constructions. Interesting. Have you some link? Bruno Most of this aspect of things is in my mind and not on paper. I am doing a lot of work with the Ryu-Takayanagi formula and the existence of quantum error correction codes (QECC). The QECC works to correct "mistakes" when quantum bits or qubits are out of order within a limited Hamming distance. We might think of this as meaning if I write instead of the word quantum I write quanyum that in the alphabet the distance between t and y is 5 and so the Hamming distance is 5 on one entry. However if i write wismyin, one one keyboard shift away, the Hamming distance is much larger on all entries. With a set of text it becomes impossible to reconstruct it with an error correction code if you have lost about half the information. This is connected to the problem of the Page time for Hawking radiation and the so called firewall obstruction. We might then think of the MH spacetimes as those that are able to overcome the limits of Gödel by super enumeration. The example of the Zeno machine that flips a switch is illustrative, which is one way that these spacetimes are thought to circumvent Gödel. However, if spacetime is built from quantum entanglements there is then a QECC limitation. This appears then to limit the prospect for MH spacetimes in their property to become hyperTuring machines. There are a number of formal approaches to this. One that I am looking at is with hyperbolic spacetimes, such as the AdS, that has recursive arcs. These are similar to Ford circles or that define Farey sequences. The information content is then a Shannon formula, which is logarithmic and nonlinear, plus a linear term that multiplied by a sort of Lagrange multiplier on a pure quantum solution space. These linear terms on each wedge then connect in a tensor network (similar to MERA nets etc), and these have relationships to each other that may be defined by Diophantine equations. I will have to dig more on this. Note that when we take the computationalist first person indeterminacy into account, (the fact that no machine can know which machine she is, nor which computations in arithmetic she is supported by) leads to an inflation of possibilities, and the real mystery in the computationalist mind-body problem is why the physical reality seems so much computable. Now, even with computationalism, this can be exploited (the consequence of computationalism are still obtained if the brain was a quantum computer). Many think that computationalism entails that matter and consciousness are Turing emulable, but they are not, a bit like most arithmetical truth are far beyond the computable. Bruno Scott Aaronson wrote something like this on his Shtetl Optimized website a while back. He hypothesised or conjectured that every conscious entity carries a sort of incomputable Gödel type of number that is a code for their mental space. For quantum mechanics this might be important for understanding the problem of measurement should it turn out that mental consciousness is in some ways fundamental to that problem. At this point I am rather agnostic on that stance. It appears plausible to have quantum decoherence and the selection mechanism for actual outcomes not depend on consciousness. Now as I have said decoherence occurs when a set of quantum states in a system, in an experiment set in some prepared state or so called dressed state, are encoded within quantum states associated with a reservoir of states in a macroscopic system. For a measurement this macroscopic system is the apparatus. This has always struck me as a sort of Gödelian process. With this problem the issue is how quantum mechanics puts limits on the Malement-Hogarth spacetimes so they are not able to perform hypercomputations or serve as hyper-Turing machines. Hawking radiation puts a serious limit I think with Kerr-Newman and Reissnor-Nordstrom spacetimes. The inner horizon of these black holes has a pile up of states from the null infinity I^+ that makes this a Cauchy horizon. It is Cauchy because there is a pile up of null geodesics approaching the horizon. This means an observer can witness a Zeno machine computation that can circumvent the Gödel-Turing limit. However, spacetime is responsive to quantum fluctuations and this means eternal black holes do not exist. It would then seem quantum mechanics prevents physical systems from hypercompute. In a funny way general relativity does this as well. The Zeno machine formed by a switch that opens and closes each half shortened time interval could do much the same. However, the UV chirp in frequency is such there is an ultimate limit where this would become a black hole. Similarly a ball that bounces losing height can't bounce an infinite number of times because after a finite number of bounces there are quantum interval cut-offs, and even before that thermal fluctuations as well. It then appears that physics operates to protect the Gödel-Turing result. I do think that these situations do adjust Chaitan's halting probability. A system might not physically be able to hypercompute, but if it can compute "better" then there is maybe an enhanced probability for hitting the right answer, even if uncomputable. Cheers LC -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. 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