Hi George, Hi Hal, Hi All, I take opportunity of a remark by Hal Ruhl for giving you the crux of my arithmetical UDA in a nutshell.
Hal Ruhl wrote: >If one takes the position that "logical proof" is not universally relevant >to the evolution of universes within the Everything and the determination >of the sequence of successive states of a universe is replaced with >concepts such as "computable" or my "acceptable" what role does "logical >consistency" play? > >No matter how you rank the possibility of splitting universes, each >particular daughter universe is associated with a given state to state >transition [the transition is what I consider to be the observable] by >default. > >Logical consistency would seem to play no role. The transition is merely >"acceptable". More than one "acceptable" successor state enables splitting. To say that p is consistent is the same as saying that -p is not provable. Your critics on the usefulness of the notion of consistency is similar to the critics by Schmidhuber on the use of provability. You still seem to believe that there is a particular evolving formal systems capable of generating "our (perhaps splitting) universe". In particular you seem to believe that a pure third person description of that evolving system is enough. No problem. Perhaps you are right. But in that case comp is false. And you will need some strong non computationalist hypothesis. And you take the risk of hiding the mind-body problem in the process. (This follows from UDA and/or the movie-graph argument or Maudlin's one). With comp you have no choice, for isolating the laws of physics, other than to compute the measure on your computational (DU-accessible) consistent continuations as seen by some "first person point of view". That is what the UDA argument explicitely shows, isn'it? Some universal prior can still play a role (perhaps). I am not saying that your approach, or Schmidhuber one, is *completely* foolish. Just strongly incomplete. A Universal prior can play a role in the explanation of some aspect of our mean cosmological history, I think. Let us just look at the shape of the space of "probability 1 proposition" on the DU accessible computational extensions/continuations. A naive idea would be to try to define "Prob(p) = 1" by "p is provable" which I write []p, p arithmetical or simply p representable in the language of a sound machine. (Recall a machine is sound iff it proves only true sentences. Of course a sound machine is consistent). But Godel-Lob-Solovay have succeed in finding the complete modal axiomatisation of the provable "provability sentence", and even the true (but unprovable) provability sentence; respectively by the system G and G*. And G proves <>p -> -[]<>p. In Kripke terms, that means that from each transitory world you can access to a terminal world. The basic modal axioms for probability, which is []p -> <>p is not verified. The Kripke semantics for []p -> <>p needs ideal frames: frames without any terminal worlds! And also, the provability [] is typically third person propositions. So a less naive idea would be to define "Prob(p) = 1" by "p is provable and p is true". We define a new box [o] whith [o]p = []p & p. Of course the machine is sound, so for each arithmetical p we have that [o]p <-> []p. But the machine cannot know that! G does not prove [o]p <-> []p. Only G* proves it! So the nuance remains and that move is quite interesting. It lead us to a knowledge theory formalised by S4Grz. The Kripke frame of that theory is antisymmetrical and S4Grz, it can be, argued, gives a theory of subjective temporal knowledge very close to the time-consciousness theory of Brouwer. And S4Grz makes possible to isolate an arithmetical interpretation of intuitionist logic. But there is no probability theory attach to S4Grz. And worse! If we limit ourself to the verifiable (DU-accessible) arithmetical propositions, then the box and the diamond collapse and we get ... propositional calculus! (Actually the "first person" we get with S4Grz is so solipsistic that it cannot understand even the beginning of the UDA argument). So we need a still less naive move. Actually that moves is the one quite in the spirit of the everything list. The probability need to be taken on the *consistent* extensions. Why not just ask, for having "Prob(p) = 1", that p is provable *and* consistent. By incompleteness that is quite different than taking just p true. We define a new box [€]p by []p & <>p. We consider all and only the consistent extension (p DU-accessible). This is really what the UDA argument forces us to do for getting the physical. Note that G* still proves that [€]p is equivalent with []p. But still the machine cannot know it, so that the machine perspective is quite different. Now it can be shown that this last move gives a logic having both []p->p and p->[]<>p. That is G* proves [€]p ->p and G* proves p->[€]<€>p, with p \Sigma_1 (DU-accessible). This is nothing but LASE (the little abstract Schroedinger equation). It can be shown indeed that modal system axiomatised by []p->p and p->[]<>p gives modal description of (minimal) quantum logic. That is, not only we get probabilities, but we get quantum probabilities. Unfortunately, in the process, we loose the necessitation rule. So we cannot use directly the result in the litterature for isolating an arithmetical interpretation of quantum logic. Open problem remains. I conjecture (and I argue technically) that we get the quasi-orthomodular laws, the violation of Bell inequalities, the existence of incompatible propositions, etc. So you see Hal, perhaps provability and consistency are not interesting but now, you should realise that the incompleteness phenomena (as captured by G and G*) force us to realise the non triviality of the conjonction of provability and consistency. Applied to the UD-accessible propositions, it gives us a sort of quantum probabilities, and this happens exactly where we expect a phenomenological account of the structure of the set of physical propositions (according to the UDA argument). If you define the quantization of p by [€]<€>p, like Rawling and Selesnick (Journal of the ACM, vol.47, 2000, page 737) then you get an arithmetical representation of quantum computing circuit. The difference between what does G* say, and what does G say, gives a mean to distinguish between the "physical propositions" which have probability one and are communicable and those which have probability one and are not communicable. That is close to the difference between experiment and experience. To sum up and putting the things in a more simple way, just remember that the UDA shows that comp makes physics a branch of psychology, and the simplest psychology of the ideally sound machine is just its provability (consistency) logics including the intensional variants (the new boxes) which are quite nuanced thanks to the incompleteness phenomena. The machine cannot prove it has any consistent extension (by Godel'second incompleteness theorem), but the machine can prove that the logical structure of those consistent extension (if that exists) is described by a quantum-sort of ortholattice. Bruno PS Now I have read Ziegler page on quantum logic. It is very good and Ziegler explains the relation between quantum logic and quantum probability in a rather clear way. Only modal quantum logic is missing.
