On 22 Nov 2012, at 18:38, Stephen P. King wrote:




    How exactly does the comparison occur?

By comparing the logic of the observable inferred from observation (the quantum logic based on the algebra of the observable/linear positive operators) and the logic obtained from the arithmetical quantization, which exists already.



How does the comparison occur? I will not ask what or who is involved, only how. What means exists to compare and contrast a pair of logics?


The logic exists, because, by UDA, when translated in arithmetic, makes a relative physical certainty into a true Sigma_1 sentence, which has to be provable, and consistent. So the observability with measure one is given by []p = Bp & Dt & p, with p arithmetical sigma_1 (this is coherent with the way the physical reality has to be redefined through UDA). Then the quantum logic is given by the quantization []<>p, thanks to the law p -> []<>p, and this makes possible to reverse the Goldblatt modal translation of quantum logic into arithmetic. Comparison is used in the everyday sense. Just look if we get the quantum propositions, new one, different one, etc.






Comp seems to necessitate all possible physical worlds in an equiprobable way.

?

    Does not comp require all possible 1p to exist?

Comp makes all possible 1p existing in arithmetic, from the possible arithmetical pov.





There is a deep problem with notions of priors as it seems that we cannot escape from the problem of subjectivity as we see in the (so-called) anthropic principle: each observer will necessarily find itself in a world what has laws compatible with its existence. It seems to me that the observational act itself is a breaking of the perfect symmetry of equiprobability of possible worlds.

?




But this claim implies violence to the idea of a 3p.
I found at http://higgo.com/qti/Mallah.htm an exchange between Mallah and Standish that seems to illustrate this problem:

"Russell Standish: The predictions can easily depend of the 'picture' but must be consistent with each other. Let me give a simple example: In one picture, observer A decides to measure the spin of an electron in the x direction. In the other, observer B decides to measure the spin of the electron in the y direction. Observer A will see the spin of the electron aligned with x axis, and Observer B will see it aligned with the y axis. Both observations are correct in the first person picture of that observer. A "person" with the third person perspective, sees observers A and B as inhabiting separate `worlds' of a multiverse, each with appropriate measure that can be computed from Quantum Mechanics. Jacques Mallah: On the contrary, this is a textbook example of the way I said it works. The theory predicts some measure distribution of observers; an individual observer sees an observation drawn from that distribution. There are no different sets of predictions for different pictures, just the measure distribution and the sample from it. Russell Standish: It sounds to me like you don't think the prediction changes according to what the observer chooses to observe? An electron cannot have its spin aligned with the x axis and the y axis at the same time. Once the experimenter has chosen which direction to measure the spin, the history of that particular is observer is constrained by that fact, and the predictions of QM altered accordingly. This is true both in MWI and the Copenhagen interpretation, and is the "spooky" nature of QM. I used to think that QM gave predictions in terms of distributions, and that because one didn't see isolated particles, rather ensembles of such particles, I didn't see a problem. The properties of an ensemble are well defined. However, the ability of experimenters to isolate a single particle, such as a photon, or an atom, means we have to take this "spookiness" seriously."

The idea of a 3p cannot be applied consistently to the notion of a 'person' or observer if one is considering the 1p of observers in separate 'worlds' of a multiverse unless, for example, A and B have observables that mutually commute and thus have some chance of being mutually consistent and capable of being integrated into a single narrative. I think that this problem is being overlooked because the problem of Satisfiability is being ignored.


?







I hope that we can agree that there is at least an illusion of a physical world that 'we' - you, me, Russell, .... can consider... Is it necessarily inconsistent with comp?

? ? ?

Not at all. The whole point of UDA is in explaining why the physical reality is unavoidable for the dreaming numbers, and how it emerges from + and * (in the "number base"). It is indeed a first person plural product, with the persons being all Löbian machines, etc.

I am coming at the idea of a 'physical reality' as an emergent structure and not some pre-defined ordering.

Good.






Comp gives the complete algorithm to extract bodies and physical laws, making comp testable, even if that is technically difficult,

I claim that it is not even technically difficult; it is impossible for the simple reason that there does not exist a unique Boolean algebra for all possible 1p.

? (I agree such BA does not exist, but this is exactly what we need to find a measure theorem à-la Gleason). We need a sufficiently good quantum logic, and up to now the comp quantum logic fits rather well.


    Gleason's theorem is interesting: 
http://en.wikipedia.org/wiki/Gleason%27s_theorem

"For a Hilbert space of dimension 3 or greater, the only possible measure of the probability of the state associated with a particular linear subspace a of the Hilbert space will have the form Tr(μ(a) W), the trace of the operator product of the projection operator μ(a) and the density matrix W for the system."

We sidestep the problem of how we define the transition from pure states to density matrices. Andrew's discussion might be seen as addressing this...

OK.






Why? Because it cannot be proven to be satisfiable(aka globally self-consistent) by any finite sequence of algorithms. Completeness and consistency for such cannot be assumed a priori.

?

    Do you ever address the question of satisfiability?


Which satisfiability? I use it all the time. p->p is satisfiable by all interpretation, and this is used all the time. I do not use the complexity of satisfiability, as if this needed to be used, it has to be justified by the modal logic extracted from self-reference.









but up to now, it fits remarkably, and that would not have been the case without QM. That would not have the case if "p->[]<>p" was not a theorem of the Z1* logics (matter).

Your reasoning is correct only because you are assuming the impossible to be true a priori: that there exists a solution to the Satisfiability problem

It exists. "Satisfability" is non tractable, not insoluble. The first persons don't care "waiting exponential time" by the invariance of first person experience on delays.

Of course, but an infinite BA requires eternity (infinitely many steps) to solve its satisfiability problem.

But no machine ever need to do that (and can't). The BA might be infinite, but not the proposition, unless you are using infinitary logic, which does not play a big role in comp up to now.




I am not claiming non-solubility; I am pointing out that the computation of satisfiability must run to obtain a solution,

The 1p depends on truth, not on proof.



otherwise it is false to claim that the solution is accessible.


The UD does "prove", or arithemtic proves, all the true sigma_1 sentences, which is enough for the computations to be emulated. then the 1p are distirubuted non constructively on that, independently of the complexity of the proofs. Without this, no measure problem. And with no measure problem, you lost the reduction of physics to computer science.




It is a profound mistake to claim that the existence of the largest prime number defines the exact sequence of numerals that would enumerate that prime number.

You need to decide in which base you write it, and then it is defined. But we don't need this.



Similarly, the mere possibility of satisfiability of a BA

Satisfiability concerns sentences, not BA.



cannot be used to argue about the particular distribution of propositions of the BA. You are considering first persons in the eternal and ideal case, but that does not connect omniscient machines to finite human brains.

The connection is explained by the UDA.



This is the challenge to Plato and Parmenides, how do we bridge between the Realm of Truth and the world of appearances?

By the realtion between machines' belief and reality. With comp, today, we can use the work of Tarski and others.




We could make claims forever but showing a proof requires physical effort.

And time, money, if not a sense of public relation. But that is relevant at some meta-meta-level.



There are no shortcuts to knowledge. You seem to be OK with the idea that knowledge can obtain 'for free'.

Free of physics, yes. Free of math? No. You need to postulate enough to get Turing universality.



Perhaps I am mistaken, but it seems that you are assuming the impossible to be real.

I don't. Unless you come back with the idea that 1+1=2 requires a physical world, or thing like that.






*and* that it is accessible for any finitely expressible logical structure.

It is accessible, but then I don't see at all the relevance of this.

    Please explain how it is accessible.

You were using the term. I am the one asking the question here.

Bruno


http://iridia.ulb.ac.be/~marchal/



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