On 19 Nov 2012, at 19:46, meekerdb wrote:

On 11/19/2012 8:33 AM, Bruno Marchal wrote:

and the math shows that this will include some continuous/analog observable.

How does it show that?

Intuitively: by dovetailing on each programs coupled with real numbers. Each computations are done again with all possible streams of real numbers, oracles, etc. Yes the UD is that dumb. But this is probably needed for the measure question. We can even bet that such a coupling has to be exploited by the "winner program" in some special way, if it exists, because even the white rabbit realities are multiplied into a continuum by the existence of that coupling (which is unavoidable: you can't diagonalize against the UD to build a UD avoiding those couplings).

Formally: the existence of such a semantics based on a continuum is reflected in the possible topological semantics of the material hypostases S4Grz1, Z1* and X1*, which gives rise to the arithmetical quantum logics. The formal reasons are different for S4Grz1, and the Z and X logics. The topology is intrinsic to the S4Grz type of semantics, but it might still be discrete at that level. For the X1* and Z1*, it comes in part from the lack of necessitation, and the necessity to have infinite sequences of neighborhood structures à la Scott-Montague. If he quantum logic would have given only by S4Grz1, that would have been an argument for loop gravity, and the continuum would have been restricted to the frequency-statistical operator (like in Preskill and Hartle or Graham). If it would have appeared only in the Z and X logic, that would have suggested that String theory might be the correct comp physics. Amazingly the "arithmetical/comp" quantum logics seems to appear in the three possible candidates (S4Grz1, Z1*, X1*).

Needless to say this is suggestive and it remains quite a lot of open problems in logic to proceed. What I hope is that the arithmetical quantum logics will give the quantum logic searched, but not found, by von Neumann, which have the property that the probabilities can be derived from the constrains given by the laws for the case of P(x) = 1. If this does not work, it means that we might use a stronger definition of knowledge than the one given by Theaetetus.



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