On 14 Aug 2012, at 06:33, Jason Resch wrote:

On Mon, Aug 13, 2012 at 10:53 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

The choice of the initial universal system does not matter. Of course it does matter epistemologically. If you choose a quantum computing system as initial system, the derivation of the physical laws will be confusing, and you will have an hard time to convince people that you have derived the quantum from comp, as you will have seemed to introduce it at the start. So it is better to start with the less "looking physical" initial system, and it is preferable to start from one very well know, like number + addition and multiplication.

So, let us take it to fix the thing. The theory of everything is then given by the minimal number of axioms we need to recover Turing universality.

Amazingly enough the two following axioms are already enough, where the variable are quantified universally. I assume also some equality rules, but not logic!

x + 0 = x
x + s(y) = s(x + y)

x * 0 = 0
x*s(y) = (x *y) + x

This define already a realm in which all universal number exists, and all their behavior is accessible from that simple theory: it is sigma_1 complete, that is the arithmetical version of Turing- complete. Note that such a theory is very weak, it has no negation, and cannot prove that 0 ≠ 1, for example. Of course, it is consistent and can't prove that 0 = 1 either. yet it emulates a UD through the fact that all the numbers representing proofs can be proved to exist in that theory.

Now, in that realm, due to the first person indeterminacy, you are multiplied into infinity. More precisely, your actual relative computational state appears to be proved to exist relatively to basically all universal numbers (and some non universal numbers too), and this infinitely often.

So when you decide to do an experience of physics, dropping an apple, for example, the first person indeterminacy dictates that what you will feel to be experienced is given by a statistic on all computations (provably existing in the theory above) defined with respect to all universal numbers.

Is every program given equal weight in this theory, or might programs that run more efficiently, longer, or appear more frequently (as embedded sub-programs) have greater weight in setting the probability of future first person extensions?

Only "appear more frequently in the UD*" can play a role, by the invariance of the probabilities for the first person indeterminacies.

Does the universal system have any bearing on the above? For example, intuitively it seems to me that when considering two universal systems, say Java, and FORTRAN, that due to syntactical differences, different programs might appear more or less often or easily.

The UD in Java, and the UD in FORTRAN will generates all possible UDs. If one particular one win the measure game, one (or many) special universal systems will play bigger role than other, but that has to be proved starting from any initial UD. So your question depends on the points of view taken. Ontologically, the answer is no. Epistemologically, the answer is yes, but that has to be deduced from the ontology (and the definition of person, belief, knowledge, observation, etc.). The theoretical result is that quantum universal system wins (as proved by the fact that arithmetical observation leads to an arithmetical quantization), and this is confirmed, retrospectively, by the existence of the quantum features in Nature.

Perhaps all universal systems compete amongst each other, based not only on the frequency of their programs, but how easily that universal system is realized in some meta-system.

Only if that easiness entails a bigger measure. Then yes, and that is even quite plausible, if not empirically obvious for the high physical reality levels.

Keep in mind that if you are duplicated in virtual dreams in W and M, and that in W you are executed with a quantum efficient computer, and in M you are executed by monks playing inefficiently (but correctly) with pebbles, comp entails that *you* will not feel any difference, as you cannot be directly aware of the universal level which execute you at (or below) your substitution level.


So if comp is correct, and if some physical law is correct (like 'dropped apples fall'), it can only mean that the vast majority of computation going in your actual comp state compute a state of affair where you see the apple falling. If you want, the reason why apple fall is that it happens in the majority of your computational extensions, and this has to be verified in the space of all computations. Everett confirms this very weird self-multiplication (weird with respect to the idea that we are unique and are living in a unique reality). This translated the problem of "why physical laws" into a problem of statistics in computer science, or in number theory.

Now, instead of using the four axioms above, I could have started with the combinators, and use the two combinator axioms:

((K x) y) = x
(((S x) y) z) = ((x z) (y z))

This define exactly the same set of "all computations", and the same statistical measure problem, and that is what I mean by saying that the initial axioms choice is indifferent as long as you start from something which define a UD, or all computations (that is: is Turing or sigma_1 complete).

Now, clearly, from the first person points of view, it does look like many universal system get relatively more important role. Some can be geographical, like the local chemical situation on earth (a very special universal system), or your parents, but the point is that their stability must be justified by the "winning universal system" emerging from the competition of all universal numbers going through your actual state. The apparent winner seems to be the quantum one, and it has already the shape of a universal system which manage to eliminate abnormal computations by a process of destructive interferences. But to solve the mind body problem we have to justify this destructive interference processes through the solution of the arithmetical or combinatorial measure problem.



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