On 13 Mar 2011, at 18:32, David Nyman wrote:

On 13 March 2011 17:26, Bruno Marchal <marc...@ulb.ac.be> wrote:

But then, consciousness might be the 'mental' state of a universal number when it believes in a reality. Science might begin when it questions that reality, and consciousness might reappear as the unquestionable part of that reality. It can be related with a form of ignorance awareness. It can get a
role of relative self-accelerator.

Bruno, I'm not sure I understand what you mean by "relative
self-accelerator" in the above.

It is related to two other theorems of Gödel:
The completeness theorem
and the speed-up theorem.

The completeness theorem says that a theory is consistent if and only if the theory has a model (a mathematical structure) verifying (making true) its propositions. Like the structure (N, + *) make true the sentence "1+8 = 9". Let us call that a reality. Consistent means that the theory does not prove the falsity. Modally ~Bf, or equivalently Dt. You can read Dt, by completeness, as the existence of a reality. (OK, I can add explanations if you ask)

I define consciousness by believing in a reality. This is almost like BDt. But by incompleteness BDt -> Bf, and for the correct machine, Bf - > f (damned!).

But the machine/theory can also just add the arithmetical form of Dt (consistency) as a new axiom. It is a inferential process, and not only it does not lead to inconsistency, but the speed-up theorem of Gödel (and a similar for universal machine) is that such process can, shorten is some quasi arbitrary way the length of the proofs. Likewise, Blum speed-up theorem shows that universal machine (and weaker interesting sub-universal one) can be sped up arbitrarily, except for finite number of exception (it concerns the computation which are correct above some argument, that is sufficiently big inputs).

So, it is suggested that the belief in a reality speed-up your computational power. This I think can be seen by introspection. If you drive a car, the more you take seriously your local reality (the road, the other cars), the more you can actually speed-up. If you don't believe in any reality, you could as well stop your car, even in the middle of the drive-way.

Such a theory would endow consciousness (self-consistency inference) to self-moving entity, which will better anticipate the now moving environment (from their perspective) by taking such environment including one self seriously enough.

I can speculate that consciousness might be the ultimate accelerator. The reason why bodies move at all. That would make it a sort of fundamental force. By Newton force are accelerator. But this *is* speculation, or analogy.

.  What is "accelerating" what?


A universal machine M accelerates (= diminishes the length of personal computations) a universal machine M (itself) relatively to arithmetic, or to some other universal machine U computing M. In that case the machine M transforms itself into a new machine "M+consistent(M)", and the process can continue in the constructive transfinite (and are known as autonomous progressions). They cannot reach arithmetical truth, but they can approach it, and in the long term it is, from their points of view, self-accelerating. The logic of the 8 hypostases remains invariant, despite their arithmetical interpretation (the arithmetical B and thus D) evolves.

The length of proof theorem of Gödel paper is in the book "The undecidable" edited by Martin Davis (the comp bible!) Torkel Franzen wrote a little chef d'oeuvre on the autonomous progressions: "Inexhaustibility, A Non exhaustive treatment".

Bruno



On 12 Mar 2011, at 18:18, Ismail Atalay wrote:   [in the FOR list]

So we should not be in a position to say "for the essence of consciousness to exist, this type of physical/mathematical features should be present in
this universe". Physical/mathematical features are required for its
manifestation, implementation and realization.

You might conceive that the manifestation, implementation and realization are concept definable relatively to universal numbers which are numbers
coding universal partial computable function relatively to elementary
arithmetic. So a minimal amount of arithmetic is required indeed. Universal
numbers exists, and their many interactions are already emulated,
atemporally, by the laws of addition and multiplication of non negative integers. This makes things complex because Universal numbers reflect each
other including themselves.
But then, consciousness might be the 'mental' state of a universal number when it believes in a reality. Science might begin when it questions that reality, and consciousness might reappear as the unquestionable part of that reality. It can be related with a form of ignorance awareness. It can get a
role of relative self-accelerator.
I agree with you that free will is not an illusion, because that ignorance is real, and *that* awareness is correct (trivially so for the ideal simple machine I study). Choice is not an alternative. There is an entertaining
novel by Smullyan illustrating this in the book Mind's I. (*)
*A* non compatibilist notion of free will is correct, but it is the one where non compatibilism is relative to the personal point of view of the universal number, which cannot, indeed, entirely determinate itself, and yet, in the usual computations, has to decide of some action, relatively to
some other universal numbers in its neighborhood.
Universal numbers are driven by self-satisfaction, I think, but in each universal numbers there are conflicts between lower self and higher self
satisfaction.
Higher self satisfaction is better in the long run, but opposes itself with short term lower self satisfaction, and free will comes from the fact that
we can chose in between. Perhaps. I mean it is hard not to talk under
torture, a situation which usually maximizes the conflict between the short
and long term satisfactions.
Bruno Marchal
(*) (edited by Dennet and Hofstadter, which is, BTW, an excellent
introduction to Digital Mechanism. Dennett come close to the first person
indeterminacy indeed)
http://iridia.ulb.ac.be/~marchal/


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