On 01 Oct 2011, at 09:31, Russell Standish wrote:

On Thu, Sep 22, 2011 at 07:02:28PM +0200, Bruno Marchal wrote:

OK. But note that in this case you are using the notion of 3-OM (or
computational state), not Bostrom notion of 1-OM (or my notion of
first person state).
The 3-OM are countable, but the 1-OMs are not.

Could you explain more why you think this? AFAICT, Bostrom makes no
mention of the cardinality of his OMs.

I don't think that Bostrom mentions the cardinality of his OMs, indeed. I don't think that he clearly distinguish the 1-OMs and the 3- OMs either. By "3-OM" I refer to the computational state per se, as defined relatively to the UD deployment (UD*). Those are clearly infinite and countable, even recursively countable.

The 1-OMs, for any person, are not recursively countable, indeed by an application of a theorem of Rice, they are not even 3-recognizable. Or more simply because you cannot know your substitution level. In front of some portion of UD*, you cannot recognize your 1-OMs in general. You cannot say "I am here, and there, etc." But they are (non constructively) well defined. "God" can know that you are here, and there, ... And the measure on the 1-OMs should be defined on those unrecognizable 1-OMs.

Are the 1-OMs countable? In the quote above, I say that they are not countable. What I meant by this is related to the measure problem, which cannot be made on the states themselves, but, I think, on the computational histories going through them, and, actually, on *all* computational histories going through them. This includes the dummy histories which duplicate you iteratively through some processes similar to the infinite iteration of the WM self-duplication. Even if you don't interact with the output (here: W or M) or the iteration, such computations multiplies in the non-countable infinity. (I am using implictly the fist person indeterminacy, of course). Those computation will have the shape:

you M
you M
you W
you M
You W
You W
You W
You M
ad infinitum

This gives a white noise, which is not necessarily available to you, but it still multiplies (in the most possible dumb way) your computational histories. Such infinite computations, which are somehow dovetailing on the reals (infinite sequence of W and M) have a higher measure than any finite computations and so are good candidates for the "winning" computations. Note that such an infinite background noise, although not directly accessible through your 1-OMs, should be experimentally detectable when you look at yourselves+neighborhood below the substitution level, and indeed QM confirms this by the many (up + down) superposition states of the particles states in the (assumed to be infinite) multi-universes.

This might be also confirmed by some possible semantics for the logic of the first person points of view (the quantified logic qS4Grz1, qX1* have, I think, non countable important models).

3-OMs are relatively simple objects, but 1-OMs are more sophisticated, and are defined together with the set of all computations going through their correspondent states.

To be sure, I am not entirely persuaded that Bostrom's 1-OMs makes sense with digital mechanism, and usually I prefer to use the label of first person experiences/histories. With the rule Y = II, that is: a bifurcation of a computations entails a doubling of the measure even on its "past" (in the UD steps sense), this makes clear that we have a continuum of infinite histories. Again, this is made more complex when we take amnesia and fusion of histories) into consideration.

I hope this helps a bit. In my opinion, only further progress on the "hypostases" modal logics will make it possible to isolate a reasonable definition of 1-OMs, which obviously is a quite intricate notion.



Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au

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