Le 31-juil.-07, à 14:47, Russell Standish a écrit :

> On Tue, Jul 31, 2007 at 04:06:10PM +0200, Bruno Marchal wrote:
>> Le 31-juil.-07, à 00:08, Russell Standish a écrit :
>>> On Mon, Jul 30, 2007 at 11:47:48AM +0200, Bruno Marchal wrote:
>>>> If this is not relevant in this context, I ask what is relevant ... 
>>>> ?
>>>> The problem you mention is at the cross of my work and the 
>>>> everything
>>>> list. Now, as I said some days ago, I think that a way to link more
>>>> formally my work and the everything discussion can consist in 
>>>> defining
>>>> a notion of basic atomic third person observer moment. The UDA, plus
>>>> Church thesis + a theorem proved in Boolos and Jeffrey (but see also
>>>> and better perhaps just Franzen's appendix A) makes it possible to
>>>> define the comp third person OMs by the Sigma1 sentences of
>>>> arithmetical language. Those have the shape ExF(x) with F(x)
>>>> decidable.
>>>> For example ExPrime(x) (a prime number exists), Ex(x = code of
>>>> triple(a,b,c) and machine a gives c on argument b), ... This last
>>>> example show that the notion of Sigma1 sentences is rather rich and
>>>> encompasses full computability. So the very restricted notion of
>>> Interesting. Since an observer moment contains all information that 
>>> is
>>> known about the universe,
>> ? I guess you mean ... about the observer.
> No, I mean all information known by the observer (including, but not
> exclusively information know by the observer about erself).

OK, but then adding "about the universe" is confusing at this stage. 
You interpret the quantum state as describing knowledge. (And then I am 
not sure I follow what you mean by quantum state: you are supposing the 
quantum hyp. here, aren't you (or perhaps your linearity hyp. only? 
Again where would that linearity come from?).

>>> this led me to identify the observer moment
>>> and the quantum state vector.
>> ... and the partial relative quantum state vector corresponding to the
>> observer. OK, but at this stage this would be cheating. We can not yet
>> explain why the quantum histories wins over the comp/number relations.
> Well I have my own reasons, considering knowledge acquisition as an
> evolutionary process. But I disagree about it being cheating, because
> I don't a priori assume quantum states are elements of a Hilbert
> space. That is a derived property.

So, how do you define quantum state?

>>> This is not incompatible with with your
>>> notion of the OM being a Sigma1 sentence, but it places severe
>>> restrictions on the form of the quantum state vector.
>> The OM are the Sigma1 sentences, when they are considered as third
>> person constructs.
> Third person is that which is accessible to all observers.

? (This correspond more to the first person plural notion as I have 
defined it in most of my papers: observers appeared in the fourth and 
fifth hypostases, and perhaps already a part of it appears in the third 
one; but there are no observer in the second or first hypostases).

1  p  (truth, 0-person)
2 Bp (provable, 3-person)
3 Bp & p  (knowable, 1-person)
4 Bp & Dp (observable, measurable; 1-plural-person)
5 Bp & Dp & p (sensationalisable, feelable, personally 
observable/measurable, 1 person again)

> Do you mean
> 0th person perhaps?

? (actually with comp this is defensible, but then you take the risk to 
give a name to the unnameable, indeed you could call Him/Her/It, 
FORTRAN, or JAVA, etc.). It is risky and I prefer not, even if with the 
comp hyp, that moves could be consistent (but could be eventually false 
although irrefutable). Risky.

>> Those are really the states accessible by the UD. To
>> get the quantum we have to reconsider those OMs from the points of
>> view. In the arithmetical comp setting this corresponds to looking to
>> the views expressed by the intensional variants of the logic of
>> prpvability (p, Bp, Bp & p, Bp & Dp, Bp & Dp & p, ...) with p
>> restricted to the (arithmetical) Sigma1 sentences. This gives the
>> second row of the 16 hypostases described in your book, page ? (my
>> exemplar is at home!).
>>> There can only
>>> be aleph_0 of them for instance.
>> Not really because the Sigma1 sentences are (a priori) weighted by the
>> computations going through, including those who does not terminate, if
>> only because they dovetail on the reals, and this is enough to suspect
>> that there could be  a continuum (aleph1). Of course it could be less
>> by the existence of some yet unknown equivalence relations (which I
>> succeeded not using thanks to the lobian interview). More on this when
>> David is back.
> Alright, but it would be nice to know. There are only a countable
> number of machines, so I thought there'd only be a countable no. of
> Sigma1 sentences.

You are right, but they have different weight (and different relative 
weight) due to their belonging to different histories, and they are 
(obviously ?) 2^aleph_0 histories, most of them never computably 
generated except in the sense of the universal dovetailing. The UD 
generates indeed all computable sequences mixed in all possible ways 
with all uncomputable sequences. OK (this point is often misunderstood, 
see for instance, my old conversation with Schmidhuber on the list).

>>> Perhaps these restrictions are
>>> testable? Perhaps there is something wrong with identifying the state
>>> vector with the OM?
>> Comp is really "I am a machine", and not at all "the universe is a
>> machine". The UDA shows that, unless "I am the universe", the
>> proposistion "I am a machine" and "the physical universe is a machine"
>> are incompatible. Indeed the UDA forces the physical laws to emerge
>> locally from *all computations"? A priori again this makes the 
>> universe
>> a non computational  object, it seems to me (by UDA).
> But the OM is actually the "universe", or at least a snapshot
> thereof. So we would expect it to be uncomputable. Is that also the
> case of the Sigma1 sentences?

The OM, or more exactly the 3 person OM are machine states. Those are, 
like the Sigma1 sentence, finite objects. All true Sigma1 sentences are 
provable by any universal machine (up to some translation). For example 
I can build a universal game-of-life pattern capable of dovetailing on 
the proof of all true Sigma_1 sentences. It gives a universal 
dovetailer. Universal with respect to the computable, but 
extraordinarily weak with respect to the provable.
Actually I do define, now, a universal machine as a machine which can 
prove all the true Sigma_1 sentence. Equivalently, it means all he 
sentences "p -> Bp" are true *about* the machine, with p any Sigma1 
sentence. A machine is then lobian just in case she has the ability to 
*prove* "p -> Bp" with p any Sigma1 sentence. Not only for p sigma1 "p 
-> Bp" are true about the machine, but are provable by the machine.
Grosso modo, a machine is lobian when she knows, in a very weak sense, 
that she is universal. Such machine are essentially undecidable in the 
sense that any effective consistent extension of them remains 
undecidable (admits true but unprovable sentences).



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