Re: on formally describable universes and measures

From: <>
Date: Tue, 30 Jan 2001 11:02:38 +0100

>From Bruno Marchal, Sat Jan 27:

> Let us suppose I am duplicated. I am annihilated at Brussels and
> recontituted at both Washington and Moscow.
> By comp I survive. I cannot predict with certainty where I will feel
> myself (1-person point of view) after the experiment.
> So there is an uncertainty on the domain of reconstitution. OK ?

Why repeat this over and over? This is the very reason why one has to
look at possible probability distributions over possible futures to
quantify the uncertainty.

If the distribution is computable or at least describable then you
get results such as in
Otherwise you cannot even describe it. Case closed.

> Now let us run the UD. And let us call S your current computational
> state.
> Let us first suppose we run the UD for n steps, where
> n is a finite very big number (to fix our mind).
> From the invariance lemma it follows that
> the domain of uncertainty is given
> by the set of all the virtual reconstitutions of
> your state S which
> has occured in that running. Let us call that set S_n
> Now suppose we let the UD never being stopped. Then,
> thanks to the invariance lemma, the uncertainty is given by
> the union of all the S_n, n positive integer.
> It is an easy exercise to show that we will not succeed to
> quantify the uncertainty by putting equiprobable weight on all the
> equivalent state S. This entails white noise.

In case you just mean there is no way of assigning equal
nonvanishing probability to infinitely many objects - sure.

> We must put the weight on the (relative) computationnal
> histories going through S.


> And the set of all infinite computational histories has
> the cardinality of the continuum.

This is THE major error you keep making. It is the set of all infinite
histories which has the cardinality of the continuum. But most of them
are not computable. So they cannot be _computational_ histories, which
are countable - the set of all infinite _computational_ histories does
NOT have the cardinality of the continuum.

You cannot generalize from countably many, constructively enumerable
elements in the union of all your S_n above to a non-constructive
continuum. If your claims depend on the continuum stuff then I can safely
ignore them in the future. I am just hoping for you that this is not
the basis of your entire PhD thesis (what did the referees have
to say about this?).

The problem is right here. You are misunderstanding the difference
between the things "computable in the limit" and the elements of the
continuum. In case you want to reply please just focus on this issue
but do not repost the stuff about Washington and Moscow etc.

> (It contains the ``stupid" dovetailing on the reals
> described above!).
> You must not confuse the 3-person point of view
> concerning the actual running of a UD, with the first
> person point of view, which is defined on states independently
> of any time for reaching the states by the UD. That is why the
> union of the S_n is taken on all the natural (N), and that
> is why we must take the uncountable set of all the
> infinite histories for the domain of the ``probability or
> credibility (whatever) distribution".

Computation is incompatible with wild ideas such as uncountable
time, no matter who's the observer. Any possible future of an observer
within a computer-generated universe (your 1st person, I believe) must be
among the rather few computable ones. This is true from the point of
views of both outside and inside observers. Computability is objective
- there is nothing that is computable inside some computable universe
but not outside of it. You cannot switch from countable to incountable
by stepping inside.

> I am not actually saying that the UD build a non computable
> object or enumerate an uncountable set, I am merely
> saying that with comp the UD makes sets of non computable
> object playing a role in the distribution of possible
> histories *from* the first person point of view of the machines.

Playing a role? This is so vague.

> I agree that I say something shocking. At each instant
> I am not multiplied by 10^100 like in deWitt's view of Everett
> formulation of QM, I show that with comp we are multiplied
> a priori by 2^aleph_0, at each instant ...
> I agree it is weird. But is it weirder than Feynman Integral.
> For me Feynman Integral is still weirder ...
> I'm afraid you don't take the difference
> between first and third person into account, or am I wrong ?

It is not shocking or weird at all. It is at best unclear or just plain
wrong. Clearly, your 2^aleph_0 has nothing to do with the concept of
computation. To repeat, the difference between first and third person
does not make a difference between countable and uncountable, or between
computable or noncomputable.

> Remember also that, in fine, I use arithmetical realism
> (the UD does not need to be runned).
> This is not constructive. I am not pretending having
> prove that self-aware machine face the continuum, I am

This is so vague - what do you mean by "in fine" and by "face the
continuum"? Your posts are full of unclear expressions like this.

> merely proving that if we are machine, and if arithmetical
> truth (the continuum of models of arithmetical theories)
> exists independently of me, then self-aware machine
> will be in front of that continuum when trying to
> quantify their own (comp) undeterminacy.

So what you are claiming actually depends on noncomputable stuff, which
does not exist from any constructive or computational point of view.

Your expression "comp" has mislead me. I strongly recommend to rename
this, since your "comp" has nothing to do with computation. I also
recommend replacing other misleading computation-oriented expressions such
as "dovetailer." Use "arithmetical realism" or whatever; please do not
misuse terminology with a well-defined meaning in computability theory.

Constructivists can ignore your claims as there is no compelling
reason to assume the "existence" of something like the continuum.

> I am at a billion miles isolating the right measure (if it
> exists) but the interview of the sound UTM gives an
> original and pure (not empiricaly influenced) way to begin
> with. Finding a semantics for the Z logics should clear
> the way.

Why not just assume the measure is constructively describable? This
apparently harmless assumption by itself has major consequences:
On the other hand, if we cannot even describe our
measure then we must shut up.

> >Here is the best you can achieve with a dovetailer. You can
> >extract representations of all the countably many computable reals.
> >How? Systematically enumerate all possible input strings, viewing them as
> >starts of programs to be executed in dovetail fashion. This excludes
> >many strings that cannot be input programs because one of their prefixes
> >already is a self-delimiting program that will never request additional
> >bits. Some input strings, however, are valid, finite, possibly
> >nonhalting programs (such as the algorithm computing the decimal expansion
> >of 1/3). They generate possibly infinite objects (such as 0.33333...).
> >Make a list containing all current programs or program prefixes.
> >Update the list whenever one of its elements requests a new bit. This
> >will eventually give you finite representations of each computable real.
> I doubt that. The set of computable real is not recursively enumerable.
> But OK in the limit. It is recursive in the halting set ...

Doubt what? I did not say anything about recursive enumerability.

> As Gilles Levy pointed out the efficiency of the UD is not relevant, for
> the sharing space-time emerges from the statistics of the whole set
> of finite and infinite computationnal histories, from the first person
> point of view or first person plural point of view in the case of
> bifurcation of deep computational histories shared by many.

Of course efficiency is essential as soon as it leads to countable
vs uncountable time. Countable time is within the constructive realm,
uncountable time is not.

Where are the "statistics" you are talking about? You never define
a probability distribution on the possible futures. No distribution,
no statistics.

On the other hand, the constraints imposed by the assumption of constructively
describable distributions do yield nontrivial consequences:

> >Once more: The concepts of dovetailing and continuum are incompatible!
> Yes. In a trivial sense from a third person point of view. But about
> the question "in which computational history am I", well, if I am
> a machine, then a priori I can belong to 2^aleph_0 histories.

things such as 2^aleph_0 are completely beyond computation and
constructivism and machines. Ok, the flaw in your line of reasoning
has become clear to me. I hope your PhD thesis does not stand or fall
with this, and that I won't have to post anything else on this.

Received on Tue Jan 30 2001 - 02:04:24 PST

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