Re: on formally describable universes and measures

From: Marchal <marchal.domain.name.hidden>
Date: Sat Jan 27 06:33:26 2001

Hi Juergen,

You wrote:

>> [Bruno:]
>> ... but the dovetailer generates, for each real, all its
>> bigger and bigger prefixes, and that is called traditionnaly,
>> generating the real. And the dovetailer do that for each real,
>> and so generates all the uncountably many reals.
>
>I am afraid this is nonsense. Obviously I can count the outputs of
>your dovetailer. I can count the time steps it consumes. Hence
>the dovetailer cannot possibly generate uncountably many things.

By generating a real I mean that the program generates bigger and
bigger portion of that real. And for *each* real the dovetailer
generates those portions. For the same reason the simpler program
which generates, at step 0

0,0
0,1

at step 2

0,00
0,01
0,10
0,11

at step 3

0,000
0,001
0,010
0,011
0,100
0,101
0,110
0,111

will *generate*, if it doesn't stop, each positive real written
in binary and less than 1. Of course at each step the line
are approximation of uncountably many reals. Sure! I am not
pretending the UD *enumerates* the set of all the reals.

All what I say is that for each real, the dovetailer
will generate all its prefixes in succession. We can argue
it is a bad idea to use the word ``generate", but the important
point is the following derivation. (Because the fact that ALL
the reals play a role is linked to the FIRST person points of
view. I explain below).

Let us suppose comp. That is let us suppose there is a level such that
I survive a digital substitution. (OK with that ?)
And then I am duplicable. (3-duplicable).

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. If I predict
Washington the one in Moscow knows that I was wrong of being sure. Idem
if I predict Moscow. I cannot predict I will feel being at both place,
(because I will not *feel* being at both places, or comp is false),
and if I predict nowhere, then by definition I will die and comp is
false again.

So there is an uncertainty on the domain of reconstitution. OK ?
Self-duplicability entails first person indeterminacy.
(You can formalize that with duplication of inference inductive
machines + Kolmogorof complexity). I mean there is no magic
use of ``consciousness" in the first person concept). OK with that ?
(It correspond to the use of the word ``subjective" in Everett's paper).

The ``time" invariance lemma says: for all ways I choose to quantify
that first person undeterminacy, the number (probabilities,
credibilities, whatever ...) remains unchanged if we add
arbitrary (but finite) delays in the reconstitution (because the first
person has no way to be aware of these delays). OK ?

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
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.

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.
(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".

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.

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 ?

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
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.

In my paper I give a reason to expect the solution of the
UD explosion of possible futurs could be a Quantum Universal
Dovetailer (Feynman lesson + Everett insight: the white rabbits
disappear not because they are rare, but because they are terribly
numerous and the average white rabbits annihilate themselves
with their *minus white rabbits*).

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.

I hope not having been to long, but please, Juergen, tell
me exactly where in the above derivation-sketch you want
to disagree.

Bruno
Received on Sat Jan 27 2001 - 06:33:26 PST

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