Juergen Schmidhuber wrote:
>Which are the logically possible universes? Tegmark mentioned a
>somewhat
>vaguely defined set of ``self-consistent mathematical structures,''
>implying provability of some sort. The postings of Marchal also focus
>on what's provable and what's not.
>
>Is provability really relevant? Philosophers and physicists find it
>sexy for its Goedelian limits. But what does this have to do with the
>set of possible universes?
Because I interpret "possible universe" or "possible model of p" by "p
belongs to a model" or "p belongs to a consistent extension", which I
translate in arithmetic by "p is consistent" that is by "-p is not
provable" (-[]-p). (By Godel's *completeness* theorem it is extensionnaly
equivalent).
>The provability discussion seems to distract from the real issue. If we
>limit ourselves to universes corresponding to traditionally provable
>theorems then we will miss out on many formally and constructively
>describable universes that are computable in the limit yet in a certain
>sense soaked with unprovability.
But *all* our consistent extensions are soaked with unprovability.
And the UDA shows *all* of them must be taken into account.
'Probability one of p' will be when 1) and 2)
1) []p, provable('p'), = p is true in all, if any, "possible universe".
2) <>p consistent('p') = there is a possible universe with p.
(I use Godel *completeness* theorem for first order logic, here, but
a more sophisticated presentation can be made (G and G*
works on second order arithmetic as well, but I don't want to be too
technical)).
So I translate probability one of p by "[]p & <>p". The
incompleteness theorems makes that moves no less trivial than
the translation of "knowing p" by "provable(p) & p" which gives
an arithmetical interpretation of intuitionist logic.
The comp restriction to UD*, the trace of the UD, is translated
in arithmetic by a restriction of p by \Sigma_1 sentences p (this
includes universal \Sigma_1 sentences). Universality (with
comp) is \Sigma_1 completeness (+ oracle(s), etc.).
This gives the quasi Brouwersche modal logics Z1*. (I get
interesting quantum smelling logics also with the translation of
"feeling really p" by "[]p & <>p & p", the X1* logic.
I conjecture Z1* gives an arithmetical quantum logic.
(X1*, Z0*, XO* are interesting variant, the X* minus X
are expected to play the role of a qualia logic, (for the
true mesurable but unprovable atomic propositions).
>Example: a never ending universe history h is computed by a finite
>nonhalting program p. To simulate randomness and noise etc, p invokes
>a short pseudorandom generator subroutine q which also never halts. The
>n-th pseudorandom event of history h is based on q's n-th output bit
>q(n)
>which is initialized by 0 and set to 1 as soon as the n-th statement in
>an ordered list of all possible statements of a formal axiomatic system
>is proven by a theorem prover that systematically computes all provable
>theorems. Whenever q modifies some q(n) that was already used in the
>previous computation of h, p appropriately recomputes h since the n-th
>pseudorandom event.
>
>Such a virtual reality or universe is perfectly well-defined. We can
>program it. At some point each history prefix will remain stable
>forever.
>Even if we know p and q, however, in general we will never know for sure
>whether some q(n) that is still zero won't flip to 1 at some point,
>because of Goedel etc. So this universe features lots of unprovable
>aspects.
Sure.
>But why should this lack of provability matter?...
Because UDA shows that our future is a sort
of mean on the unprovable things (our consistent extensions,
unprovable by incompleteness).
The miracle is the sharing and cohering part of those
extensions. I don't pretend having explained that miracle.
But UDA says where you must look for the explanation, and
the translation of the UDA in arithmetic gives an encouraging step
toward quantum logic, and even quale logics.
(Linking J.S. Bell and J.L. Bell btw, ref in my thesis).
>...Ignoring this universe
>just implies loss of generality. Provability is not the issue.
You are postulating a computable universe. And you believe, like
some physicist that an explanation of that universe is an explanation
for everything (discarding completely the "mind-body" problem).
I am NOT postulating a computable universe. I am not even
postulating a universe at all.
I postulate only *physicists*, in a very large sense
of the word. I postulate that they are (locally) computable.
I postulate only arithmetical truth and its many consistent
internal interpretations.
Provability is not really the issue indeed. It is the border between
provability and unprovability which is the issue.
G* \ G gives insightful informations about that, including the
'probability on the uncomputable' issue.
Although I give theorem provers for the Z logic, question like
what is the status of the Arithmetical Bell's Inequality:
[]<>A & []<>B -> []<>(([]<>A & []<>C) v
([]<>B & []-[]<>C))
remains open. (Here []p = provable(p) and consistent(p), A, B, C
are \Sigma_1 sentences).
Perhaps my discovery (proof of the arithmetical B) is just a
mathematical mirage (although what a coincidence, with respect
to the intuitive UDA!). Perhaps comp, "my" (psycho)logical
comp is wrong. But don't tell me I'm vague, 'cause I'm utterly
precise.
I recall you the crux of the crux (the proof of the modal
symmetry axiom B "p->[]<>p") is given (in english) at
http://www.escribe.com/science/theory/m2855.html
Hoping that helps.
Bruno
PS I have presented the UDA in 11 questions to Joel Dobrzelewski.
Have you follow that issue? At which step would you say "no".
Universal Dovetailer Argument (step by step):
UDA step 1
http://www.escribe.com/science/theory/m2971.html
UDA step 2-6
http://www.escribe.com/science/theory/m2978.html
UDA step 7 8
http://www.escribe.com/science/theory/m2992.html
UDA step 9 10
http://www.escribe.com/science/theory/m2998.html
UDA last question
http://www.escribe.com/science/theory/m3005.html
Joel 1-2-3
http://www.escribe.com/science/theory/m3013.html
Re: UDA...
http://www.escribe.com/science/theory/m3019.html
George'sigh
http://www.escribe.com/science/theory/m3026.html
Re:UDA...
http://www.escribe.com/science/theory/m3035.html
Joel's nagging question
http://www.escribe.com/science/theory/m3038.html
Re:UDA...
http://www.escribe.com/science/theory/m3042.html
http://iridia.ulb.ac.be/~marchal
Received on Thu Nov 01 2001 - 10:05:12 PST