Provable vs Computable

From: <>
Date: Fri, 4 May 2001 12:04:19 +0200

Which are the logically possible universes? Max Tegmark mentioned
a somewhat vaguely defined set of ``self-consistent mathematical
structures,'' implying provability of some sort. The postings of Bruno
Marchal and George Levy and Hal Ruhl 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?

I believe the provability discussion distracts a bit 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 unprovable aspects.

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 element
of an ordered list of all possible program prefixes halts. 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. 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.

But why should this lack of provability matter? It does not do any harm.

Note also that observers evolving within the universe may write
books about all kinds of unprovable things; they may also write down
inconsistent axioms; etc. All of this is computable though, since the
entire universe history is. So again, why should provability matter?

Juergen Schmidhuber
Received on Fri May 04 2001 - 03:05:51 PDT

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