Cannibalizing previous thread "Provable vs Computable:"

Which are the logically possible universes?  Tegmark mentioned a
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?

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.

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
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
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

But why should this lack of provability matter?  Ignoring this universe
just implies loss of generality.  Provability is not the issue.

Juergen Schmidhuber   

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