Dear Juergen: I am not so much interested in provability as I am in whether or not the "noise" in a universes history is pseudorandom or random and forging an .
At 5/4/01, you wrote: >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 http://www.idsia.ch/~juergen/toesv2/
