Cannibalizing previous thread "Provable vs Computable:" 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? 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 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. But why should this lack of provability matter? Ignoring this universe just implies loss of generality. Provability is not the issue. Juergen Schmidhuber http://www.idsia.ch/~juergen/ http://www.idsia.ch/~juergen/everything/html.html http://www.idsia.ch/~juergen/toesv2/