>>Jonathan Colvin At first glance that would seem to be the case. But isn't there a >> problem? >> If we consider worlds to be the propositions of formal >systems (as in >> Tegmark), then by Godel there should be unprovable propositions (ie. >> worlds >> that are never instantiated). This seems in direct contradiction to >> the actual existence of everything conveivable, does it not? > > >Bruno: Are you sure Tegmark identify worlds with propositions of FS?
Perhaps I should have said, to be precise, "If we consider worlds to consist of the sets of consistent propositions of formal systems". Just being lazy. I'm not aware that anyone else has yet identified worlds with the propositions of FS, but I am identifying them as such. It seems reasonable, since the Ultimate Ensemble is simply the set of all formal systems. >Anyway, what logicians (and modal logicians in particular) are >used to do is to identify worlds with maximal consistent sets >of propositions (or sentences). >Then you >can extract from Godel that any FS can be instantiated in >alternative worlds. >For example if you take a typical FS like Peano Arithmetic, >the proposition that PA is consistent is undecidable. This >means that there is at least two maximal consistent sets of >propositions extending the set of theorems of PA: >one with the proposition that PA is consistent and one with >the proposition that PA is inconsistent. In that sense the non >provable propositions are instantiated in worlds. > In general >worlds are not effective >(computable) objects: >we cannot mechanically (even allowing infinite resources) >generate a world. Hmmm..but then if such worlds are not effective objects, how can they be said to be "instantiated"? If we extend this to Tegmark, this implies that even given infinite time, a world can never be "complete" (fully generated). Which implies that even given infinite time, not everything that *can* happen *will* happen; which was my argument to begin with. Jonathan Colvin