On Oct 8, 2014, at 6:48 AM, Sy David Friedman wrote:
I am most pessimistic about Type 1 truth (Thin Realism). To get any useful conclusions here one would not only have to talk about “good Set Theory” but about “the Best Set Theory”, or at least show that all forms of “good Set Theory” reach the same conclusion about something like CH. Can we really expect to ever do that? To be specific: We’ve got an axiom proposed by Hugh which, if things work out nicely, implies CH. But then at the same time we have all of the “very good Set Theory” that comes out of forcing axioms, which have enormous combinatorial power, many applications and imply not CH. So it seems that if Type 1 truth will ever have a chance of resolving CH one would have to either shoot down Ultimate-L, shoot down forcing axioms or argue that one of these is not “good Set Theory”. Pen, how do you propose to do that? Forcing axioms are here to stay as “good Set Theory”, they can’t be “shot down”. And even if Ultimate-L dies, there will very likely be something to replace it. Why should we expect this replacement for Ultimate-L to come to the same conclusion about CH that forcing axioms reach (i.e. that CH is false)?
I do not see this at all, In fact, not surprisingly, I completely disagree.
If V = Ultimate L (and there are large enough cardinals) then one will have inner models containing the reals in which the forcing axioms hold (including Martin’s Maximum). Thus the theorems of Martin’s Maximum for say all apply to the objects in such inner models.
For example, consider Farah’s result that all automorphisms of the Calkin Algebra are inner automorphisms assuming MM. Any inner model containing the reals correctly computes the Calkin Algebra, so Farah’s result applies equally well to automorphisms which belong to such inner models.
One also has inner models containing the reals of the Pmax-axiom and inner models containing the reals for all of its variations. These axioms are much powerful at the level of .
This is completely analogous to the theory of determinacy which flourishes in the Axiom of Choice universe through the study of inner models of AD which contain the reals.
Finally note there there is a fundamental asymmetry here. Assuming V = Ultimate L one can have inner models containing the reals of say MM. But assuming MM one cannot have an inner model containing the reals which satisfies V = Ultimate L.