Dear Geoffrey and Sol,
The issue is whether “the totality of all full models of third-order number theory” is a legitimate vehicle for establishing that statements like CH have a definite truth value.
I was trying to shake Geoffrey’s supreme confidence that CH has a definite truth value by appealing to statements outside set theory that clearly do not have a definite truth value.
So that’s why we went outside math into science, and also to
- Mozart is a better musician than Beethoven.
- Geoffrey is bald.
All that we have about sets of integers is that they are extensions of ARBITRARY predicates on the integers. That’s not much of an explanation. It does tell us that if we come across any way of giving a predicate then we can form the set. The full impredicative comprehension axiom scheme looks very compelling – although it can be criticized in. Nothing like full comprehension, even on the reals, has the power to decide CH. Any concept of “set theoretic universe” makes for a far far more complex story. The notion of arbitrary ordinal looks much harder to argue is definite in the appropriate sense that even “set of reals”.
Experience shows that almost no one who has thought about these matters has ever changed their mind based on an argument even as well thought out as Sol’s. So the kind of interchange Geoffrey and I have been having does seem to be relevant.
I tend to think that we are all spoiled by just how amazingly far we can go in the finite with very minimal principles, and also even just how far ZFC goes. We think that if we can go so amazingly far, then it is because of some underlying Platonic reality, or at least a way of doing set theory that is unequivocally forced on us. It seems likely that we are mistaken.
But you still have to tell a convincing story. For skeptics, the story is only interesting to the extent that is simple and compelling.
Again, I never see anybody change their minds on the “definite truth value” issue. But there is a place where maybe we can agree. That is on the issue of whether CH research is a relatively promising area of research in the foundations of mathematics. I read Solovay as saying “no”, even though he a “committed Platonist”. What does Geoffrey think?