On Tue, 14 Oct 2014, W Hugh Woodin wrote:
On Oct 14, 2014, at 3:50 PM, Harvey Friedman wrote:
Hugh just wrote:
“I would argue instead that this is simply a sort of coming of age for Set Theory; i.e. we can now pose simple questions about models of Set Theory which seem completely out of reach.
Number Theory is full of such problems. And one can easily generate such problems in a foundational framework”
I and I think many readers of this discussion, would very much like to see such “simple questions about models of Set Theory which seem completely out of reach” explained in generally understandable terms, nicely laid out in one message. I especially would like to see the simplest such question you have in mind.
Suppose M is a ctm and M \models ZFC. Must M have an outer model of ZFC which is cardinal preserving and not a set forcing extension?
I don’t know the answer to this interesting question, but I’m pretty sure how it’s going to go. In my old paper
I introduced a cardinal-preserving method for forcing clubs through with finite conditions, even without CH in the ground model. My motivation was to try to do this for instead of for just , preserving the powerset axiom. (I was looking for a new characterisation of 0#.) So I was essentially asking your question back then. Unfortunately there were 2 obstacles: I didn’t even know how to do this for or for without killing CH (see the last 2 questions in my paper cited above). The good news is that Krueger and Mota recently solved the latter problem; we are currently thinking about . So my conjecture is: There is a cardinal-preserving class-forcing with finite conditions that does not reduce to a set-forcing and preserves ZFC. I admit that this is very hard, but there is no hint of an obstruction to it. At the same time, I confess that I don’t know how to do it.