Re: Paper and slides on indefiniteness of CH

Dear Sy,

I introduced a cardinal-preserving method for forcing clubs through \omega_2 with finite conditions, even without CH in the ground model. My motivation was to try to do this for \text{Ord} instead of for just omega_2, 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 \omega_3 or for \omega_2 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 \omega_3. 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.

I agree that may be how the question go. Actually I think Aspero has the best partial results now, he can prove that for each n there is a cardinal preserving forcing which has a new subset of \aleph_n which is not \aleph_n-cc generic over V.

However to generalize the method it looks like one might need square at \alpha_{\omega} etc. So there may be some rather serious obstructions.


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