# 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.

Regards,
Hugh