# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Sun, 31 Aug 2014, W Hugh Woodin wrote:

The condensation principle that you force (local club condensation) actually does not hold in fine-structural models for the finite levels of supercompact which have been constructed (assuming the relevant iteration hypothesis). There are new fine-structural phenomena which happen in the long-extender fine structure models and which do not have precursors in the theory of short-extender models. (These models are generalizations of the short-extender models with Jensen indexing, the standard parameters are solid etc.)

When you say “Jensen indexing” do you mean the one that I proposed: index at the successor of the image of the critical point?

At the same time these models do satisfy other key condensation principles such as strong condensation at all small cardinals (and well past the least weakly compact). I believe that it is still open whether strong condensation can be forced even at all the $\aleph_n$’s by set forcing. V = Ultimate L implies strong condensation holds at small cardinals and well past the least inaccessible.

Very interesting! I guess we provably lose strong condensation at the level of $\omega$-Erdős, but it would of course be very nice to have it below that level of strength.

Finally the fine structure models also satisfy condensation principles at the least limit of Woodin cardinals which imply that the Unique Branch Hypothesis holds (for strongly closed iteration trees) below the least limit of Woodin cardinals. If this could be provably set forced (without appealing to the $\Omega$ Conjecture) then that would be extremely interesting since it would probably yield a proof of a version of the Unique Branch Hypothesis which is sufficient for all of these inner model constructions.

For the uninitiated I think we should be clear about the difference between what has been proved and what has only been conjectured. As I understand it:

1. The Friedman-Holy models actually exist (they can be forced) and fulfill John’s 3 conditions.
2. The models you are discussing are only conjectured to exist. In the final paragraph above you hint at a way of actually producing them.
3. If your models do exist then they also fulfill John’s 3 conditions but have condensation properties which are rather different from those of the Friedman-Holy models.

But don’t your models, if they exist, have some strong absoluteness properties that the Friedman-Holy models are not known to have? That’s why I suggested that John’s list of 3 conditions may have been incomplete.

Hugh, it is wonderful that you have the vision to see how inner model theory might go, and the picture you paint is fascinating. But as you know the history of inner model theory has been full of surprises, and in particular we can’t just assume that the iterability hypotheses will be verified. For this reason, I do think it important to be clear about what has been proved and what has only been conjectured.

Thanks,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

I guess I need to weigh in here on your message to Steel.

You then advertise Hugh’s new axiom as having 3 properties:

1. It implies core existence.
2. It suggests a way of developing fine-structure theory for the core.
3. It may be consistent with all large cardinals.

Surely there is something missing here! Look at my paper with Peter Holy: “A quasi-lower bound on the consistency strength of PFA, to appear, Transactions American Mathematical Society“. (I spoke about it at the 1st European Set Theory meeting in 2007.)

We use a “formidable” argument to show that condensation with acceptability is consistent with essentially all large cardinals. As we use a reverse Easton iteration the models we build are the “cores” in your sense of their own set-generic multiverses. And condensation plus acceptability is a big step towards a fine-structure theory. It isn’t hard to put all of this into an axiom so our work fulfills the description you have above of Hugh’s axiom.

The condensation principle that you force (local club condensation) actually does not hold in fine-structural models for the finite levels of supercompact which have been constructed (assuming the relevant iteration hypothesis). There are new fine-structural phenomena which happen in the long-extender fine structure models and which do not have precursors in the theory of short-extender models. (These models are generalizations of the short-extender models with Jensen indexing, the standard parameters are solid etc.)

At the same time these models do satisfy other key condensation principles such as strong condensation at all small cardinals (and well past the least weakly compact). I believe that it is still open whether strong condensation can be forced even at all the $\aleph_n$’s by set forcing. V = Ultimate L implies strong condensation holds at small cardinals and well past the least inaccessible.

Finally the fine structure models also satisfy condensation principles at the least limit of Woodin cardinals which imply that the Unique Branch Hypothesis holds (for strongly closed iteration trees) below the least limit of Woodin cardinals. If this could be provably set forced (without appealing to the $\Omega$ Conjecture) then that would be extremely interesting since it would probably yield a proof of a version of the Unique Branch Hypothesis which is sufficient for all of these inner model constructions.

Regards,
Hugh