# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

I don’t think that we have a big disagreement here. The Friedman-Holy models are certainly not canonical and I agree that a key question is whether there are canonical, fine-structural inner models for large cardinals.

But no unverified hypotheses are needed to create the Friedman-Holy models and they witness the compatibility of arbitrary large cardinals with a key component of fine-structure theory: acceptability. We also force Global Square and I suspect that these models can be built as inner models using “internal consistency” arguments for reverse Easton forcings. (Of course the real motivation for the models was to use ideas of Neeman to show that a degree of supercompactness is a “quasi” lower bound on the consistency of the Proper Forcing Axiom).

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

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.

I do not agree here. I do not think that the Friedman-Holy models are a starting point for fine-structure. That was the point I was trying to make.

For example, (global) strong condensation does not even imply $\square_{\omega_1}$ and $\square$ is a central feature of fine-structure.

For me anyway, fine-structure is a feature of canonical models and it is the canonical models which are important here. Identified instances of fine-structure can be forced but this in general is a difficult problem.  Could one create a list of such features, force them all while retaining all large cardinals, and finally argue that the result is a canonical model?

This seems extremely unlikely to me.

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.

You are giving “my” models far more relevance here than they deserve.

Very technical point here: The models only reach the finite levels of supercompact and so do not fulfill John’s conditions. In fact, fine-structural extender models can never work since any such model is always a generic extension once one is past the level of one Woodin cardinal. One needs the hierarchy of fine-structural strategic-extender models. Though the latter occur naturally they have been much more difficult to explicitly construct. For example it is not yet known (as far as I know) if there can exist such an inner model at the level of a Woodin limit of Woodin cardinals no matter what iteration hypothesis and large cardinals one assumes in V.

If the Ultimate L Conjecture is true then V = Ultimate L meets John’s conditions. If this conjecture is false or more generally if there is an anti-inner model theorem (say at supercompact) then the Friedman-Holy models and their generalizations may be the best one can do (and to me this is the essence of the the inner model versus outer model debate).

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.

Of course I agree with this last point. The entire of theory of iterable models at the level of measurable Woodin cardinals and beyond could be vacuous and not because of an inconsistency. But it has not yet happened that a developed theory of canonical inner models as turned out to be vacuous. Will it happen? That is an absolutely key question right now.

Regards,
Hugh

# 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