Tag Archives: Fine structure

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.