# Re: Paper and slides on indefiniteness of CH

Dear Sy,

What form of ordinal maximality are you using? In my paper with Arrigoni I had a weaker form, with Honzik a stronger one. In the latter version, a countable ordinal maximal universe remains ordinal maximal in any outer model of V.

The notion of ordinal maximality to which I was referring was that in the bulletin paper and that which is used to formulate IMH* there.

Indeed, I do strongly endorse the consistency of AD and much more. I do not subscribe to the view that we need the inner model programme to justify the consistency of large cardinals. I think that there is enough evidence from the role of large cardinals in establishing consistency results to justify their own consistency, and indeed I would go further and assert their existence in inner models.

I really do not understand the basis for your conviction in the consistency of PD (or AD or ZFC + $\omega$ many Woodin cardinals).

Consider the Extended Reimann Hypothesis (ERH). ERH passes all the usual tests cited for PD (or AD or ZFC + $\omega$ many Woodin cardinals) as the basis of its consistency. Tremendous structure theory, implications of theorems which are later proved by other means etc.

Yet there does not seem to be any conviction in the Number Theory community that even the Reimann Hypothesis is true (and of course RH follows from the consistency of ERH).  Look at the statement on the rules of the Millennium Prizes. A counterexample to RH is not unconditionally accepted as a solution. If there was any consensus that RH is true this escape clause would not be in the stated rules.

Further the structure theory you cite as evidence for Con PD is in the context of PD etc. If one rejects that context then how can one maintain the conviction that Con PD is true?

Rephrased: The fact that the models of T (if any exist) have a rich internal theory is not evidence that there are any models of T. Something else is needed.

I think this is another example of the fundamental difference in our points of view. Yes, “iterable and correct” inner models are important for the relationship between large cardinals and descriptive set theory. But the fundamental concept of inner model is simply a transitive class containing all the ordinals and modeling ZFC, there is no technical requirement of ‘iterability’ involved. Thus again we have the difference between the interpretation of a basic notion in (a particular area of) set-theoretic practice and its natural interpretation in discussions of set-theoretic truth. And there is no hope of producing useful inner models which are correct for 2nd order arithmetic without special assumptions on V, such as the existence of large cardinals. And even if one puts large cardinal axioms into the base theory one still has no guarantee of even Sigma-1-3 correctness for outer models which are not set-generic. So to say that large cardinals “freeze projective truth” is not accurate, unless one adopts a set-generic interpretation of “freezing.”

I completely agree this is the basic issue over which we disagree.

The position that all extensions, class or set generic, on on an equal footing is at the outset already a bias against large cardinals. The canonical objects identified by large cardinals, such as the generalizations of $0^\#$, can disappear (i.e. cease to be recognized as such) if one passes to a class forcing extension.

Rephrased: The claim that an inner model is just a proper class is a bias against large cardinals. Once one passes the level of one Woodin cardinal the existence of proper class inner models becomes analogous to the existence of transitive set models in the context of just ZFC. It has no real structural implications for V particularly in the context of for example IMH (which are not already implied by the existence of an inner model of just 1 Woodin cardinal). This fact is not irrelevant to HP since it lies at the core of the consistency proof of IMH.

Let me explain further and also clarify the relationship between $\Omega$-logic and set forcing. For this discussion and to simplify things grant that the $\Omega$ Conjecture is provable and that the base theory is now ZFC + a proper class of Woodin cardinals.

To a set theorist, a natural variation of ordinal maximality, let’s call this strong rank maximality, is that there are rank preserving extensions of M in which large cardinals exist above the ordinals of M (and here one wants to include all “possible large cardinals” whatever that means).

Question 1:  How can we even be sure that there is no pairwise incompatibility here which argues against the very concept of the $\Pi_2$ consequences of strong rank maximality?

Question 2:  If one can make sense of the $\Pi_2$ consequences of strong rank maximality and given that M is strongly rank maximal, can the $\Pi_2$ consequences of this for M be defined in M?

Here is the first point. If there is a proper class of X-cardinals  (and accepting also that an X-cardinal is preserved under forcing by partial orders of size less than the X-cardinal), then in every set-generic extension there is a proper class of X-cardinals and so in every set-generic extension, the sentence  $\phi$ holds where

$\phi$ = “Every set $A$ belongs to a set model with an X-cardinal above $A$.”

$\phi$ is a $\Pi_2$-sentence and therefore by the $\Omega$ Conjecture this \Pi_2-sentence is $\Omega$ provable. Further these are arguably exactly the $\Pi_2$ sentences which generate the $\Pi_2$ consequences of strong rank maximality.

Here is the second point. If $M_1$ is a rank initial segment of $M_2$ then every sentence which is $\Omega$-provable in $M_2$ is $\Omega$-provable in $M_1$. $\Omega$ proofs have a notion of (ordinal) length and in the ordering of the $\Omega$-provable sentences by proofs of shortest  length, the sentences which are $\Omega$-provable in $M_2$ are an initial segment of the sentences which are $\Omega$-provable in $M_1$ (and they could be the same of course).

Putting everything together, the $\Pi_2$-consequences of the strong rank maximality of a given model $M$ makes perfect sense (no pairwise incompatibility issues) and this set of $\Pi_2$-sentences is actually definable in $M$.

This connection with $\Omega$-logic naturally allows one to adapt strong rank maximality into the HP framework, one restricts to extensions in which the $\Omega$-proofs of the initial model are not de-certified in the extension (for example if a $\Pi_2$ sentence is $\Omega$-provable in the initial model $M$, it is $\Omega$-provable in the extension).

This includes set-forcing extensions but also many other extensions. So in this sense $\Omega$-logic is not just about set-forcing. $\Omega$-logic is about trying to clarify (or even make sense of) the $\Pi_2$ consequences of large cardinals (and how is this possibly not relevant to a discussion of truth in set theory?).

My concern with HP is this. I do not see a scenario in which HP even with strong rank maximality can lead anywhere on the fundamental questions involving the large cardinal hierarchy.  The reason is that strong rank maximality considerations will force one to restrict to the case that PD holds in V at which point strong rank maximality notions require consulting at the very least the $\Omega$-logic of V and this is not definable within the hyper-universe of V.

Granting this, genuine progress on CH is even less plausible since how could that solution ever be certified in the context of strong rank maximality? A solution to CH which is not compatible with strong rank maximality is not a solution to CH since it is refuted by large cardinals.

You will disagree and perhaps that is the conclusion of this discussion, we simply disagree.

But here is a challenge for HP and this does presuppose any conception or application of a notion of strong rank maximality.

Identify a new family of axioms of strong infinity beyond those which have been identified to date (a next generation of large cardinal axioms) or failing this, generate some new insight into the hierarchy of large cardinal axioms we already have. For example, HP does not discriminate against the consistency of a Reinhardt cardinal. Can HP make a prediction here? If so what is that prediction?

Regards,
Hugh