Woodin's unpublished proof of the global failure of GCH

Question

An unpublished result of Woodin says the following:

Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$

In the paper "The generalized continuum hypothesis can fail everywhere, Ann. Math. 133 (1991), 1–35" it is stated that Woodin has proved this result assuming the existence of a $P^{2}(\kappa)-$hypermeasurable (or equivalently a $\kappa+2-$strong) cardinal $\kappa$.

But using core model theory we now that more than a $P^{2}(\kappa)-$hypermeasurable cardinal $\kappa$ is required for this result (though a $P^{3}(\kappa)-$hypermeasurable cardinal $\kappa$ is sufficient).

So my question is

Question. 1-What large cardinal assumption is used in the proof of the above Theorem.

2-Does anyone know the proof? Does it use the supercompact Radin forcing as in the paper stated above, or it uses the ordinary Radin forcing (like Cummings paper "A model in which GCH holds at successors but fails at limits").

Answer 1

As you say, Hugh's precise result is unpublished. I have not seen any written reports of it, so I do not know the precise hypotheses it uses. For purely historical reasons, I would be interested if someone has such a report.

I assume the Cummings-Woodin unpublished manuscript (mentioned here and here) was meant to present an account of the argument but, at least the version I have, stops before getting there (in a chapter titled "Modified Prikry forcing, Part I").

(By the way, send me an email if you'd like a copy and haven't been able to contact James for one.)

In any case, Hugh is arguing with hypermeasurables, and using the corresponding version of Radin forcing, instead of the original supercompact-based version. The argument in the Foreman-Woodin paper (that proves the consistency of the weaker statement "$\mathsf{GCH}$ fails everywhere", and uses a variant of the supercompact-based version) is actually much older. As they say in the paper,

This work was done in 1979 while both authors were students at the University of California at Berkeley.

On the other hand, there is at least one published proof of the result, see

Carmi Merimovich. A Power Function with a Fixed Finite Gap Everywhere, The Journal of Symbolic Logic, 72 (2), (2007), 361-417. MR2320282 (2008k:03101).

(A preliminary version is available at the arXiv.)

Merimovich uses extender based Radin forcing. This is Merimovich's extension of a technique originally developed by Gitik and Magidor. The introduction to the paper gives a good account of the history of the problem, and of the techniques it uses, and appropriate references can be found there. For further (later) refinements of the technique, see

Carmi Merimovich. Extender-based Magidor-Radin forcing, Israel J. Math., 182, (2011), 439–480. MR2783980 (2012c:03146).

His argument can give a model of $\forall\lambda\,(2^\lambda=\lambda^{+n})$ for any fixed $n$, $1<n<\omega$, (call this statement $\mathsf{GCH}^{+n}$), though he presents the details for $n=3$. He assumes $\mathsf{GCH}$ and the existence of a cardinal $\kappa$ that is what is either called $(\kappa+n)$-strong, or $\kappa^{+n+1}$-strong, that is, there is an elementary embedding $j:V\to M$ with $\mathrm{cp}(j)=\kappa$ and $V_{\kappa+n}\subset M$. Using extender based Radin forcing at $\kappa$, he obtains an extension where $\kappa$ is still inaccessible, and $V_\kappa$ satisfies $\mathsf{GCH}^{+n}$. The final model is then $V_\kappa$.

(The Foreman-Woodin argument is similar in this respect, they begin with a supercompact $\kappa$ with infinitely many inaccessible cardinals above, and their final model is the $V_\kappa$ of the forcing extension. Again, $\kappa$ is inaccessible in this extension, and in fact significantly more. I do not know how to arrange such detailed global behavior of the continuum function, without cutting the universe at some point.)

Naturally, since the proof produces a set model of the desired statement, this means that Merimovich's assumptions are an overkill, but I do not know what the large cardinal companion of $\mathsf{GCH}^{+n}$ is, or even of better upper bounds. This seems a rather delicate and attractive problem (then again, this is an area I have always found very appealing.).

Answer 2

At the present there are at least three published paper concerning the global behavior of the power function:

1-Foreman-Woodin, The generalized continuum hypothesis can fail everywhere.

2- Cummings, A model in which GCH holds at successors but fails at limits.

3-Merimovich, A power function with a fixed finite gap everywhere.

The first paper uses a supercompact cardinal with infinitely many inaccessibles above it, while the last two papers use strong cardinals. Let me mention that:

(*) In all of these models cofinalities are changed and in the last two models cardinals are also collapsed.

(**) All of these models are obtained in two steps: At the first step a reverse Easton iteration is done which blows up the power of some cardinals below $\kappa$ ($\kappa$ is the large cardinal which we are using to exist in the ground model) and in the extension some guiding generics are constructed for the use of second step. In the second step a variant of Radin forcing (usually with interleaved collapsed,...) is used. Note that changing cofinalities and collapsing cardinals are presented in this step. At the end $\kappa$ remains inaccessible and below $\kappa$ we have the behavior of the function as we requested.

I would like to mention two recent results of Sy Friedman and I:

Theorem 1. Assuming the existence of a $\kappa+3-$strong cardinal $\kappa,$ there exists a pair $(W,V)$ of models of $ZFC$ such that:

1- $W$ and $V$ ahve the same cardinals,

2-GCH holds in $W$,

3-$GCH$ fails everywhere in $V$,

4-$V=W[R]$ for some real $R$.

The above theorem says that it is possible to kill the GCH everywhere just by adding a single real. It answers an open question of Shelah-Woodin "Forcing the failure of CH by adding a real" (I like the above result a lot).

Theorem 2. Assuming the existence of a $\kappa+4-$strong cardinal $\kappa$ it is consistent to have a pair $(W,V)$ of models of $ZFC$ such that:

1-$W$ and $V$ have the same cofinalities,

2-GCH holds in $W$,

3-$V\models \forall \lambda, 2^{\lambda}=\lambda^{+3}.$

Thus it is possible to kill the GCH everywhere by a cofinality preserving forcing.

I also should mention that Moti Gitik and Carmi Merimovich are doing a project in which they are planning to obatain the global failure of GCH from the optimal hypotheses. If I remeber their result correctly it says something like this:

Theorem. The following are equiconsistent:

1-For any $\alpha,$ there are stationary many cardinals $\kappa$ with $O(\kappa)=\kappa^{++}+\alpha,$

2-GCH fails everywhere,

3-$\forall \lambda, 2^{\lambda}=\lambda^{++}.$

Gitik told me that their proof uses the Radin forcing and they need to embed many models into each other. My guess is that the paper " Gitik, Moti; Merimovich, Carmi Power function on stationary classes" is related to their work.