Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ordinals $\alpha$. It is known of course that $F(\alpha) = \alpha+1$ is a complete resolution of GCH (in the positive) that is relatively consistent with ZFC. I read that it's an unpublished theorem of Woodin that $F(\alpha) = \alpha+2$ is a complete resolution of GCH that is relatively consistent with ZFC plus some large cardinal hypothesis. My questions are: (1) What's the weakest known complete resolution of GCH in consistency strength other than $F(\alpha) = \alpha+1$ and what large cardinal axiom is required for it? (2) What are some other complete resolutions of GCH that are known to be consistent relative to specific large cardinal hypotheses, what are their respective large cardinal hypotheses, and how do these consistency strengths relate to one another?
Complete Resolutions of GCH
continuum-hypothesislarge-cardinalsset-theory
Related Solutions
As you say, Hugh's precise result is unpublished.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I assume the Cummings-Woodin unpublished manuscript (mentioned here and here) was meant to present an account of the argument but, at least the version I have, stops before getting there (in a chapter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.).
In the following answer, by Foreman-woodin model, I mean the model constructed by them in the paper "The generalized continuum hypothesis can fail everywhere. Ann. of Math. (2) 133 (1991), no. 1, 1–35. "
Questions 1 and 3 have positive answer: In Foreman-Woodin model for the total failure of GCH the following hold:
1) For all infinite cardinal $\kappa, 2^{\kappa}$ is weakly inaccessible, and hence a fixed point of the $\aleph-$function,
2) If $\kappa \leq \lambda< 2^{\kappa},$ then $ 2^{\lambda}= 2^{\kappa}.$
In this model for all infinite cardinals $\kappa, 2^{\kappa}=\aleph_{ 2^{\kappa}}$ in particular for all fixed points $\kappa$ of the $\aleph-$function, $2^{\aleph_\kappa}=\aleph_{ 2^{\kappa}}$. Also note that in this model for all infinite cardinals $\kappa,$ if we let $\lambda=2^{\aleph_\kappa},$ then $\lambda \geq 2^\kappa,$ and $2^{\aleph_\kappa}=\aleph_\lambda.$ So both of questions 1 and 3 have a positive answer.
For your question 2, $\delta$ can be arbitrary large: Start with GCH+there exists a supercompact cardinal $\kappa$+ there are infinitely many inaccessibles above it. Now let $\delta$ be any ordinal $<\kappa.$ Force with Foreman-Woodin construction above $\delta$ (in the sense that let the first point of the Radin club added during their forcing construction be above $\delta$). In their final model (which is $V_\kappa$ of some extension of the ground model) for all infinite cardinals $\lambda <\kappa, 2^\lambda \geq \lambda^{+\delta}$. So if $F$ is defined in the ground model by $F(\kappa)=\kappa^{+\delta},$ then the acceleration rank of $F$ is $\delta$ (using GCH), and in the finial model for all infinite cardinals $\kappa, 2^\kappa \geq F(\kappa).$
Remark. I may note that we can not define the function $F$ in the ground model, such that it is the realization of power function in the extension, but we can find some inner model of the final extension in which $GCH$ holds and such a function $F$ is definable.
Best Answer
One candidate answer scheme might be the following: if $F$ is any (sufficiently absolute) definable function on the class of regular alephs such that $\kappa < \lambda \Rightarrow F(\kappa) \leq F(\lambda)$ and $\operatorname{cf}(F(\kappa)) > \kappa$, then ZFC + $(\forall \kappa = \operatorname{cf}(\kappa))(2^\kappa = F(\kappa))$ + SCH is consistent, where SCH is the Singular Cardinals Hypothesis or, in an equivalent form, the Gimel Hypothesis, due to Solovay, asserting $(\forall \kappa > \operatorname{cf}(\kappa))( \kappa^{\operatorname{cf}(\kappa)} = \max(2^{\operatorname{cf}(\kappa)}, \kappa^+))$, and no large cardinals are required.
Knowledge of the gimel function $\gimel(\kappa) = \kappa^{\operatorname{cf}(\kappa)}$ suffices to determine cardinal exponentiation recursively (for example, see P. Komjath, V. Totik, (Problems and Theorems in Classical Set Theory): chapter 10, problem 26, sets this out). So it is natural to explore the gimel function in greater depth. Writing a singular $\kappa$ as the limit of an increasing sequence $a$ of smaller regular cardinals leads to the observation that the deeper problem concerns the cofinality $\operatorname{cf}(([\kappa]^{\leq \lambda}, \subseteq))$ of the partial order $([\kappa]^{\leq \lambda}, \subseteq)$ for regular $\lambda < \kappa$. In this direction, one comes eventually to pcf theory, which offers an analysis of the puppet master $\operatorname{pcf}(a)$ rather than his troupe of erratic marionettes $\langle 2^\lambda : \lambda \in Card \rangle$.