What is the (currently known) consistency strength of global failure of the GCH?


I do not have access to the exact statement of the original Foreman-Woodin result. My searches seem to indicate that they used an assumption at the region of a supercompact, although I have seen comments stating that the result has been improved to require something in the region of a hypermeasurable. Is this correct? What this exact upper bound?


Thanks a lot.



Failure of the GCH
The following quotations are taken from Matthew Foreman and W. Hugh Woodin, "The generalized continuum hypothesis can fail everywhere," *Ann. Math.* **133** (1991), 1–35.



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> THEOREM. Let $\kappa$ be a supercompact cardinal with infinitely many inaccessible cardinals above $\kappa$. Then there is a partial ordering $\mathbf P$ such that in $V^{\mathbf P}$, $V\_\kappa \models ZFC + \forall \lambda: 2^\lambda > \lambda^+$.
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> In fact we can arrange by our choice of partial orderings that $V^{\mathbf P}\models$ $\kappa$ is $\beth\_n(\kappa)$-supercompact. Solovay has shown that if $\kappa$ is supercompact then $2^{\beth\_\omega(\kappa)} = \beth\_\omega(\kappa)^+$; hence this is near best possible. Woodin extended this result to get:
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> THEOREM (Woodin). If there is a supercompact cardinal then there is a model of ZFC in which $2^\kappa = \kappa^{++}$ for each cardinal $\kappa$.
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The last sentence of the paper reads:



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> The second author has also reduced the consistency strength of "$ZFC + \forall\kappa: 2^\kappa > \kappa^+$" and "$ZFC + \forall\kappa: 2^\kappa = \kappa^{++}$" to that of a ${\mathscr P}^2(\kappa)$-hypermeasurable.
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It's not clear to me if the proofs of the two theorems attributed to Woodin have ever been published.


