Failure of the GCH 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.
 A: It looks like the other answers only deal with upper bounds, so I thought I'd point out that by a result of Gitik,
http://www.sciencedirect.com/science/article/pii/016800729190016F,
$\exists \kappa\; o(\kappa) = \kappa^{++}$ is a lower bound for $\neg$SCH and therefore also for the global failure of GCH.  (But we still haven't answered the question of the exact consistency strength of the latter.  Is there a better lower bound out there?)
A: see here:
http://dx.doi.org/10.2178/jsl/1185803615
A: 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.

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 $V^{\mathbf P}$, $V_\kappa \models ZFC + \forall \lambda: 2^\lambda > \lambda^+$.
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:
THEOREM (Woodin).  If there is a supercompact cardinal then there is a model of ZFC in which $2^\kappa = \kappa^{++}$ for each cardinal $\kappa$.

The last sentence of the paper reads:

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.

It's not clear to me if the proofs of the two theorems attributed to Woodin have ever been published.
A: By work of Gitik-Mitchell a $(\kappa+2)$-strong cardinal $\kappa$ is required, and by work of Merimovich a  $(\kappa+3)$-strong cardinal $\kappa$ (in fact a cardinal $\kappa$ with $o(\kappa)=\kappa^{++}+\kappa^{+}$) is enough. Gitik and Merimovich have a project to get the total failure of $GCH$ from optimal hypotheses. It's yet incomplete. If I remember it 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-$\forall \lambda, 2^{\lambda}=\lambda^{++}.$
