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.
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.
The unpublished proofs are by now standard. You can get $ZFC+\forall\kappa(2^\kappa=\kappa^{+n})$ for any fixed natural $n>0$ from ${\mathcal P}^n(\kappa)$-hypermeasurables. The arguments in, for example, J. Cummings paper (on violating SCH at all limits) or the many papers by Gitik and his students (including his very nice Handbook article) should explain how to fill in any missing details.
– Andrés E. Caicedo Nov 03 '11 at 19:12By 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-$\forall \lambda, 2^{\lambda}=\lambda^{++}.$
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?)
