# Different Methods for Forcing Total Failure of Generalized Continuum Hypothesis

It seems there are few papers on total failure of Generalized Continuum Hypothesis.

As an example Merimovich in 2007 constructed a model of ZFC such that GCH fails everywhere assuming consistency of ZFC and strong cardinals.

My question is that how many essentially different methods are known for forcing total failure of GCH?

Any reference to a paper on the subject is also welcome.

• I think asking for the number of known methods hardly makes sense. -- Voted to close. – Stefan Kohl Jan 28 '14 at 17:11

To my knowledge, there are at least three published methods for killing the $GCH$ everywhere:
1) forcing with supercompact Radin forcing (by Foreman-Woodin) . In fact Foreman and Woodin introduced an intermediate submodel of the supercompact Radin forcing in which the $GCH$ fails everywhere. They used a supercompact cardinal with infinitely many inaccessibles above it.
2) forcing with ordinary Radin forcing using guiding generics. See for example Cummings paper "A model in which $GCH$ holds at successors but fails at limits". The proof uses a strong cardinal (in fact a ($\kappa+3)-$strong cardinal $\kappa$ is sufficient)
3) forcing with extender based Radin forcing (Merimovich). The large cardinal assumption is the existence of a measurable cardinal $\kappa$ of Mitchell order $O(\kappa)=\kappa^{++}+\kappa^+.$