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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.

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    $\begingroup$ I think asking for the number of known methods hardly makes sense. -- Voted to close. $\endgroup$
    – Stefan Kohl ♦
    Commented Jan 28, 2014 at 17:11

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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 $O(\kappa)=\kappa^{++}+\kappa^+.$

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  • $\begingroup$ What about killing GCH everywhere with a single real? :-) $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 28, 2014 at 8:00
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    $\begingroup$ We used a modified version of Merimovich construction (the reason is that in Merimovich construction cardinals are collapsed, but we were interested in preserving all cardinals, so we modified the Merimovich construction simply by replacing collapsing forcings with suitable Cohen forcings), but there the point is that we need to code a lot of information into a single real over a core model, which is the main part of the paper $\endgroup$
    – Mohammad Golshani
    Commented Jan 28, 2014 at 8:03

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