# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

Sorry for my delayed response.

For the purposes of this discussion:

The $\textsf{IMH}^\#$ is the statement that V is #-generated and if a sentence holds in a #-generated outer model then it holds in an inner model. It is consistent with all large cardinals.

The (crude, uncut) $\textsf{SIMH}^\#$ is the statement that V is #-generated and if a sentence with absolute parameters holds in a cardinal-preserving, #-generated outer model then it holds in an inner model. It implies a strong failure of CH but is not known to be consistent.

The reason that the $\textsf{IMH}^\#$ can be shown to be consistent is that Jensen coding works for #-generated models.

The reason that the $\textsf{SIMH}^\#$  is not known to be consistent is that Jensen coding will collapse cardinals if GCH fails.

I hope that this clarifies the situation.