# Re: Paper and slides on indefiniteness of CH

Dear Sy,

So what is $\textsf{SIMH}^\#(\omega_1)$? You wrote in your message of Sept 29:

The $\textsf{IMH}^\#$ is compatible with all large cardinals. So is the  $\textsf{SIMH}^\#(\omega_1)$

A second question. The version of  $\textsf{SIMH}^\#$ you specified in your next message to me on Sept 29:

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.

does not even obviously imply $\textsf{IMH}^\#$.  Perhaps you meant, the above together with $\textsf{IMH}^\#$? If not then calling it $\textsf{SIMH}^\#$ is rather misleading. Either way it is closer to $\textsf{IMH}^\#(\text{card})$.

Anyway this explains my confusion, thanks.

Regards,
Hugh