# 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

# Re: Paper and slides on indefiniteness of CH

Dear all,

I believe this concludes the discussion that Sy and I were having on $\textsf{IMH}^\#$.  I would like to note one more thing and here the proof does likely requires the machinery Sy was discussing.

Let $\textsf{IMH}^\#(\text{card})$ be $\textsf{IMH}^\#$ together with if there is a cardinal preserving $\#$-generated extension in which $\varphi$ holds then there is a cardinal preserving inner model in which $\varphi$ holds. Similarly define $\textsf{IMH}^\#(\text{card-arith})$ (see earlier in the thread) by restricting to $\#$-generated extensions and inner models, which preserve all cardinals and cardinal arithmetic.

$\textsf{IMH}^\#(\text{card-arith})$ is consistent relative to a bit more than one Woodin cardinal, in fact it holds if every real has a sharp and there is a countable ordinal which is a Woodin cardinal in an inner model. But the proof requires the adaptation of Jensen coding to $\#$-generated models etc.  There is a natural guess as to the statement of that theorem and I am assuming it holds: If M is an inner model of GCH which is $\#$-generated then there is a real $x$ such that M is a definable inner model of $L[x]$, $x^\#$ exists, and such that $L[x]$ is a cardinal preserving extension of M.

As for $\textsf{IMH}(\text{card-arith})$, a natural conjecture is that $\textsf{IMH}^\#(\text{card-arith})$ implies GCH.

I have no idea about the consistency of $\textsf{IMH}^\#(\text{card})$ which looks like the problem of showing $\textsf{SIMH}$ is consistent.

Now a variant of a problem which I raised much earlier in this email-thread seems quite relevant. The new problem is: Can there exist a $\#$-generated M such that if N is a $\#$-generated extension which is cardinal preserving then every set of N is set-generic over M.

Regards,
Hugh