# Re: Paper and slides on indefiniteness of CH

Dear Sy,

I guess I have misunderstood something. This would not be the first time.

I thought that M witnesses $\textsf{SIMH}^\#(\omega_1)$ implies that  if there is a $\#$-generated extension of M preserving $\omega^M_1$ in which there is a definable inner model in which $\varphi$ holds of $\omega^M_1$, then in M there is a definable inner model in which $\varphi$ holds of $\omega^M_1$.  Maybe this implied by $\textsf{SIMH}^\#(\omega_2)$ and what I thought was $\textsf{SIMH}^\#(\omega_1)$ is really  $\textsf{SIMH}^\#(\omega_1+1)$.

In any case this in turn implies that in M there is a real $x$ such that $\omega_1 = \omega_1^{L[x]}$.  So unless I am really confused the existence of a real $x$ such that $\omega_1 = \omega_1^{L[x]}$ follows from $\textsf{SIMH}^\#$ which still makes my point.

So I guess it would be useful to have precise statements (in terms of countable models etc) of $\textsf{SIMH}^\#$ and $\textsf{SIMH}^\#(\kappa)$ that we all can refer to.

Regards.
Hugh

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

The disadvantage of your formulation of $\textsf{IMH}^\#$ is that it is not even in general a $\Sigma^1_3$ property of M and so it is appealing in more essential ways to the structure of the “hyperuniverse”.  This is why the consistency proof of $\textsf{SIMH}^\#(\omega_1)$ uses substantially more than a Woodin cardinal with an inaccessible above,  unlike the case of $\textsf{IMH}$ and $\textsf{SIMH}(\omega_1)$.

OK, It seems we will just have to agree that we disagree here.

I think it is worth pointing out to everyone that $\textsf{IMH}^\#$, and even the weaker \textsf{SIMH}(\omega_1)\$ which we know to be consistent, implies that there is a real $x$ such that $x^\#$ does not exist (even though $x^\#$ exists in the parent hyperuniverse which is a bit odd to say the least in light of the more essential role that the hyperuniverse is playing). The reason of course is that $\textsf{SIMH}(\omega_1)$ implies that there is a real $x$ such that $L[x]$ correctly computes $\omega_1$.

This is a rather high price to pay for getting not-CH.

Thus for me at least, $\textsf{SIMH}^\#$ has all the problems of $\textsf{IMH}$ with regard to isolating candidate truths of V.

Regards,
Hugh