# 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