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

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