Tag Archives: IMH#(card)

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