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

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