Tag Archives: Real without a sharp

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