Dear Sy,

The disadvantage of your formulation of is that it is not even in general a 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 uses substantially more than a Woodin cardinal with an inaccessible above,  unlike the case of  and .

OK, It seems we will just have to agree that we disagree here.

I think it is worth pointing out to everyone that , and even the weaker \textsf{SIMH}(\omega_1)$ which we know to be consistent, implies that there is a real such that does not exist (even though 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  implies that there is a real such that correctly computes .

This is a rather high price to pay for getting not-CH.

Thus for me at least, has all the problems of with regard to isolating candidate truths of V.

Regards,
Hugh