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