Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 30, 2014, at 6:28 AM, Sy David Friedman wrote:

Dear Hugh,

Regarding your “HOD Conjecture”: I look at it very differently. My guess is that it is true, but this only means that maximality (V far from HOD) implies that extendible cardinals don’t exist! Probably this can be improved to “supercompacts don’t exist”!

So one could reasonably take the view that the HOD Conjecture is as misguided now as would have been the conjecture that L is close to V given the Jensen Covering Theorem. (Let’s revise history and pretend that Jensen’s Covering Theorem was proved before measurable cardinals etc. had been defined and analyzed).

Unless you can somehow get extendible cardinals into the picture, what you call the HOD conjecture is indeed misguided, as Cummings, Golshani and I have shown.

The \text{HOD} Conjecture asserts there is a proper class of regular cardinals which are not \omega-strongly measurable in \text{HOD}. Your results here in no way show this is misguided and moreover, while interesting, these results are completely irrelevant to the \text{HOD} Conjecture. I have already pointed this out to you several times.

Why?

1) It is not known (without appealing to Reinhardt cardinals) if there can exist even 4 regular cardinals which are \omega-strongly measurable in \text{HOD}, even getting 3 requires \textsf{I}0 and the \Omega-Conjecture.

2) It is not known if the successor of a singular strong limit of uncountable cofinality can be \omega-strongly measurable in \text{HOD}.

To refute the \text{HOD} Conjecture one must produce a model in which all sufficiently large regular cardinals are \omega-strongly measurable in \text{HOD}.

Regards,
Hugh

PS: \kappa is \omega-strongly measurable in \text{HOD} if there exists \lambda < \kappa such that (2^{\lambda})^{\text{HOD}} < \kappa and such that
there is no partition of S = {\alpha < \kappa: \text{cf}(\alpha) = \omega} into \lambda many sets \langle S_{\alpha}: \alpha < \lambda\rangle \in \text{HOD} such
each set S_{\alpha} is stationary in V.

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