# Re: Paper and slides on indefiniteness of CH

Dear Bob,

I guess I have used it both ways. But also I am most interested in (A) but in the form ZFC + extendible proves the formal statement of the HOD Conjecture i.e. that there is a proper class of regular cardinals which are not $\omega$-strongly measurable in HOD.

I suppose I should have called the formal statement of the HOD Conjecture, the HOD Hypothesis; and then defined the HOD Conjecture as the conjecture that ZFC (or ZFC + extendible) proves the HOD Hypothesis. Probably it is too late to make that change.

Here is a simple version the HOD Dichotomy theorem:

Theorem. Suppose $\delta$ is extendible. Then the following are equivalent.

1. HOD Hypothesis.
2. There is a regular cardinal above $\delta$ which is not $\omega$-strongly measurable in HOD.
3. There is a regular cardinal above $\delta$ which is not measurable in HOD.
4. For every singular cardinal $\gamma > \delta$, $\gamma$ is singular in HOD and $\gamma^+$ is the $\gamma^+$ of HOD.
5. $\delta$ is supercompact in HOD witnessed by the restriction to HOD of supercompactness measures in $V$.

Regards, Hugh