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

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>