Re: Paper and slides on indefiniteness of CH

Dear Sy,

I am a bit bewildered by your criticism. The scenario I described rests largely on one single conjecture, the Ultimate L Conjecture. While the motivations for this conjecture are complicated, the statement uses just basic notions in modern set theory: HOD, some specific large cardinal notions, universally Baire sets, and L relativized to sets $A \subset \mathbb R$.

If this conjecture is refuted then the scenario collapses. If this conjecture is proved then a substantial portion of the scenario will be verified.

As for your “conjecture” and I assume you make this conjecture only for the purposes of your criticism:

How’s this for a conjecture: It is consistent to have a supercompact but none in an inner model of HOD. Do you have more evidence against that conjecture than for its negation?

This conjecture is not relevant. There is no reason I should even have an opinion on this.

Technical aside for those who are interested:

Sy’s “conjecture” is focusing on the “resource problem” within inner model constructions. If there is an inner model construction for a $\Phi$-cardinal, the resource problem is whether that construction be implemented if one only assumes the existence of a $\Phi$-cardinal. There are examples where the inner model constructions use a bit more and the reduction in what is required is subtle.

If one has “more” than a $\Phi$-cardinal then there is a trivial solution to the problem of finding an inner model of HOD with a $\Phi$-cardinal. For example if there is a supercompact cardinal with a measurable cardinal above then there is an inner model for a supercompact cardinal within HOD. This why simply asking whether there is an inner model of a $\Phi$ cardinal contained within HOD is not so interesting — just assume more in V and it has a trivial solution.

For Sy’s “conjecture” a positive answer, while certainly surprising, is not obviously in any way (to me at least) an anti-inner model theorem (the Ultimate L Conjecture could still be true), and a negative answer does not require solving the inner model problem for one supercompact (since that inner model theory only has to work in the situation that there is a supercompact cardinal and no inner model with a supercompact cardinal and a measurable cardinal above). This why it is not relevant to the Ultimate L Conjecture.

Finally for many axiomatic schemes of large cardinal beyond supercompact (so $\Phi$ is now a scheme) the inner model of HOD problem likely has a solution (without assuming anything more in V) which again does not require the fine-structure models — this is the point of the HOD Conjecture and the solution is HOD itself. A precursor to this is the theorem (outright and not requiring any conjectures) that if there is a supercompact cardinal in V then there is a measurable cardinal in HOD.

Regards,
Hugh