Re: Paper and slides on indefiniteness of CH

Dear Sy,

I’d like to take a crack at

Can you or John tell me what evidence you have that the iterability problem will be solved positively to enable the construction of an inner model for a supercompact in the foreseeable future?

There’s a lot of evidence that there are canonical inner models with supercompacts, and that their canonicity derives from a form of iterability. The evidence consists mostly of approximations to such a result that have already been proved. The large cardinals/determinacy/inner models theory has a great deal of internal unity, and people keep extending it to reach higher consistency strengths. It’s not going to break down between many Woodin cardinals and supercompacts. “Subtle is the lord, but he is not malicious.”

Whether and when humanity will prove that there are canonical, iterable inner models with supercompacts is another question. I would guess that there will always be at least a few people interested in the higher reaches of the consistency strength hierarchy, and so eventually some (perhaps AI-enhanced) human will prove it. I would also guess that it won’t happen in the next few years. I think we see a road, one that involves developing the theory of HOD in models of AD.

Best,
John

Re: Paper and slides on indefiniteness of CH

If certain technical conjectures are proved, then “V=ultimate L” is an attractive axiom.

The paper “Gödel’s program“, available on my webpage, gives an argument for that. I would guess the technical conjectures are true; in any case, large cardinals should decide them.

“V = Ultimate L” implies CH.

I don’t think we should care how hard it is to understand the statement of  “V = Ultimate L”. But in fact, one semester of graduate set theory is all you need to understand it. The conjectures are just that it implies GCH, and is consistent with the existence of supercompacts. One graduate semester is enough to understand the conjectures.

Best,
John