I realize that I am not sure how you use the word “conjecture”. Here are two sample readings of this word in your “Hod Conjecture”. (I realize this does not exhaust the possibilities.)
A) It is provable in ZFC that if $kappa$ is a huge cardinal, then the HOD conjecture holds in .
B) It is simply true that the HOD conjecture holds. No implication concerning provability is intended.
(I realize position B is incomprehensible (and/or absurd) to the Friedman brothers.) My position that CH is false (and that is weakly inaccessible) is much like this suggested alternative B.
In the last conjecture of your letter what does ” rigid” mean?
I’m repeating my query for the whole mailing list:
I confess I’m not sure precisely what you are claiming to prove. Let
us consider the case when we do not assume “inner model theory for a measurable Woodin”. Your theorem would say precisely: V is a class generic extension of a model of GCH if …?
Can someone (Sy? Hugh?) explain the precise meaning of the phrase “sharp generated model”?
— Bob Solovay
Dear Sy & Hugh,
I also would like to follow your discussion with Hugh on the topic he just raised.
What is the precise definition of “maximality”. Is it evident from
that definition, that maximality implies there are reals not in HOD?
If not, can you give a cite as to where this is proved?
This is in response to the following quote of Harvey:
Incidentally, do you agree with me that CH research is not a relatively promising area of f.o.m. research? I tend to believe that people think CH research is a promising area of f.o.m. research if and only if they subscribe to “CH has a determinate truth value”.
I am a convinced platonist and fully subscribe to the proposition that CH has a definite truth value.
I am quite sceptical about the prospects of determining CH by any approach currently on the horizon. In particular, I doubt that anything growing out of the work of Sy Friedman or any of the work of Woodin, past or present, that I know about will lead to any determination that I find in the least convincing.
I have enormous respect for Woodin’s mathematical achievements, even though I do not think they have any prospect of leading to a solution of the continuum problem.