On Tue, 28 Oct 2014, W Hugh Woodin wrote:
My point is that the non-rigidity of HOD is a natural extrapolation of ZFC large cardinals into a new realm of strength. I only reject it now because of the Ultimate L Conjecture and its implication of the HOD Conjecture. It would be interesting to have an independent line which argues for the non-rigidity of HOD.
This is the only reason I ask.
Please don’t confuse two things: I conjectured the rigidity of the Stable Core for purely mathematical reasons. I don’t see it as part of the HP. Indeed, I don’t see a clear argument that the nonrigidity of inner models follows from some form of maximality.
But I still don’t have an answer to this question:
What theory of truth do you have? I.e. what do you consider evidence for the truth of set-theoretic statements?
But I did answer your question by stating how I see things developing, what my conception of V would be, and the tests that need to be passed. You were not happy with the answer. I guess I have nothing else to add at this point since I am focused on a rather specific scenario.
That doesn’t answer the question: If you assert that we will know the truth value of CH, how do you account for the fact that we have many different forms of set-theoretic practice? Do you really think that one form (Ultimate L perhaps) will “have virtues that will swamp all the others”, as Pen suggested?
PS: With regard to your mail starting with “PS:”: I have worked with people in model theory. When we get an idea we sometimes say “but that would give an easy solution to Vaught’s conjecture” so we start to look for (and find) a mistake. That’s all I meant by my comments: What I was doing would have given a “not difficult solution to the HOD conjecture”; so on this basis I should have doubted the argument and indeed I found a bug.