You were asked to say more about your “conception of V in which PD holds”. What you wrote said little about that and instead went into the stratosphere, with in my view an unwarranted fantasy based on very thin evidence. I too could present such a fantasy about a vesion of my Stable Core with GCH and even fine structure which is rigid and witnesses large cardinals, over which weak covering holds in the presence of large cardinals and over which V is generic. I could go on about the internal structure of this Refined Stable Core, blah, blah, blah. The point is that I would never advertise such a fantasy as I can’t back it up with enough solid evidence.
A key question is simply ignored when you write:
Continuing, one is led to degrees of supercompactness (the details here are now based on quite a number of conjectures, but let’s ignore that).
Frankly, this is quite ridiculous. The iterability problem for developing an inner model with a supercompact has been open for many years. It is the main open problem of inner model theory. So the real question for the first line of your story is: what evidence and what understanding do we have of this problem? I already tried to make the point that in inner model theory there is a history of things not going as predicted, so I do not find it credible to build a picture based on the blanket assumption that this will go the way we now expect it to go. 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 is not a question of a possible inconsistency in large cardinal theory! It is a question of whether our understanding of inner model theory at the level of Woodin cardinals has anything to do with inner model theory for a supercompact. 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?
Now if we don’t have a solution to this problem then your comment
One is quickly led to the theorem that the existence of the generalization of L to the level of exactly one supercompact cardinal is where the expansion driven by the horizontal maximality principles stops.
is vacuous, as there might not be a “generalization of L to the level of exactly one supercompact cardinal” in the first place! It is hard to appreciate an implication when the hypothesis is so debatable.
On the positive side, I do agree that your work on as an analogue of is impressive and very suggestive, and should be part of the final picture. But speculation about Ultimate L seems premature to me, once again stacked on top of a pile of unproved conjectures.
What are we to make of all this? You give the feeling that you are appearing at the finish line without running the race. We would all love to see “the final picture” but the rest of us understand that we have to be realistic about what has been established and what is just conjecture. There is solid stuff in your picture: core model theory, universally Baire sets and . But the gaps in the rest are so huge that it is not helpful to be presented with such a picture. It gives the false impression that you have figured everything out, while in fact there is a lot not yet understood even near the beginning of your story. Many of us pay close attention to what you say as you have done such great work; could you please stop rushing ahead and stay closer to the frontiers of what we actually know?