Hugh has talked about how things might go if various conjectures fall in a particular direction: there’d then be a principle ‘V=Ultimate L’ that would at least deserve serious consideration. That’s far short of ‘endorsement’, of course. Can you point to an HP-generated principle that has that sort of status?
and you responded:
I can come close. It would be the . But it’s not really analogous to Ultimate L for several reasons:
- I hesitate to “conjecture” that the is consistent.
- The in its crude, uncut form might not be “right”. Recall that my view is that only after a lengthy exploratory process of analysis and unification of different maximality criteria can one understand the Optimal maximality criterion. I can’t say with confidence that the original uncut form of the will be part of that Optimal criterion; it may have to first be unified with other criteria.
- The , unlike Ultimate L, is absolutely not a “back to square one” principle, as Hugh put it. Even if it is inconsistent, the HP will continue its exploration of maximality criteria and in fact, understanding the failure of the will be a huge boost to the programme, as it will provide extremely valuable knowledge about how maximality criteria work mathematically.
This is a technical criticism. In brief I am claiming that based on the methodology of HP you have described (though perhaps now rejected), is not the correct synthesis of IMH and reflection. Moreover the correct synthesis, which is significantly stronger, resurrects all the issues associated with IMH regarding “smallness”.
Consider the following extreme version of :
Suppose is a ctm and . Then witnesses extreme- if:
- There is a thickening of M, satisfying ZFC, in which M is a -generated inner model.
- M witnesses in all thickenings of M, satisfying ZFC, in which M is a -generated inner model.
One advantage to extreme- is that the formulation does not need to refer to sharps in the hyperuniverse (and so there is a natural variation which can be formulated just using the -logic of M). This also implies that the property that M witnesses extreme- is as opposed to which is not even in general .
Given the motivations you have cited for etc., it seems clear that extreme- is the correct result of synthesizing IMH with reflection unless it is inconsistent.
Thm: Assume every real has a sharp and that some countable ordinal is a Woodin cardinal in a definable inner model. Then there is a ctm which witnesses that extreme- holds.
However unlike , extreme- is not consistent with all large cardinals.
Thm: If M satisfies extreme- then there is a real in M such that in M, does not exist.
This seems to be a bit of an issue for the motivation of and . How will you deal with this?