On Nov 5, 2014, at 7:40 AM, Sy David Friedman wrote:
1. Your formulation of is almost correct:
M witnesses if
1) M is weakly #-generated.
2) If holds in an outer model of M which is weakly
#-generated then holds in an inner model of M.
But as we have to work with theories, 2) has to be: If for each countable , holds in an outer model of M which is generated by an -iterable presharp then holds in an inner model of M.
Let’s call this New-.
Are you sure this is consistent?
Assume coding works in the weakly #-generated context:
Coding Assumption: if M is weakly #-generated then M can be coded by a real in an outer model which is weakly #-generated.
Theorem. Assume PD. Then there is a real such that for all ctm M, if x is in M then M does not satisfy New-.
(So in any case, one cannot get consistency by the determinacy proof).
2. Could you explain a bit more why V = Ultimate-L is attractive?
Shelah has the informal notion of a semi-complete axiom.
V = L is a semi-complete axiom as is in the context of etc.
A natural question is whether there is a semi-complete axiom which is consistent with all large cardinals. No example is known.
If the Ultimate L Conjecture is true (provable) then V = Ultimate L is arguably such an axiom and further it is such an axiom which implies V = HOD (being “semi-complete” seems much stronger in the context of V = HOD).
Of course this is not a basis in any way for arguing V = Ultimate L. But is certainly makes it an interesting axiom whose rejection must be based on something equally interesting.
You said: “For me, the “validation” of V = Ultimate L will have to come from the insights V = Ultimate L gives for the hierarchy of large cardinals beyond supercompact.”
But why would those insights disappear if V is, for example, some rich generic extension of Ultimate L? If Jack had proved that does not exist I would not favour V = L but rather V = some rich outer model of L.
I think if our evolving understanding of the large cardinal hierarchy rests primarily on the context of V = Ultimate L then very likely the rich generic extensions are not playing much of a role in understanding the large cardinal hierarchy.
This for me would build the case for V = Ultimate L and against these rich extensions. It would then take something quite significant in the theory of the rich extensions to undermine that.
But such speculations seem very premature. We do not even know if the HOD Conjecture is true. If the HOD Conjecture is not true then the entire Ultimate L scenario fails.
3. I told Pen that finding a GCH inner model over which V is generic is a leading open question in set theory. But you gave an argument suggesting that this has to be strengthened. Recently I gave a talk about HOD where I discussed the following four properties of an inner model M:
Genericity: V is a generic extension of M.
Weak Covering: For a proper class of cardinals , .
Rigidity: There is no nontrivial elementary embedding from M to M.
Large Cardinal Witnessing: Any large cardinal property witnessed in V is witnessed in M.
(When 0# does not exist, all of these hold for M = L except for Genericity: V need not be class-generic over L. As you know, there has been a lot of work on the case M = HOD.)
Now I’d like to offer Pen a new “leading open question”. (Of course I could offer the PCF Conjecture, but I would prefer to offer something closer to the discussion we have been having.) It would be great if you and I could agree on one. How about this: Is there an inner model M satisfying GCH together with the above four properties?
Why not just go with the HOD Conjecture? Or the Ultimate L Conjecture?
There is is another intriguing problem which has been suggested by this thread.
Suppose is not correctly computed by HOD for any infinite cardinal .Must weak square hold at some singular strong limit cardinal?
This looks like a great problem to me and it seems clearly to be a new problem.