Ok we keep going.
On Oct 31, 2014, at 3:30 AM, Sy David Friedman wrote:
With co-authors I established the consistency of the following Maximality Criterion. For each infinite cardinal , of HOD is less than .
Both Hugh and I feel that this Criterion violates the existence of certain large cardinals. If that is confirmed, then I will (tentatively) conclude that Maximality contradicts the existence of large cardinals.
It seems that you believe the HOD Conjecture (i.e. that the HOD Hypothesis is a theorem of ZFC). But then HOD is close to V in a rather strong sense (just not in the sense of computing many successor cardinals correctly). This arguably undermines the whole foundation for your maximality principle (Maximality Criterion stated above). I guess you could respond that you only think that the HOD Hypothesis is a theorem of ZFC + extendible and not necessarily from just ZFC.
If the HOD Hypothesis is false in V and there is an extendible cardinal, then in some sense, V is as far as possible (modulo trivialities) from HOD. So in this situation the maximality principle you propose holds in the strongest possible form. This would actually seem to confirm extendible cardinals for you. Their presence transforms the failure of the HOD Hypothesis into an extreme failure of the closeness of V to HOD, optimizing your maximality principle. So in the synthesis of maximality, in the sense of the failure of the HOD Hypothesis, with large cardinals, in the sense of the existence of extendible cardinals, one gets the optimal version of your maximality principle.
The only obstruction is the HOD Conjecture. The only evidence I have for the HOD Conjecture is the Ultimate L scenario. What evidence do you have that compels you not to make what would seem to be strongly motivated conjecture for you (that ZFC + extendible does not prove the HOD Hypothesis)?
I find your position rather mysterious. It is starting to look like your main motivation is simply to deny large cardinals.