On Wed, 29 Oct 2014, W Hugh Woodin wrote:
My question to Sy was implicitly: Why does he not, based on maximality, reject HOD Conjecture since disregarding the evidence from the Inner Model Program, the most natural speculation is that the HOD Conjecture is false.
1. The HP is concerned with maximality but does not aim to make “conjectures”; its aim is to throw out maximality criteria and analyse them, converging towards an optimal criterion, that is all. A natural maximality criterion is that V is “far from ” and indeed my work with Cummings and Golshani shows that this is consistent. In fact, I would guess that an even stronger statement that V is “very far from ” is consistent, namely that all regular cardinals are inaccessible in and more. What you call “the Conjecture” (why does it get this special name? There are many other conjectures one could make about !) presumes an extendible cardinal; what is that doing there? I have no idea how to get extendible cardinals from maximality.
2. Sometimes I make conjectures, for example the rigidity of the Stable Core. But this has nothing to do with the HP as I don’t see what non-rigidity of inner models has to do with maximality. I don’t have reason to believe in the rigidity of (with no predicate) and I don’t see what such a statement has to do with maximality.