Dear Hugh,

In light of your HOD Dichotomy I interpreted the HOD conjecture to say that if there is an extendible delta then HOD correctly computes successors of singulars above correctly. All I meant was that if you drop the extendible then this conclusion need not hold. I am guessing (I really don’t know) that if there is an extendible then this conclusion does hold (and hence the HOD Conjecture is true).

Unless you can derive extendibles from some form of maximality the consequence I would draw from the HOD conjecture would be that maximality violates the existence of extendible cardinals.

Best, Sy