Quick answer to the question directed to me below. —Hugh
On Aug 20, 2014, at 2:49 AM, Sy David Friedman
As I understand it (please correct me) your [Solomon Feferman's] valid point is that by for example taking “set” to mean “constructible set” we have violated the intrinsic feature of “maximality”, a feature which the concept of set is meant to exhibit. (Aside: I can then well imagine that on similar grounds you would hesitate to accept an axiom called V = Ultimate-L! But perhaps Hugh will clarify that this need not even imply V = HOD, so it should not be regarded as an anti-maximality statement.)
The axiom V = Ultimate L implies V = HOD and that V is not a set-generic extension of any inner model.