Dear Hugh,

The axiom V = Ultimate L implies V = HOD

So V = Ultimate L cannot be “true” because it violates the maximality of the universe of sets. Recall Sol’s comment about “sharpenings” of the set concept that violate what the set concept is supposed to be about. Maximality implies that there are sets (even reals) which are not ordinal-definable.

Best,

Sy

PS: This is of course not to say that V = Ultimate L is mathematically uninteresting or cannot play a role in the formulation of some future “true” axiom of set theory.

PPS: Since the Reinhardt fiasco I think it would be best to refer to statements that are not known to be consistent (relative to LCs) as “hypotheses” and not as “axioms”, especially in the context of a discussion over truth in set theory.