Dear Bob,

What is the precise definition of “maximality”. Is it evident from that definition, that maximality implies there are reals not in HOD? If not, can you give a cite as to where this is proved?

I do not know of a precise definition of “maximality”. Rather I regard “maximality” as an “intrinsic feature” of the universe of sets via what Pen has referred to as “the usual kind of conceptualism: there is a shared concept of the set-theoretic universe (something like the iterative conception); it’s standardly characterized as including ‘maximality’, both in ‘width’ (Sol’s ‘arbitrary subset’) and in the ‘height’ (at least small LCs).” [Pen: I dropped the bit about "reflection" as I wasn't sure what you meant; but I don't think that its omission will affect this discussion.]

“Maximality” is indeed formulated mathematically in a number of different ways in the HP, but I don’t know if there is an ultimate mathematical formulation which fully captures it and therefore cannot claim that there will be a precise definition of this intrinsic feature.

Nevertheless I do regard the existence of reals not in HOD to be derivable from “maximality” for the following reason, which I expect to be shared by others who share the maximal iterative conception: Part of this conception is that the powerset of omega consists of arbitrary subsets of omega. This is violated by V = L, which insists that the only subsets of omega are those which are predicatively definable relative to the ordinals, and also by V = HOD, which insists that the only subsets of omega are those which are definable relative to ordinals.

Returning to V = Ultimate L: I would expect that anyone claiming this to be true would also claim V = L to be true had Jack succeeded in showing that cannot exist. But V = L is in my view clearly wrong (irregardless of whether can exist), as well-expressed by Gödel:

“From an axiom in some sense opposite to [V=L], the negation of Cantor’s conjecture could perhaps be derived. I am thinking of an axiom which … would state some maximum property of the system of all sets, whereas [V=L] states a minimum property. Note that only a maximum property would seem to harmonize with the concept of set …”

This argument seems to refute V = Ultimate L.

Best,
Sy