Claudio Ternullo wrote

The HP is about the collection of all c.t.m. of ZFC (aka the “hyperuniverse” [H]). A “preferred” member of H is one of these c.t.m. satisfying some H-axiom (e.g., IMH).

Your coauthor has not explained why HP doesn’t carry the name CTMP = countable transitive model program. That is my suggestion and has been supported by Hugh. Why not?

What does the choice of a countable transitive model have to do with “(intrinsic) maximality in set theory”?

At a fundamental level, what does “(intrinsic) maximality in set theory” mean in the first place?

Which axioms of ZFC are motivated or associated with “(intrinsic) maximality in set theory”? And why? Which ones aren’t and why?

What is your preferred precise formulation of IMH? E.g., is it in terms of countable models?

What do you make of the fact that the IMH is inconsistent with even an inaccessible (if I remember correctly)? If this means that IMH needs to be redesigned, how does this reflect on whether CTMP = HP is really being properly motivated by “(intrinsic) maximality in set theory”?

What is the simplest essence of the ideas surrounding “fixing or redesigning IMH”? Please, in generally understandable terms here, so that people can get to the essence of the matter, and not have it clouded by technicalities.

Overall, it would be particularly useful to avoid quoting complicated technicalities and adhere to generally understandable considerations. After all, CTMP = HP is being offered as some sort of truly foundational program. Legitimate foundational programs lend themselves to generally understandable explanations with overwhelmingly attractive features.

I have not been able to engage your coauthor in this way, so perhaps this is going to fall on you. Sorry about that (smile).

Harvey