Dear Hugh,

So during the course of this entire email thread, HP seems to have evolved from declaring as you did in your first summary message to Sol of August 12:

I conjecture that CH is false as a consequence of my Strong Inner Model Hypothesis (i.e. Levy absoluteness with “cardinal-absolute parameters” for cardinal-preserving extensions) or one of its variants which is compatible with large cardinals existence.

(Aside: This is an extraordinary claim which you amplify in writing on Oct 13: “Further, as I said, I think there is a good chance of arriving at an optimal HP criterion”)

I stand by both of these claims, there has been no change.

to the point that it is reduced to a collection of extremely difficult problems in the structure theory of countable transitive models of ZFC; i.e. the only principle left standing seems to be (but maybe that has also fallen or been reduced to a rough uncut principle), and this has been formulated to make any near term analysis impossible. You seem to feel this is a positive attribute of HP.

Now you have lost me. You quoted a crucial phrase in my message:

Although the process exhibits genuine progress, converging closer and closer to the kind of maximality principle Max is looking for, the challenge of providing consistent forms of maximality that will answer Max’s fair questions is enormously difficult.

Getting as far as the is genuine progress. I provided a direction for further progress in my Maximality Protocol. Be patient, Hugh! Working out the mathematical features of Maximality will take time, and the programme has only just begun.

I would argue instead that this is simply a sort of coming of age for Set Theory; i.e. we can now pose simple questions about models of Set Theory which seem completely out of reach.

I am sorry that you use the phrase “which seem completely out of reach”. I would prefer to say “which demand the development of radically new techniques”. I am optimistic.

As Pen has implied, it is good to have different programmes in set theory, whether they be motivated by sophisticated issues emanating from large cardinal theory and descriptive set theory, like your Ultimate L programme, or by an “intrinsic heuristic” like the Maximality of V. Your programme is also extremely hard, but I would not fault it for that reason and I hope that it works out as hoped.

I look forward to seeing a reasoned account of HP with many of the issues that have been raised addressed. I hope that such an account has an initial list of axioms whose selection is well motivated based on the then current state of HP, i.e. why instead of Strong- etc.

Please re-read the Maximality Protocol: Height Maximality, Cardinal Maximality, Width Maximality, in that order. I gave precise suggestions for Height and Cardinal Maximality; Width Maximality is obviously trickier but at least I made a tentative proposal with the . The problem with Strong- was given toward the end of my Max story (Max isn’t happy with being captured by a single real).

I’m not sure what you are asking for: To sort out this Maximality business we (and I can’t do it alone) need to explore the different possibilities and see what makes the most sense. Do you really expect me to give a precise definition of “what makes the most sense” in advance? You ask too much. It will take time, we need to map out the possibilities first.

Best of all of course would be at least one specific conjecture strongly motivated by HP ideas, whose proof confirms HP, and whose refutation is a serious setback. But I acknowledge this may not be a reasonable expectation at this preliminary stage.

I am happy to read this last sentence. What I have been trying (without much success) to say is that there is no “back to square one” conjecture here.

I have yet to see anything that suggests that HP is anything more than part of the structure theory of countable transitive models (I like Harvey’s idea; rename HP as CTMP).

??? This is very harsh. Please re-read my 13.October e-mail to Pen (the first, long one, not the second, short one). It’s about the move from Maximality features of V (as explained to Geoffrey on 24.September in terms of lengthenings, without thickenings) to the Hyperuniverse.

Best,

Sy