Re: Paper and slides on indefiniteness of CH

Dear Pen and Geoffrey,

I don’t really understand what you are after with this multiverse vs. single-universe (augmented with height potentialism) discussion. As I see it:

If we are talking about ST in terms of its role as a foundation for or subfield of mathematics (Types 1 and 2) then we needn’t trouble ourvselves with this discussion of universes for ST and can hang our hats on what axioms of set theory are advantageous for the development of set theory and mathematics, as was done with AC and the Axiom of Infinity, for example. With Thin Realsim, who cares about our conception of the universe of sets if it’s fixed by set-theoretic and mathematical practice?

But if we are interested in the maximality of the set-concept, more specifically the maximality of the universe of sets in height and width, then indeed this multiverse vs. single-universe discussion kicks in. With regard to the HP I see two different ways to handle this, one with an extreme multiverse view (resulting from radical potentialism/Skolem worshipping) and the other from a single-universe view (stroked a bit by height potentialism):

1. Extreme Multiverse view. We have no single V but a wealth of different possible V’s. This wealth is so wealthy that any particular V can be thickened or lengthened (no quotes!) and shockingly, made countable by going to a larger V. So there is no absolute notion of cardinality, only distinct notions of cardinality within each of the possible V’s. OK, now when talking about maximality of a possible V we simply mean that lengthening or thickening  V will not reveal new properties that we couldn’t already see in V. (Note: One could go further and even look at blowups of V which see V as countable, but mathematically this doesn’t seem to add much.) Then when we talk of a first-order statement like not-CH being a consequence of maximality we mean that it holds in all of the possible V’s which are maximal.

Frankly speaking, the Extreme Multiverse View is my own personal view of things and gives the cleanest and clearest approach to studying maximality. That’s because it allows the freedom to make all of the moves that you want to make in comparing a possible V to other possible V’s.

Note that the multiverse described above looks exactly like the Hyperuniverse of a model of ZFC. In other words, the Extreme Multiverse View says that whether or not we realise it, we live in a Hyperuniverse, and we are kidding ourselves when we claim that we have *truly* uncountable sets: Some bigger universe looks down at us and laughs when she hears this, knowing perfectly well that we are just playing around with countable stuff.

Now I do know that some (all?) of you don’t dig this way of doing things (do you feel insecure not knowing that there is a “true uncountable”?). I don’t see your point but I’m an accomodating guy, and so:

2. Single-Universe view (peppered by height potentialism). OK, now we want to understand the maximality of “the real V” (whatever the hell that is; OK, I’ll try to put a hold on my sarcasm). How are we going to do this, since “the real V” has all of the sets so it is totally obvious that it is “maximal”, as there are no alternatives! Well, Pen and Geoffrey have thrown us a bone and told us that it’s OK to think about lengthening V (height potentialism), quoting Hilary’s theological claim (just teasing, Hilary). Good, then we can make all sorts of moves with maximality in height by saying things like: Well surely we can lengthen V to a model of ZFC (after all, all rank initial segments of V can be so lengthened, why not V itself?), but if we can do that then surely we can lengthen that lengthening again to a 2nd model of ZFC, again and again, and then apply reflection to say that first-order statements about V which hold in such lengthenings also hold about some V_alpha inside a coresponding lengthening of $V_\alpha$, … If you push hard enough you get a very strong height maximality which corresponds to Friedman-Honzik #-generation.

But what do we do about width maximality? Well, we had this discussion already, it involves V-logic. To reduce it to a sound-bite: Width maximality can be expressed in terms of “thickenings” (we need quotes!), a concept we can handle without thickening V but by just lengthening V a bit. All this stuff about IMH, $\textsf{SIMH}^\#$, unreachability, width reflection, strong absoluteness, Cardinal Maximality, … can be handled in this way.

Fine. Now the next move I made was to say that actually you don’t need to tie your hands by staying so close to “the real V”, because the mathematics you are doing would work equally well if you replace “the real V” by a countable transitive model of ZFC (“reduction to the Hyperuniverse”). You don’t need to trash “the real V” for this, I was just saying that the mathematical results would be the same if you were to do that, and then we get ourselves into a context where we can make all the moves we want to make in the comparison of universes with no awkwardness, as the universes we need for the comparisons are genuinely available.

In other words, the mathematical analyses of Maximality via the Extreme Multiverse and Single-Universe views are the same!So pick your ontology or lack thereof as you like it, it’s not going to change things mathematically!

Now I can be even more accomodating. Some of you doubters out there may buy the way I propose to treat maximality via a Single-Universe view (via lengthenings and “thickenings”) but hide your money when it comes to the “reduction to the Hyperuniverse” (due to some weird dislike of countable transitive models of ZFC). OK, then I would say the following, something I should have said much earlier: Fine, forget about the reduction to countable transitive models, just stay with the (awkward) way of analysing maximality that I describe above (via lengthenings and “thickenings”) without leaving “the real V”! You don’t need to move the discussion to countable transitive models anyway, it was just what I considered to be a convenience of great clarification-power, nothing more!

Is everybody happy now? You can have your “real V” and you don’t need to talk about countable transitive models of ZFC. What remains is nevertheless a powerful way to discuss and extract consequences from the maximality of V in height and width. Of course you will make me sad if you block the move to ctm’s, because then you strip the programme of the name “Hyperuniverse Programme” and it becomes the “Maximality Programme” or something like that. I guess I’ll get over that disappointment in time, as it’s only a change of name, not a change of approach or content in the programme. But I’ll still be disappointed, please feel sorry for me.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

On Sun, Oct 19, 2014 at 5:06 AM, Radek Honzik wrote: Dear all,

I will attempt to answer briefly the questions posted by Harvey. My view on HP is different from Sy’s, but I see HP as a legitimate foundational program.

The most accomplished contributors to this list seem to be doubtful.

Thanks for your replying. Your reply doesn’t contain a mish mash of hidden assumptions, not so hidden assumptions, question begging, ignoring criticisms, missed opportunities for joining issues, evasions, crude bragging about important busy activities, undefined terms, total lack of respect for the audience who does not keep up with specialist jargon, and a long list of other sins that make this thread an example of how not to do or even talk about foundations and philosophy.

I appreciate that you have for the most part avoided these sins.

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

Let me write IMST instead of “(intrinsic) maximality in set theory” for the sake of brevity.

I doubt IMST can mean more than “viewing sets as big as possible, without the use of considerations based on practice of set theory as the main incentive”.

I think you mean “viewing the set theoretic universe as inclusive as possible”?

Your sentence has an indirect construction that really is not necessary and slows down the reader.

“Intrinsic” is thus temporarily reduced to “non-extrinsic”; in view of the heavy philosophical discussions around this notion, I prefer to give it this more restrictive meaning. Note that “extrinsic”, unlike “intrinsic”, has a well-defined inter-subjective meaning. This leaves us with the word “big”; I guess that this is the primitive term, which cannot be defined by anything more simple — at least on the level of general discussion.

Again, I don’t think you want to use “big” here – I am suggesting something very similar – the set theoretic universe is as inclusive as possible. I regard this as only an informal starting point and the job is to systematically explore analyses of it.

Admittedly, this definition is far from informative. For me, HP is a way of explicating this definition in a mathematical framework. Making its meaning more precise, and by the same token, less general. A discussion should be if other approaches — which set out to get real mathematical results — retain more of the general meaning of the term IMST. No approach can retain all the meaning of ISMT because it is by definition vague and subjective; thus HP should not be expected to do that.

But there is the real possibility of saying something generally understandable, surprising, and robust. I haven’t seen anything like that in CTMP (aka HP).

1] Why doesn’t HP carry the obvious name CTMP = countable transitive model program.

Because the program was formulated by Sy with the aim of having wider application than the study of ctm’s.

Since the name “hyperuniverse” specifically refers to the countable transitive models of ZFC, period, it amount to nothing more than a propogandistic slogan designed to lure the listener into thinking that there is something profound going on having to do with the foundations of set theory. But since nothing yet has come out of this special study of ctms for foundations of set theory, even propogandistic slogans about ctms are premature.

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

Countable models are a way of explicating IMST. It is a technical convenience which allows us to use model-theoretic techniques, not available for higher cardinalities.

What you have here is an unanalyzed idea of “intrinsic maximality in set theory”, and before that is analyzed to any depth, you have the blanket assumption that countable transitive models are going to be the way you can formulate what is going to become the analysis of “intrrinsic maximality in set theory”. The real agenda is a creative or novel analysis of “intrinsic maximality in set theory”, and BEFORE that is accomplished any “proof” that ctms will do by some sort of Lowenheim Skolem argument is bogus. Of course, you can set up some idiosyncratic framework, pretend that you going to make this your analysis of “intrinsic maximality in set theory”, and then cite Lowenheim Skolem. But that is bogus. After you make some creative or novel analysis, and work through the problematic issues (inconsistencies and other non robustness arising out of parameters, sets of sentences, etcetera), and have a framework that is credible and well argued, you can cite the Lowenheim Skolem theorem – if it really does apply correctly – to say that any claims of a certain form are equivalent to claims of the form with ctms. That would make some sense, but I still would not advise it since the proper framework, if there is any, is not going to be based on ctms. Ctms would only be a convenience.

This is the serious conceptual error being made in endless emails by Sy trying to justify the use of ctms. An unjustified framework for treating “intrinsic maximality in set theory” is alluded to, and to the extent that it is precise, one quotes Skolem Lowenheim to argue that one can wlog work with ctms. This is a very serious question begging sin. The issue at hand is first to have a novel or creative and well argued and thought out framework for treating “intrinsic maximality in set theory”. AFTER THAT, one can talk about the convenience – but NOT the fundamental nature of – using ctms.

This reversal of proper order of ideas – putting the cart before the horse – is a major error in work in foundations and philosophy.

IN ADDITION, on the mathematical level, I quote from Hugh. This indicates that even in frameworks proposed beyond IMH, there is no Lowenheim Skolem argument, and one is compelled to make the move that ctms are fundamental, rather than just a convenience. Here is the exchange:

Sy wrote:

More details: Take the IMH as an example. It is expressible in V-logic. And V-logic is first-order over the least admissible (Goedel-) lengthening of V (i.e. we go far enough in the L-hierarchy built over V until we get a model > of KP). We apply LS to this admissible lengthening, that’s all.

Hugh wrote

This is of course fine for IMH. But this does not work at all for $\textsf{SIMH}^\#$. One really seems to need the hyperuniverse for that. Details: $\textsf{SIMH}^\#$ is not in general a first order property of $M$ in $L(M)$ or even in $L(N,U)$ where $(N,U)$ witnesses that $M$ is #-generated.

MY COMMENT: So we may be already seeing that in some of these approaches being offered, one must buy into the fundamental appropriateness of ctms in the philosophy, and not just an automatic freebie from the Lowenheim Skolem theorem. FURTHERMORE, IMH, where reduction to ctm makes sense through Skolem Lowenheim, has not even been seriously analyzed as an “intrinsic maximality in set theory” by serious foundational and philosophical standards. There is a large array of issues, including inconsistencies and non robustness involving parameters and sets of sentences, and so forth.

Aside: I do not quite understand why the discussion rests so heavily on this issue: everyone seems to accept it readily when we talk about forcing (I know it can be eleminated in forcing, but the intuition — see Cohen’s book — comes from countable models). Would it make a difference if the models had cardinality omega_1, or omega and omega_1, or should they be proper classes etc? Larger cardinalities would introduce technical problems which are inessential for the aims of HP.

The crucial issue can be raised as follows. Do we or do we not want to take the structure of ctms as somehow reflecting on the structure of the actual set theoretic universe?

I am interested in seeing what happens under both answers. What is totally unacceptable is to make the hidden assumption of “yes we do” while pretending “no we are not because of the Löwenheim Skolem theorem”. That is just bad foundations and philosophy.

I am going to explore what happens when we UNAPOLOGETICALLY say “we do”. No bogus Löwenheim Skolem.

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

IMST by historical consensus includes at this moment ZFC. “Historical consensus” for me means that many people decided that the vague meaning of IMST extends to ZFC. I do think that this depends on time (take the example of AC). HP is a way to raise some new first-order sentences as candidates for this extension.

Then what is all this talk on the traffic doubting whether AxC is supported by “intrinsic maximality of the set theoretic universe?”

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

Yes.

5] 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”?

I view the process of obtaining results in HP like an experiment in explicating the vague meaning of IMST. It is to be expected that some of the results will be surprising, and will require interpretation.

This is a good attitude. However, there is still not much that has come out, and it is still unclear whether this will change. So declaring it a “program” without having the right kind of ideas in hand, and coining a jargon name, is way premature.

6] 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.

It is a creative process: explicate IMST by principle $P_1$ — after some mathematical work, it outputs varphi (such as $P_$1 = IMH, $\varphi$ = no inaccessible). Then try $P_2$, etc ($P_2$ can be a “redesigned”, or “modified” version of $P_1$). Of course, one hopes that his/her understanding of set theory will be helpful in identifying P’s which have potential to output nice (good, deep) mathematics. It is essential that the principles $P$‘s should be as practice-independent as possible (= intrinsic, in my reading); that is what makes the program foundational (again, in my more narrow sense).

Taking into account what you are saying, and the difficulties that Hugh has been pointing out about the post IMH proposals, this does not have enough of the features of a legitimate foundational program at this stage. It has the features of a legitimate exploratory project without a flowery name and pretentious philosophy. We don’t know if it is going to develop into a legitimate foundational program, which would justify flowery names and pretentious philosophy.

Harvey

Re: Paper and slides on indefiniteness of CH

Sy wrote:

In other words, we can discuss lengthenings and shortenings of V without declaring ourselves to be multiversers. Similarly we can discuss “thickenings” in quotes. No multiverse yet. But then via a Loewenheim-Skolem argument we realise that it suffices to work with a countable little-V, where it is natural and mathematically extremely useful to regard lengthenings and “thickenings” as additional universes. Thus the reduction of the study of Maximality of V to the study of mathematical criteria for the selection of preferred “pictures of V” inside the Hyperuniverse, The Hyperuniverse is of course entirely dependent on V; if we accept a new axiom about V then this will affect the Hyperuniverse. For example if we accept a little more than first-order reflection then a consequence is that the Hyperuniverse is nonempty.

If Sy would slow down and carefully explain in universally understandable terms just what he is talking about, we would all probably recognize that the use of the “Löwenheim-Skolem argument” is bogus.

I’m not sure that Sy is aware that there are some standards for doing philosophy and foundations of set theory (or anything else). Perhaps Sy believes that with enough energetic offerings of slogans, and enough seeking of soundbites from philosophers (which he has found are not all that easy to get), you can avoid having to come up with real foundational/philosophical ideas that work.

I am not aware of a single person on this email list who is inclined to believe that CTMP (aka HP) constitutes any kind of legitimate foundational program for set theory – at least on the basis of anything offered up here. (CTMP appears to be a not uninteresting technical study, but even as a technical study, it currently suffers from a lack of systemization – at least judging by what is being offered up here).

If there is a single person on this email list who thinks that CTMP (aka HP) constitutes any kind of legitimate foundational program for set theory, I think that we would all very much appreciate that they come forward and say why they think so, and start offering up some clear, deliberate, and generally understandable answers to the questions I raised a short time ago. I copy them below.

Now I am not primarily here to tear down silly propoganda. Enough of this has already been done by me and others. I am making efforts to steer this discussion into productive channels that meet that great standard: being generally understandable to everybody, with no attempt to mask flawed ideas — or seemingly unsound ideas — in a mixture of technicalities, slogans, and propoganda I invited Sy to engage in a productive discussion that would meet at least minimal standards for how foundations and philosophy can be discussed, and he has refused to engage dozens of times.

So again, if there is anybody here who thinks that CTMP (aka HP) is a legitimate foundational program for set theory, please say so, and engage in the following questions I posted recently: In the meantime, I am finishing up a wholly positive message that I hope you are interested in.

QUESTIONS – lightly edited from the original list

Why doesn’t HP carry the obvious name CTMP = countable transitive model program. That is my suggestion and has been supported by Hugh.

What does the choice of a countable transitive model have to do with “(intrinsic) maximality in set theory”? Avoid quoting complicated technicalities, meaningless slogans, or idiosyncratic jargon and adhere to generally understandable considerations.

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 or idiosyncratic jargon 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.

Harvey

Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Sat, 27 Sep 2014, Penelope Maddy wrote:

Dear Sy,

I fear that height actualism is not dead; surely there must be even a few Platonists out there, and for such people (they are not “nuts”!) I’d have to work a lot harder to make sense of the HP. Is the Height Actualism Club large enough to make that worth the effort? It would help a lot to know how the height actualists treat proper classes: are they all first-order definable? And how do they feel about “collections of proper classes”; do they regard that as nonsense?

I have no strong commitment to height actualism, but I did once think about proper classes as something other than what looks like just another few ranks in the hierarchy — something more like extensions of properties, so that they could be self-membered, for example.  My goal was to understand some of Reinhardt’s arguments this way, but it didn’t work for that job, so I left it behind.

So you generated IMH first, then developed the HP from it? Where did IMH come from?

I launched the (strictly mathematical) Internal Consistency Programme. A first-order statement is “internally consistent” if it holds in an inner model (assuming the existence of inner models with large cardinals). To be “internally consistent” is stronger than to be just plain old consistent, so new methods are needed to show that consistent statements are internally consistent (sometimes they are not) and there’s also a new notion of “internal consistency strength” (measured by large cardinals) that can differ from the usual notion of consistency strength. All of this work was of course about what first-order statements can hold in inner models so it was an obvious question to ask if one could “maximise” what is internally consistent. That is exactly the inner model hypothesis.

I see.  Thank you.

Can you remind us briefly why you withdrew your endorsement of IMH?

Because it only takes maximality in width into account and fails to consider maximality in height!

It this the problem of IMH implying there are no inaccessibles?

Yes, exactly!

We’re now out of my depth, though, so I hope we might hear others on this. E.g., it seems the countable models and the literal thickenings (as opposed to imaginary ‘thickenings’) have both dropped out of the picture.  ??

No, otherwise it wouldn’t be the Hyperuniverse Programme! (Recall that the Hyperuniverse is the collection of countable transitive models of ZFC.)

An important step in the HP for facilitating the math is the “Reduction to the Hyperuniverse”. Recall that we have reduced the discussion of “thickenings” of $V$ to a magic theory in a logic called “$V$-logic” which lives in a slight “lengthening” $V^+$ of $V$, a model of KP with $V$ as an element. In other words, the IMH (for example) is not first-order in $V$ but it becomes first-order in $V^+$. But now that we’re first-order we can apply Loewenheim-Skolem to $V^+$! This gives a countable $v$ and $v^+$ with the same first-order properties as $V$ and $V^+$. What this means is that if we want to know if a first-order property follows from the IMH it suffices to show that it holds just in the countable $v$‘s whose associated $v^+$‘s see that $v$ obeys the IMH. The move from $V$ to $v$ doesn’t change anything except now our “thickenings” of $v$ with quotes are now real thickenings of $v$ without quotes! So we can discard the $v^+$‘s with their magic theories and just talk boldly and directly about real thickenings of countable transitive models of ZFC. Fantasy has become reality.

In summary the moves are as follows: To handle the “thickenings” needed to make sense if the IMH we create a slight lengthening $V^+$ of $V$ to make the IMH first-order, then apply Loewenheim-Skolem to reduce the problem of deriving first-order properties from the IMH to a study of countable transitive models together with their real thickenings. So in the end we get rid of “thickenings” altogether and can work the math on countable transitive models of ZFC, nice clean math inside the Hyperuniverse!

The above applies not just to the IMH but also to other HP-criteria.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Penny,

On Tue, 5 Aug 2014, Penelope Maddy wrote:

Thank you for the plug, Sol.  Sy says some interesting things in his BSL paper about ‘true in V':  it doesn’t ‘reflect an ontological state of affairs concerning the universe of all sets as a reality to which existence can be ascribed independently of set-theoretic practice’, but rather ‘a façon de parler that only conveys information about set-theorists’ epistemic attitudes, as a description of the status that certain statements have or are expected to have in set-theorist’s eyes’ (p. 80). There is ‘no “external” constraint … to which one must be faithful’, only ‘justifiable procedures’ (p. 80); V is ‘a product of our own, progressively developing along with the advances of set theory’ (p. 93).  This sounds more or less congenial to my Arealist (a non-platonist):   in the course of doing set theory, when we adopt an axiom or prove a theorem from axioms we accept, we say it’s ‘true in V’, and the Arealist will say this along with the realist; the philosophical debate is about what we say when we’re describing set-theoretic activity itself, and here the Arealist denies (and the realist asserts) that it’s out to discover the truth about some objectively existing abstracta.  (By the way, I don’t think ‘truth-value realism’ is the way to go here.  In its usual form, it avoids abstract entities, but there remains an external fact-of-the-matter quite independent of the practice to which we’re supposed to be faithful.)

My apologies here. In my reply to Sol I only made reference to truth-value realism for the purpose of illustrating that one can ascribe meaning to set-theoretic truth without being a platonist. Indeed my view of truth is very far from the truth-value realist, it is entirely epistemic in nature.

Unfortunately the rest of my story of the Arealist as it stands won’t be much help because the non-platonistic grounds given there in favor of embracing various set-theoretic methods or principles are fundamentally extrinsic and Sy is out to find a new kind of intrinsic support.

Yes. I am trying to make the case that there are unexplored intrinsic sources of evidence in set theory. Some have argued that we must rely solely on extrinsic sources, evidence emanating directly from current set-theoretic practice, because intrinsic evidence cannot take us past what is derivable from the maximal iterative conception. I do agree that this conception can lead us no further than reflection principles compatible with V = L.

But in fact my intuition goes further and suggests that no intrinsic first-order property of the universe of sets will enable us to resolve problems like CH. We have to examine features of the universe of sets that are only revealed by comparing it to other possible universes (goodbye Platonism) and infer first-order properties from these “higher-order” intrinsic features of V (a name for the epistemically-conceived universe of sets).

Obviously a direct comparison of V with other universes is not possible (V contains all sets) so we must instead content ourselves with the comparison of pictures of V. These pictures are perfectly provided by the hyperuniverse (also conceived of non-platonistically). And by Löwenheim-Skolem we lose none of the first-order features of V when we model it within the hyperuniverse.

Now consider the effect that this has on the principle of maximality. Whereas the maximal iterative concept allows us to talk about generating sets inside V by iterating powerset “as long as possible”, the hyperuniverse allows us to express the maximality of (a picture of) V in a more powerful way: maximal means “as large as possible in comparison to other universes” and the hyperuniverse gives a precise meaning to this by providing those “other universes”. Maximality is no longer just an internal matter regarding the existence of sets within V, but is also an external matter regarding the largeness of the universe of sets as a whole in comparison to other universes. Thus the move from the concept of set to the concept of set-theoretic universe.

Now comes a crucial point. I assert that maximality is an intrinsic feature of the universe of sets. Certainly I can assert that there is a rich discussion of maximality in the philosophy of set theory literature with some strong advocates of the principle, including Goedel, Scott and yourself (correct me if I am wrong).

Maximality is not the only philosophical principle regarding the set-theoretic universe that drives the HP but surely it is currently the most important one. Another is omniscience (the definability in V of truth across universes external to V). Maybe there will be more.

I’m probably insufficiently attentive, or just plain dim, but I confess to being confused about how this new intrinsic evidence is intended to work.   It isn’t a matter of being part of the concept of set, nor is it given by the clear light of mathematical intuition.  It does involve, quoting from Gödel, ‘a more profound understanding of basic concepts underlying logic and mathematics’, and in particular, in Sy’s words, ‘a logical-mathematical analysis of the hyperuniverse’ (p. 79).  Is it just a matter of switching from the concept of set to the concept of the hyperuniverse?  (My guess is no.)  Our examination of the hyperuniverse is supposed to ‘evoke’ (p. 79) certain general principles (the principles are ‘based on’ general features of the hyperuniverse (p. 87)), which will in turn ‘suggest’ (pp. 79, 87) criteria for singling out the preferred universes — and the items ultimately supported by these considerations are the first-order statements true in all preferred universes. One such general principle is maximality, but I’d like to understand better how it arises intrinsically out of our contemplation of the hyperuniverse (at the top of p. 88).  On p. 93, the principle (or its more specific versions) is said to be ‘the rigorous expression of what it means for an element of the hyperuniverse, i.e., a countable transitive model of ZFC, to display “maximal properties”‘.  Does this mean that maximality for the hyperuniverse derives from a prior principle of maximality inherent in the concept of set?

You ask poignant questions; I hope that what I say above is persuasive!

Many thanks for your interest, and very best wishes,
Sy