# Re: Paper and slides on indefiniteness of CH

Dear Pen,

This is why I asked Claudio if a potentialist (like you) counts as a multiverser.  In practice, it doesn’t seem there’s a lot of difference between your potentialist multiverser and a universer who says:  there’s a single fixed universe, but we can’t describe it completely; we have to keep adding more large cardinal axioms.  If the algebraist comes to the set theorist in his foundational role and asks a question turns out to hinge on, say, inaccessibles (as apparently in Wiles’ original proof), you’d say, ‘no problem, what you want lives in this end-extention’, and my universer would say, ‘no problem, there are inaccessibles’.

But I was imagining that Claudio’s multiverse would be more varied that that.  So I floated a couple of possibilities:

You might say to the algebraist:  there’s a so-and-so if there’s one in one of the universes of the multiverse.  Or you might say to the universer that her worries are misplaced, that your multiverse view is out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.

(Claudio seemed to opt for the second, but ultimately rejected it; I’m not sure what he thinks about the first.)

Given the current purposes and features of HP, I believe my multiverser would probably tell the algebraist that there’s a way to extract something like a preferred theory of sets out of the hyperuniverse, but she may change her mind, should the unification process turn out not to be a prevailing goal of the programme. I know this does not clarify the situation in full, but it’s probably too early to state something definitive about the final version of HP.
Best wishes,
Claudio

# Re: Paper and slides on indefiniteness of CH

Dear Pen and Geoff,

I’ll leave it to Sy to say the last word on the issue of unification in connection to HP. Although I believe that it could be a fully legitimate and valid goal within the programme, I’d find it philosophically problematic for the reasons I’d pointed out in my previous emails. Therefore, I’d rather talk about reducing variance of truth across the hyperuniverse as a consequence of the existence of convergent consequences.

On Oct 21, 2014, at 6:42 PM, Geoffrey Hellman wrote:

I’m also at a loss, for similar reasons, and have been for a while. I’d have thought that a true “multiverser” would want to replace all talk of “V” –understood as the universe of [absolutely] all ordinals (and sets, etc.)–with some more benign term, such as some very large, (perhaps maximally) fat, transitive model of (here a ref to ZFC + some very large card axioms).

Geoff, I’m not sure that we really need this. If a multiverser doesn’t believe in an absolute V, but rather in a collection of different pictures of V, what he refers to when he’s talking about V is just one of those pictures. In particular, as far as the ontological level is concerned, the hyperuniverser know that what he’s really talking about is just some c.t.m in the hyperuniverse. However, this doesn’t mean that he may not use his internal picture(s) of the universe of all sets to find truths living in c.t.m. of the hyperuniverse (this somehow account for my use of the label dualism with reference to this conception in previous exchanges and that’s also why I said that the real V notion has a role to play within HP).

Best wishes,

Claudio

# Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Oct 19, 2014, at 6:36 PM, Penelope Maddy wrote:

Dear Claudio,

Pen, in a sense you’re right, the hyperuniverse “lives” within V (I’d rather say that it “originates from” V) and my multiverser surely has a notion of V as anybody else working with ZFC.

OK, but now I lose track of the sense in which yours is a multiverse view: there’s V and within V there’s the hyperuniverse (the collection of ctms). Any universer can say as much.

I understand my language was ambiguous. I wasn’t claiming that the whole hyperuniverse is within V. That is simply impossible, insofar as there are members of H which satisfy CH and others which don’t, some which satisfy IMH and some which don’t and so on. However, it is always possible (and logically necessary) to see any member of H as living in V. Any multiverser may concede that universes, say, mutually differring set-generic models, are in V, but this doesn’t commit her to be a universer.

I take the universer to believe that there’s one single (or definitive) V we perceive approximately and incompletely and/or we may gradually determine uniquely, whereas I take the multiverser to simply affirm that V is a collection of different set-theoretic universes each of which is endowed with differring properties. That is why a multiverser cannot pursue unification: universes contain information which cannot be amalgamated into one single universe (an ultimate V). We may say that V is, for the multiverser, an archetypal universe, not a unique or unifying reality.

Now, HP thinks that an amalgamation (via the convergence phenomenon) might be, in principle, possible, but my construal of it is that, even at that point, there’d be no need to go back to a universe-view.

(I confess this is a disappointing. I was hoping that a true multiverser would be joining this discussion.)

I’m not sure that Sy entirely agrees with me on this point, but to me HP implies an irreversible departing from the idea of finding a single, unified body of set-theoretic truths. Even if a convergence of consequences of H-axioms were to manifest itself in a stronger and more tangible way, via, e.g., results of the calibre of those already found by Sy and Radek, I’d be reluctant to accept the idea that this would automatically reinstate our confidence in a universe-view through simply referring back such a convergence to a pristine V.

Now I’m confused again. Here’s the formulation you agreed to:

Or you might say to the universer that her worries are misplaced, that your multiverse view is out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.

Though embracing a single universe is the most straightforward way of pursing unify, I was taking you to be pursuing it in a multiverse context (not to be embracing ‘a pristine V’). Fine with me.

But now that you’ve clarified that you aren’t really a multiverser, that you see all this as taking place within V, why reject unify now? And if you do, what will you say to our algebraist?

See above. I believe HP is a multiverse theory, therefore an HPer ought not to foster unification (but rather justified selection of universes). I’m not sure that all HP people would agree on this, though.

Moreover, HP, in my view, constitutes the reversal of the foundational perspective I described above (that is, to find an ultimate universe), by deliberately using V as a mere inspirational concept for formulating new set-theoretic hypotheses rather than as a fixed entity whose properties will come to be known gradually.

So there’s a sense in which you have V and a sense in which you don’t. If V is so indeterminate, how can the collection of ctms within it be a well-defined object open to precise mathematical investigation?

In fact, it seems to me that practice shows us that a c.t.m. is easier to deal with and, in the end, more informative than real V (incidentally, as also pointed out by Radek, that is the key to understanding forcing).

Best wishes,
Claudio

# Re: Paper and slides on indefiniteness of CH

Dear Pen and Harvey,

I believe I should address some of the issues that I had left unanswered and maybe provide some further responses to the questions you’d asked. Wrt to Harvey’s, I think that Radek has already given some persuasive answers, so I will for the time being concentrate on Pen’s questions.

Sy, please feel free at any point to add any remarks you may find useful to contribute to the discussion.

Pen, in a sense you’re right, the hyperuniverse “lives” within V (I’d rather say that it “originates from” V) and my multiverser surely has a notion of V as anybody else working with ZFC. Moreover, members of H are thought to be strongly related to V also in another way, through the satisfaction of principles which are, originally, assumed to be referring to the universe. By setting up the hyperuniverse concept and framework, however, she stipulates that questions of truth about V be dealt within the hyperuniverse itself. Again, this doesn’t imply that whatever is taken to hold across portions of the hyperuniverse is then referred back to the universe.

I’m not sure that Sy entirely agrees with me on this point, but to me HP implies an irreversible departing from the idea of finding a single, unified body of set-theoretic truths. Even if a convergence of consequences of H-axioms were to manifest itself in a stronger and more tangible way, via, e.g., results of the calibre of those already found by Sy and Radek, I’d be reluctant to accept the idea that this would automatically reinstate our confidence in a universe-view through simply referring back such a convergence to a pristine V.

Moreover, HP, in my view, constitutes the reversal of the foundational perspective I described above (that is, to find an ultimate universe), by deliberately using V as a mere inspirational concept for formulating new set-theoretic hypotheses rather than as a fixed entity whose properties will come to be known gradually. As has been indicated by someone in this thread, HP fosters an essentially top-down approach to set-theoretic truth, whose goal is that of investigating what truth about sets may be generated beyond that incapsulated by ZFC using new information about V. I used the term ideal in my previous email just to convey the contrast between a V investigated through attributing to it certain features and a real V as a fixed entity progressively determined through subsequent refinements (I guess that this ideal status of V might be construed in the light of Kant’s Grenzbegriffe working as regulatory ideals [I owe this interpretation of V to Tatiana Arrigoni]).

Coming to the other issues raised, c.t.m. have proved to be technically very expedient and fertile in terms of consequences for the purposes of HP and, moreover, they seem to capture adequately the basic intuitions at work in such techniques as forcing. And as you pointed out, the hyperuniverse might be seen as something allowing us to generate a unified conceptual arena to study a multiverse framework, so why wouldn’t the use of c.t.m. be justified precisely on these specific grounds?

To go back to a more general point, I believe that HP should be judged essentially for its merits as a dynamic interpretation of truth within a multiverse framework. In my view, its construction, thus, responds, to a legitimate foundational goal, provided one construes foundations not in the sense of selecting uniquely and determinately the best possible general axioms for the mathematics we know (including set-theoretic mathematics), but rather in that of exhibiting (and studying) evidential processes for their selection: the study of what I defined properties of an ideal V within the hyperuniverse is one such evidential process.

Now, as I said in my previous email, the programme, at least as far as its epistemological goals are concerned, is far from being perfected in all its parts, of course. As said (and requested by many people also here), it has to clarify, for instance, what further legitimate ideal properties of V there are, and what justifies their probable intrinsicness (relationship to the set concept).

Has this brief summary answered (at least some of) your legitimate concerns?

Best wishes,
Claudio

# Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Oct 17, 2014, at 9:35 PM, Penelope Maddy wrote:

Dear Claudio,

So I’m wondering, on your multiverse picture, how this would work. You might say to the algebraist: there’s a so-and-so if there’s one in one of the universes of the multiverse. Or you might say to the universer that her worries are misplaced, that your multiverse view is out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.

Is it the latter?

You’re right, that’s the latter. However, I see the potential difficulty with explaining to someone (e.g., an algebraist) who wants definite mathematical answers that there might be a *splitting* of truth in different universes, notwithstanding the indication of some preferred reality.

I doubt the algebraist will care about this. Set theory’s job is to provide a single accept theory of sets (by which I just mean a batch of axioms) that can play the role we’ve been talking about: providing a kind of certification and a shared arena. As long as you produce that, I don’t think it matters much to outsiders what set theorists say among themselves about the underlying ontology or semantics. My worry was that your multiverser wouldn’t be able to give a clean answer to the algebraist, but apparently that worry is misplaced.

So, now, when we ask the universer what set theory is up to, he says we’re out to describe V. (I’m inclined to allow the universer to go on to say that V is ‘potential’ in some way or other — would you hold that this would turn him into a multiverser?)

I see the potential philosophical subtlety (and difficulty) there. A potentialist about V might claim that different pictures of V obtained through manipulation of its height and width do not automatically force him to take up a multiverse view. I’m not completely sure that this is the case. Surely, within HP potentialism about V is, from the beginning, operationally connected to a distinctive framework, that of c.t.m. in the hyperuniverse.

For your multiverser, there is no V, but a bunch of universes, right? What does this bunch look like?

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).

Best wishes,

Claudio

# Re: Paper and slides on indefiniteness of CH

Dear Pen,

Thank you very much for your email.

On Oct 17, 2014, at 2:09 AM, Penelope Maddy wrote:

Dear Claudio,

Thank you for your rich message.  I had thought that, on Sy’s understanding, the HP isn’t really a multiverse view, but a way of discovering new things about V (like an answer to CH).  If we’re to understand it as a true multiverse view, that’s a different matter.

I believe that the HP might still provide an answer to problems like CH but, as I said, that’ll essentially depend upon whether the choice of the H-principles (or H-axioms), that is, the choice of the general properties of an ideal V within the hyperuniverse (H) will generate convergent consequences in the preferred universes. In my view, such a convergence would provide strong evidence in favour of the acceptance of an answer to CH in the standard sense (of course, modulo the acceptance of H as the only general ontological setting one cares about).
However, an alternative (non-standard and non-prescriptive) way HP provides solutions to open set-theoretic problems is through indicating ways to find answers based on a rationally justifiable selection of multiverse principles. Obviously, the non-standard way may appeal a lot less to a truth-value realist and universer than to a pluralist (and multiverser). Nonetheless, I believe that it should definitely be relevant to anyone interested in understanding the way a multiverse conceptual framework may still yield specific versions of set-theoretic truth, rather than just being an inert collection of models. In my opinion, that was the original concern which motivated Sy and Tatiana’s formulation of the hyperuniverse: to find a way to characterise dynamically the search for truth within a multiverse environment. Now, rather than specifying multiverse laws based on the mere interaction among models, the HP looks into principles which can be arguably derived from a pre-theoretic intuition of an ideal universe, but are, afterwards, embedded into members of the multiverse. (that is why, in my previous email, I was referring to a dual role of V within the programme).
I’d like to ask one preliminary question on behalf of the advocate of the ‘universe view’.  You write:

Now, there surely are reasons to believe that the universe-view has better prospects within the foundations of set theory. Some of them might be related to … more general concerns related to the foundations of mathematics as more safely couched within a single-universe rather than a plural-universe framework.
You may have something more sophisticated in mind, but this could be read as a simple worry about the ‘foundational’ role of set theory.  A universer might think that set theory arose with the proliferation of pure mathematics, for many reasons, but partly to certify the coherence of new structures and to provide a single arena for all those new structures to be studied in relation to one another — and she might think that it continues to play that role today.  (In a recent ASL talk, even Vladimir Voevodsky, advocate of ‘univalent foundations’, assigned this role to set theory, as the theory ‘used to ensure that the more and more complex languages of the univalent approach are consistent’ (from the abstract in the BSL), or ‘at least as consistent as set theory’ (from the slides).)   It appears that having one standard theory of sets is a requirement for playing this role:  when the algebraist asks whether or not there’s     a so-and-so, we look to see whether you can prove there’s a (surrogate for) a so-and-so in our accepted theory of sets.  And perhaps this lends itself to a universe-view.

So I’m wondering, on your multiverse picture, how this would work.  You might say to the algebraist:  there’s a so-and-so if there’s one     in one of the universes of the multiverse.  Or you might say to the universer that her worries are misplaced, that your multiverse viewis out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.

Is it the latter?
You’re right, that’s the latter. However, I see the potential difficulty with explaining to someone (e.g., an algebraist) who wants definite mathematical answers that there might be a  splitting of truth in different universes, notwithstanding the indication of some preferred reality. Part of the solution might lie in rescuing what you have mentioned above and explained in depth many times in your work: set theory provides a conceptual unified arena which encompasses all mathematical phenomena in a particularly convenient and elegant way. However,  whether this remarkable phenomenon would have us think that there should also be a unified ontology in the sense of a unique, static collection of sets endowed with fixed properties is far from clear, although I think there’s robust evidence that this might not be the case. Unification in this sense, if concretely possible at all and although appealing a lot to certain strands of realism, might be operationally ineffective anyway, as one might always prefer working in a different  universe and/or axiomatic framework (I fully understand this is a big philosophical issue, so please take these remarks as merely incidental). One of the possible merits of HP, therefore, might be that of trying to reach a sort of dynamic unification, through providing, to paraphrase your words, a unified multiverse arena wherein one can adequately represent the process of selection of some (ZFC-grounded) set theories based on certain kinds of intuition.
One might have legitimate concerns about HP’s notion of set-theoretic truth and background ontology but I don’t see any reason why one should not consider the assets of this programme wrt our foundational and (some) operational goals.
All best,
Pen

Thanks again and best wishes,

Claudio

# Re: Paper and slides on indefiniteness of CH

Dear all,

I’ve been collaborating with Sy on the HP for almost two years now and I believe I should add something to what he’s presented so far which may contribute to clarifying the programme’s goals and features. What follows does not necessarily coincide with Sy’s views or those of the other HP people (Radek and Carolin).

1. Universe vs multiverse view. One of the points which has been sometimes overlooked by people in this thread is that the HP aims to be a theory of the set-theoretic multiverse. This means that, however robustly or thinly realistic one’s ontological commitments are, HP assumes that it is a fact that set theory is not about a single, uniquely identifiable and describable ontological framework. By the standard semantic approach to truth, this cannot but mean that the HP commits itself to pluralism as a default position, although, as Sy has been trying to explain many times, the programme has no philosophically motivated prejudice against reducing the amount of truth-variance within the hyperuniverse (H). Some might legitimately ask whether HP aims to set forward some specific argument in favour of pluralism as a general philosophical conception, but that is exactly what was thought not to be necessary in setting up the programme: HP was born to make sense of the model-theoretic plurality of frameworks within current set theory and took it as a fact that such a plurality was there from the beginning. Now, there surely are reasons to believe that the universe-view has better prospects within the foundations of set theory. Some of them might be related to a strong commitment to thorough-going realism, some to a milder, practice-based form of realism looking favourably at confirmation coming from significant strands of practice (the latter are what I believe Hugh, as, in my opinion, a universe-view supporter, sees most favourably), some to more general concerns related to the foundations of mathematics as more safely couched within a single-universe rather than a plural-universe framework. Now, a universe-view supporter has all reasons to ask himself what the “unique” solution of open problems in set theory is and, typycally, sees as only relevant to his conception any result which reassures him in the correctness of his conception. Ok, fine. But then, the HP cannot be any help to him. The HP may foster “unique” solutions to the set-theoretic problems only as a consequence of what Sy called a sort of “convergence”, in terms of consequences, of H-principles into the same axioms (that is, their consequences in members of H). But this is not even required of the programme in itself. Therefore, my final point is: the value of the HP should not be measured on the grounds of its providing us with definitive (whatever this may mean) answers to open problems in set theory.
2. The value of HP. This clarified, we should move on and see whether HP is a valuable theory of the set-theoretic multiverse. A theory of the multiverse might be defective in different senses. Peter, for instance, has found reasons of concern in Joel’s radical conception and Hugh’s attempt at yielding one (but only to discard it as logically impossible) is partly rejected by John (Steel) and so on. Now, we believe HP has better prospects to be philosophically and foundationally more attractive. Some of the reasons are described below.a. Ontological minimality: HP identifies a core model-theoretic construct, that is, c.t.m., as the only constituent of multiverse ontology. Further, mathematical and logical, reasons for this choice have been explained at length by Sy, but I wish to recall that the main (and, to some extent, remarkable) fact is that we do not lose any information about set-theoretic truth by making this choice. The choice of c.t.m. might look illegitimate to some staunch realist, for whom ontological constructs follow our pre-theoretic grasp of them, but then I’d look forward to seeing this person argue against first-order logic as a natural consequence of this attitude. So long as we agree that the choice of logic is the correct one, then I don’t see any strong argument against restricting our attention on c.t.m.

b. Dualism about real V: this point was addressed by Peter, who thought it could be a potential weakness. In my view, this is a strength. HP has no actualist commitment to real V, but only to a post-zermelian approach to V as an endless sequence of models (although, Zermelo’s view has a commitment to width actualism, expressed by second-order versions of the axioms). Thus, the link with an actualised real V expressed by this approach is very thin. H-principles express properties of an ideal (not real) universe and, then, “collapse” the universe into some c.t.m. Consequences of H-principles are seen as local axioms in portions of the hyperuniverse. Thus, our real V works dually: first as a regulatory ideal, as if temporarily actualised for the sake of the study of its general properties (e.g., maximality as expressed by the IMH) and, secondarily, as a c.t.m. within H endowed with specific properties. There is no conflict between these two roles and, in fact, dualism seems to us to fit set-theoretic practice marvellously. Talk of the universe within practice constantly oscillates between these two poles: the universe as an actual construct surpassing any other conceivable construct and the universe as a model (a c.t.m. for the sake of our constructions, possibly extended or reduced to, respectively, an outer or inner model). HP aims to give a general conceptual explanation for the existing dualism in our practice.

c. Axiom-generating methodological approach: the HP supporter is not a truth-value realist nor does she believe that there are unique solutions to open set-theoretic problems. Accordingly, she thinks that no axioms (or collection theoreof) will give us the nicest possible set theory. But this is entirely fine with her: we have different universes, let’s live with them. However, she’s not entirely indifferent to issues of truth within the hyperuniverse and, in fact, she believes that the study of properties of an ideal V might bear on the structure and properties of members of H. She might think that the radical view is the only option available: there are different concepts of set and, thus, different universes which instantiate such concepts. But maybe an alternative way might be pursued. Again, she has a concept of a real V, which is an ideal actualisation of the universe, and she believes that we have reasons to hold that such an idealisation may play a role in the current practice. Therefore, she believes that properties of such an ideal construct may be studied within the basic ontological constructs she has on hand: c.t.m. within the hyperuniverse. The only way to study these properties is to examine their implications. But now she understands that (some of) the statements implied by properties of the universe may be nothing else but (new) axioms. At the end of the procedure she realises that the way a new axiom is generated is fully accounted for by her views concerning set-theoretic ontology (minimality above) and her pre-theoretic intuition of an idealised (and temporarily actualised) universe. As a consequence, she comes to see the HP as a tool to generate new knowledge (through generating new axioms) within set theory.

d. Explanatory (epistemic) strength: even if one does not see any reason to see c. as a legitimate procedure within set theory, HP may still stand out as a way to explain how we may come to believe new axioms. In other terms, HP may be a theory explaining the role of conjectures leading to new axioms within set theory. Sy has placed a lot of emphasis on the fact that the whole procedure qualifies as an intrinsically justified procedure. Peter raised relevant concerns about whether Sy’s claims were legitimate. In my view, the ideal V described by such principles as the IMH, $\textsf{IMH}^\#$ etc. has good prospects to be seen as related to the concept of set. Now, that is the hardest philosophical part and I acknowledge that it is far from being established with a reasonable degree of certitude. Does the concept of set imply that V (the V I have been discussing, its ideal actualisation) is maximal in the sense asserted by IMH? Why should maximality even be part of the concept of set? Now, there are arguments available in the literature providing reasons to believe that maximality plays a special role in our intuitions about sets (Gödel, Wang, in a sense Dummett (based on Russell’s self-reproductive properties), have all argued in favour of maximality as related to the set concept) but it is far from clear that all of these arguments may be used to defend principles such as IMH. Therefore, it is correct to require of the HP to provide stronger arguments in favour of the intrinsicness of IMH, $\textsf{IMH}^\#$ etc. But suppose that the kind of principles the HP is examining has been proved to be legitimately justifiable through appeal to the set concept. Then I believe that the HP might be viewed as a strong foundational approach to truth within a relevant segment of current practice in set theory (at least, the ZFC-based portion), as grounded on the study of intrinsically justified criteria. Obviously, this wouldn’t qualify HP automatically as a theory which yields new self-evident truth! (whatever the term may mean).

3. Is HP realist (in ontology)?. Now, someone might say: “this is just cheating. You’re talking about a concept of set which should be robust enough to imply certain properties rather than others, for instance, maximality as opposed to minimality, uniformity (Gödel) rather than constant alteration (however this might be formalised mathematically) etc. If you want to argue that this is the case, then you have to hold the belief that there is an ultimate universe of sets endowed with such properties. Even if you’re not able to prove that such properties are actually implied by the concept of set, something which should not be taken for granted, by asserting that there is a concept of set and an ultimate universe instantiating it, you automatically subscribe to some form of realism (in ontology).” I address this objection as I think that it is relevant to assessing the whole legitimacy of the HP. HP does not start with the idea that there is a universe of sets. HP thinks that there are different universes, which are built using the currently known procedures (set forcing, class forcing, inner models, etc.), which all point to one single ontological template, a c.t.m. However, HP also believes that the properties of such universes are dependent upon the properties of an ancestral universe (real V) which is nothing but an ideal universe. Such properties are also ideal, in the sense that they do not prescribe the existence of certain sets or certain ordinals, but only that the universe should enjoy some general property x, which, in our view, is related to the set concept. The maximal iterative conception is the concept that has been often set forth as the standard concept of set. As Sy has said many times, HP does not question this. What HP wants to do is to expand on this concept. But this expansion should not be construed as the deliverance of self-evident new principles giving ultimate truths, but rather as the addition of new features to the concept (e.g., if one adopted IMH, maximal iteration + maximal internal consistency of V). That’s it. Admittedly, there is not even any requirement that new additions give rise to the same truths across the multiverse. Does this mean that there are different concepts of set? It might be so, but this does not imply that there are different real universes. However, in my view, one rather wants to study alternative expansions of the concept with a common initial ground (the iterative notion), rather than alternative concepts of set. To sum up, HP may commit itself to some form of objectivity in the way we expand on the concept of set, but not in the idea that we have a pre-theoretic grasp of a universe of sets endowed with some properties. To use Pen’s terminology, a HP supporter could be a Thin Realist, but with an emphasis on refinements of and additions to the concept of set, rather than to universes or ontological frameworks.

Sorry for the very long email. I hope it addresses clearly at least some of the general concerns that people in this thread had raised with regard to the plausibility of HP and contributes effectively to the exciting debate going on.

Best regards,

Claudio