# Re: Paper and slides on indefiniteness of CH

Dear Peter,

My apologies for the actualism/potentialism confusion! The situation is this: We have been throwing around 3 views:

1. Actualism in height and width (Neil Barton?)
2. Actualism only in width (Pen and Geoffrey?)
3. Actualism in neither.

Now the problem with me is that I have endorsed both 2 and 3 at different times! What I have been trying to say is that the choice between 2 and 3 does not matter for the HP, the programme can be presented from either point of view without any change in the mathematics. In 3 the universes to which V is compared actually are there, as part of the background multiverse (an extreme multiverse view) and in 2 you can only talk about them with “quotes”, yet the question of what is true in them is internal to (a mild lengthening of) V.

I have been a chameleon on this: My personal view is 3, but since no one shares that view I have offered to adopt view 2, to avoid a philosophical debate which has no pracatical relevance for the HP.

It is similar with the use of countable models! Starting with view 2 I argue that the comparisons that are made of V with other “universes” (in quotes) could equally well be done by replacing V by a ctm and removing the quotes. But again, this is not necessary for the programme, as one could simply refuse to do that and awkardly work with quoted “universes” all of the time. I don’t understand why anyone would want to do such an awkward thing, but I am willing to play along and sadly retitle the programme the MP (Maximality Programme) instead of the Hyperuniverse Programme, as now the countable models play no role anymore. In this way the MP is separated from the study of countable transitive models altogether.

In summary: There is some math going on in the HP which is robust under changes of interpretation of the programme. My favourite interpretation would be View 3 above, but I have settled on View 2 to make people happy, and am even willing to drop the reduction to countable models to make even more people happy.

I am an extreme potentialist who is willing to behave like a width actualist.

The mathematical dust has largely settled — as far as the program as it currently stands is concerned –, thanks to Hugh’s contributions.

What? There is plenty of unsettled mathematical dust out there, not just with the future development of the HP but also with the current discussion of it. See my mail of 25.October to Pen, for example. What do we say about the likelihood that maximality of V with respect to HOD likely contradicts large cardinal existence? Even if the HP leads to the failure of supercompacts to exist, can one at least get PD out of the HP and if so, how?

More broadly, a lot remains unanswered in this discussion regarding Type 1 evidence (for “good set theory”): If $\text{AD}^{L(\mathbb R)}$ is parasitic on $\text{AD}$ how does one argue that it is a good choice of theory? When we climb the interpretability hierarchy, should we drop AC in our choice of theories and instead talk about what happens in inner models, as in the case of AD? Similarly, why is large cardinal existence in V preferred over LC existence in inner models? Are Reinhardt cardinals relevant to these questions? And with regard to Ultimate L: What theory of truth is to be used when assessing its merits? Is it just Thin Realism, and if so, what is the argument that it yields “the best set theory” (“whose virtues swamp all the others” as Pen would say) and if not, is there something analagous to the HP analysis of maximality from which Ultimate L could be derived?

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Neil,

On Sat, 25 Oct 2014, Neil Barton wrote:

Dear HP-ers, HP-worriers, and friends, In this thread (which I confess has been moving pretty quickly for me; I’ve read it all but do apologise if I’m revisiting some old ground) we’ve seen that the key claim is that there is a deep relationship between countable transitive models and some V, either real, ideal, or sat within a multiverse.

Well, yes, this has been something I have emphasized, the “reduction to the Hyperuniverse” which sets up a connection between the study of the maximality of V and the study of ctm’s. But please note that this is not really essential to the programme! As I said in my mail of 23.October to Pen and Geoffrey:

“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.”

I have a few general worries on this that if assuaged will help me better appreciate the view.

OK, let’s discuss the reduction to ctm’s anyway. I respond to your comments below.

I’m going to speak in “Universey” terms, just because its the easiest way I find for me to speak. Indeed, when I first heard the HP material, it occurred to me that this looked like an epistemological methodology for a Universist; we’re using the collection of all ctms as a structure to find out information (even probabilistic) about V more widely. If substantive issues turn on this way of speaking, let me know and I’ll understand better.

You are starting off just fine, this is a perfectly reasonable way to interpret the programme.

Let’s first note that in the wake of independence, it’s going to be a pretty hard-line Universist (read “nutty Universist”) who asserts that we shouldn’t be studying truth across models in order to understand V better.

Pen Maddy is not a nut! You can simply ground truth in what is good set theory and mathematics, as would a Thin Realist (right, Pen?) and not bother with all of this talk about models.

Indeed the model theory gives us a fascinating insight into the way sets behave and ways in which V might be. However, its then essential to the HPers position that it is the “truth across ctms” approach that tells us best about V, rather than “truth across models” more generally.

The reduction to ctm’s asserts that even though looking just at ctm’s appears to be more restrictive, in fact it gives the same results as if you were to consider models more generally. More on this below.

I see at least two ways this might be established: A. Ctms (and the totality of) are more easily understood than other kinds of model.

For a simple reason: You can build lengthenings and thickenings of ctm’s, while you can’t do this for utm’s (uncountable transitive models). Consider forcing: For ctm’s, forcing estensions actually exist but for utm’s they can only be “imagined”!

B. Ctms are a better guide to (first-order) truth than other kinds of model.

I wouldn’t say that; I would only say that they are just as good as using arbitrary models.

I worry that both A and B are false (something I came to worry in the context of trying to use the HP for an absolutist).

B might be false (because of the word “better”) but A is true for rather vacuous reasons (due to the ability to bend and twist models which are countable, rather than to just “imagine” doing that).

A.1. It would be good if we could show two things to address the first question: A.1.1. The Hyperuniverse is in some sense “tractable” in the sense that we can refer to it easily using fairly weak resources. A.1.2. The Hyperuniverse is in some sense “minimal”; we only have the models we need to study pictures of V. There’s no extraneous subject matter confusing things.

You are demanding! How about just: ctm’s are nice to work with and give accurate information about the maximalaty of V?

The natural way to assuage A.1.1. for someone who accepts something more than just first-order resources is to provide a categoricity proof for the hyperuniverse from fairly weak resources (we don’t want to go full second-order; it’s the very notion of arbitrary subset we’re trying to understand). I thought about doing this in ancestral logic, but this obviously won’t work; there are uncountably many members of the Hyperuniverse and the downward LST holds for ancestral logic. So, I don’t see how we’re able to refer to the hyperuniverse better than just models in general in studying ways V might be.

(Of course, you might not care about categoricity; but lots of philosophers do, so it’s at least worth a look)

Probably I miss your point here, but it may be that you have overlooked the “dualism” between the Hyperuniverse and V that Claudio has emphasized. Note that the Hyperuniverse is just as ill-defined as V itself; it depends heavily on V. The suggestion of the “reduction to the Hyperuniverse” is only that by working with the Hyperuniverse (a particular multiverse conception) one can conveniently phrase maximality issues in a way that is faithful to the meaning of maximality for V. There is no presumption that the Hyperuniverse is any more “categorical” or “tractable” than V itself.

Re: A.1.2 The Hyperuniverse is not minimal. For any complete, maximal, truth set T of first-order sentences consistent with ZFC, there’s many universes in H satisfying that truth set. So really, for studying first-order pictures of V’ there’s lots in there you don’t need.

That is the whole point! You have to start somewhere, and the Hyperuniverse is the natural starting point, as it is the arena in which the mathematics of set theory takes place (recall how forcing works: “Let M be a ctm, and P a partial order in M, we then consider P-generic extensions of M and show that they exist, also as ctm’s”). The incompleteness of ZFC is precisely manifested in the fact that the Hyperuniverse is “too big” and must be thinned out to its subcollection consisting of those universes which best exhibit maximality features.

So, I’d like to hear from the HPers the sense in which we can more easily access the elements of H. One often hears set theorists refer to ctms (and indeed Skolem hulls and the like) as “nice”, “manageable”, “tractable”. I confess that in light of the above I don’t really understand what is meant by this (unless it’s something trivial like guaranteeing the existence of generics in V). So, what is meant by this kind of talk? Is there anything philosophically or epistemically deep here?

In my view there is nothing deep here, it is only the observation that the Hyperuniverse is closed under the model-building methods of set theory: No matter what kind of forcing or infinitary logic construction we do to create new models from ctm’s, we end up again with a ctm. The deeper point is that ctm’s do a faithful job of representing what is implied by the maximality of V in height and width. This latter point is not obvious.

By the way, I have heard “nice” and “managable” but never “tractable” in reference to the Hyperuniverse.

And when one says that the Hyperuniverse is more “accessible” than V one cannot really mean this literally, as it is just as ill-defined as V and thoroughly dependent upon V. Instead it only means that in the Hyperuniverse one can stretch one’s elbows and explore many different pictures of V, something which is awkward to do sticking just with V. Again, think about forcing extensions of V; how does one gain access to those? The only way is to imagine them, as you have no context in which to build them. In the HP you observe that the properties of forcing extensions, indeed of arbitrary outer models, that you want to explore are nevertheless “almost first-order” in V (they are first-order in slight lengthenings of V) and therefore what you conclude about these properties would be the same if you were to replace V with a countable version of itself.

On to B. Are ctms a better guide to truth in V than other kinds of model? Certainly on the Universist picture it seems like the answer should be no; various kinds of construction that are completely illegitimate over V are legitimate of ctms; e.g. $\alpha$-hyperclass forcing (assuming you don’t believe in hyperclasses, which you shouldn’t if you’re a Universist).

Wait a minute, slow down! Re-read the comments of Pen and Geoffrey about this. They have entertained a height potentialism which naturally permits the addition of alpha-many new von Neumann levels on top of V! For them, there is no problem talking about “alpha-hyperclasses”” (P&G, please confirm).

So maybe you are talking about a width and height actualist? As I hvae said, the HP becomes very awkward with such a limitation.

Why should techniques of this kind produce models that look anything like a way V might be when V has no hyperclasses?

I haven’t gotten into this because I have been at least presuming height potentialism. I can say something about an HHP, a Handcuffed version of the HP, if you like, but let’s postpone that a bit.

Now maybe a potentialist has a response here, but I’m unsure how it would go. Sy’s potentialist seems to hold that it’s a kind of epistemic potentialism; we don’t know how high V is so should study pictures on which it has different heights.

Pen and Geoffrey, please help me here! You have talked favourably of height potentialism and Neil thinks there’s something afoul here.

But given this, it still seems that hyperclasses are out; whatever height V turns out to have, there aren’t any hyperclasses.

Huh? If you add one new level to V you get classes, if you add two new levels to V you get hyperclasses, if you add alpha new levels to V you get alpha-hyperclasses. Plain as pie.

Or are you really suggesting that we don’t know what the height of V is, but given any guess at that we have to stop ourselves from thinking that it could have a greater height? Now that is nutty!

If one wants to look at pictures of V, maybe it’s better just to analyse the model theory more generally with standard transitive models and a ban on hyperclass forcing?

Ban on hyperclass forcing? What? Maybe it’s time for Neil Barton to make a confession: You are an actualist, right? For you, V has a fixed height and width and it is nonsense to think about increasing height or width, right? Please come clean here, it’s OK, I will still like you.

But now you have a problem. I assume that you are OK with classes. And you know what it means for a class relation E on V to be wellfounded and extensional. So you know what it means for the structure (V,E) to be a model of ZFC which has the standard (V,epsilon) as a rank initial segment. In other words, whether you like it or not, you have no problem thinking about lengthenings of V in terms of classes. So you are being dragged kicking and screaming into the world of Hyperclasses! Do you really have a coherent explanation for why these class structures which represent lengthenings of V do not exist?

[A note; like Pen I have worries that one can't make sense of the hybrid-view. The only hybrid I can make sense of is to be epistemically hyperuniversist and ontologically universist. I worry that my inability to see the "real" potentialist picture here is affecting how I characterise the debate.]

The hybrid works perfectly well with height potentialism (“multiversism light”); starting with that the move to the rich multiverse perspective provided by the Hyperuniverse is OK and dualises nicely (as Claudio said) with the single-universe view of V (augmented by height potentialism). The only hybrid that worries me is to mix actualism about V with the HP, as actualism doesn’t allow you to make the moves you want to make that are essential to analysing maximality. But in my view, I don’t think that actualism in height (as opposed to actualism in width) is really a coherent view; it seems that at least Hilary and Geoffrey will agree.

But if you want me to take a stab at the HHP (the Handcuffed HP) I will be happy to do so.

Anyway, I’m sympathetic to the idea that I’ve missed a whole bunch of subtleties here. But I’d love to have these set to rights.

I am very grateful for your interesting comments! I have surely come to understand the HP much better as the result of comments like yours.

P.S. I’ve added my good friend Chris Scambler to the list who was interested in the discussion. I hope this is okay with everyone here. P.P.S. If there are responses I’ll try to reply as quick as I can, but time is tight currently.

I know what you mean. I can’t recall when time was not tight!

All the best, Sy

PS: We will (indeed must) have a Hyperuniverse Project meeting at the KGRC in September 2015. I hope that this can be merged with whatever plans you, Toby or others may have for meetings on the philosophy of mathematics in the second half of 2015. I haven’t forgotten about the London SOTFOM2 in January.

# 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 Sy and friends,

Firstly, let me just say that I’ve found this thread really interesting, and I’m very keen to keep the discussion together (though the technical subtleties sometimes elude me, it’s great to see the thought process).

But 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 think this very much depends on who you talk to. For example, lots of height actualists like to render proper class talk using plural reference (given in Boolos and developed in Uzquiano). Again, however, people differ on whether all “pluralities” should be first-order definable.

Another route some have taken is to regard class talk as simply shorthand for talk for first-order satisfaction in $V$, and associated formalisations of such talk in NBG (without putting words in people’s mouths, this is I believe Peter Koellner’s position, but I’m prepared to be corrected on this). This is obviously all first-order definable.

Still another way is to take Leon Horsten and Philip Welch’s approach, and regard proper classes as mereological fusions of sets that lie outside the scope of our first-order quantifiers. Again, whether or not you think such things are first-order definable depends on taste.

Finally, there’s the view that proper classes should be understood as “properties” or some other similar intensional notion. Pen’s already mentioned her 1983 paper, but it also pops up in some of Øystein Linnebo’s work from his (I beieve) pre-potentialist days (e.g. “Sets, Properties, and Unrestricted Quantification”).

Similarly whether or not one should have collections of proper classes is also going to depend on who you talk to. I’d say most actualists reject collections of proper classes, and this maps loosely on to the philosophical views above; properties/mereological fusions are fundamentally different kinds of entities from sets, and you can’t (for reasons that differ between authors) take collections of them. Similarly, in the plural case, this turns on the legitimacy of super-plural quantification, a hotly debated topic in the Philosophy of Language, Logic, and Set Theory (see, for example, the different views presented by Hanoch Ben-Yami’s “Higher-Level Plurals versus Articulated Reference, and an Elaboration of Salva Veritate” and Linnebo and Nicholas’ “Superplurals in English”). I’d say most actualists who take the plural route reject super plural quantification (usually by arguing that it is more ontologically committing than plural quantification).

There are some who have embraced collections’ of proper classes, however. In his PhD thesis Hewitt allows for finite order superplural reference and some have been tempted to use collections of proper classes in the pursuit of a class-theoretic interpretation of Category Theory (I think Muller has something like this in “Sets, Classes, and Categories”, but it’s a while since I read that – I should check the reference). Of course, the extent to which this is coherent is fiercely contested!

In short, when talking about “Actualism” we face exactly the same problem as when talking about “Potentialism”; there’s a whole gamut of positions referred to by the term. Constructing general arguments for and against these kinds of positions, and analysing the extent to which a view that falls under one of the two labels can account transparently for a piece of set-theoretic discourse is thus rather tricky.

Very Best,
Neil

# Re: Paper and slides on indefiniteness of CH

Dear Pen and Geoffrey,

On Wed, 24 Sep 2014, Penelope Maddy wrote:

Thanks, Geoffrey. Of course you’re right. To use Peter’s terminology, if you’re a
potentialist about height, but an actualist about width, then CH is determinate in the usual
way. I was speaking of Sy’s potentialism, which I think is intended to be potentialist about
both height and width.

You both say that if one hangs onto width actualism then “CH is determinate in the usual way”. I have no idea what “the usual way” means; can you tell me?

But nothing you have said suggests that there is any problem at all with determining the CH as a radical potentialist. Again:

… solving the CH via the HP would amount to verifying that the pictures of V which optimally exhibit the Maximality feature of the set concept all satisfy CH or all satisfy its negation. I do consider this to be discovering something about V. But I readily agree that it is not the “ordinary way people think of that project”.

And in more detail:

We have many pictures of V. Through a process of comparison we isolate those pictures which best exhibit the feature of Maximality, the “optimal” pictures. Then we have three possibilities:

a. Does CH hold in all of the optimal pictures?
b. Does CH fail in all of the optimal pictures?
c. Otherwise

In Case a, we have inferred CH from Maximality, in Case b we have inferrred -CH from Maximality and in Case c we come to no definitive conclusion about CH on the basis of Maximality.

OK, maybe this is not the “usual way” (whatever that is), but don’t you acknowledge that this is a programme that could resolve CH using Maximality?

I also owe Pen an answer to:

… at some point could you give one explicit HP-generated mathematical principle that you endorse, and explain its significance to us philosophers?

As I tried to explain to Peter, it is too soon to “endorse” any mathematical principle that comes out of the HP! The programme generates different possible mathematical consequences of Maximality, mirroring different aspects of Maximality. For example, the IMH is a way of handling width maximality and $\#$-generation a way of handling height maximality. Each has its interesting mathematical consequences. But they contradict each other! The aim of the programme is to generate the entire spectrum of possible ways of formulating the different aspects of Maximality, analysing them, comparing them, unifying them, … until the picture converges on an optimal Maximality criterion. Then we can talk about what to “endorse”. I conjecture that the negation of CH will be a consequence, but it is too soon to make that claim.

The IMH is significant for many reasons. First, it refutes the claim that “Maximality in width” implies the existence of large cardinals; indeed the IMH is the most natural formulation of “Maximality in width” and it refutes the existence of large cardinals! Second, it illustrates how one can legitimately talk about “arbitrary thickenings” of V in discussions of Maximality, without tying one hands to the restrictive notion of forcing extension. Third, as discussed at length in my papers with Tatiana, it inspires a re-think of the role of large cardinals in set theory, explaining this in terms of their existence in inner models as opposed to their existence in V.

But the HP has moved beyond the IMH to other criteria like $\#$-generation, unreachability and (possibly) omniscience, together with different ways of unifying these criteria into new “synthesised” criteria. It is an ongoing study with a lot of math behind it (yes Pen, “good set theory” that people can care about!) but this study is still in its infancy. I can’t come to any definitive conclusions yet, sorry to disappoint. But I’ll keep you posted.

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Thanks, Geoffrey. Of course you’re right. To use Peter’s terminology, if you’re a potentialist about height, but an actualist about width, then CH is determinate in the usual way. I was speaking of Sy’s potentialism, which I think is intended to be potentialist about both height and width.

All best,
Pen

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

you are ignoring the obvious; I explain.

this discussion is not about the philosophy of mathematics! It is about the concept of set, i.e. part of the philosophy of set theory! The obvious point you ignore is that there is a philosophy of set theory, independent of what is done in mathematics

I am claiming that the HP is a legitimate contribution to the philosophy of the set-concept, part of the philosophy of set theory. Of course there is more to the philosophy of set theory than just the philosophy of the set-concept, namely the philosophy of what is done in set theory as a branch of mathematics, but that is a different thing.

Well, as it happens, I do take set theory to be part of mathematics — and the philosophy of set theory to be part of the philosophy of mathematics.  You insist that you’re doing a kind of philosophy of set theory that isn’t part of philosophy of mathematics, and I reply that you must be attempting a short of pure philosophy, like the study of radical skepticism or a priori metaphysics.

OK!  So we have an answer to the question Peter has been asking:  are you an actualist or a potentialist?  Answer:  a potentialist.

Finally. I thought that this was already implicit (although not explicit) when you quoted Tatiana and I in your first e-mail to this group:

“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).”

It is hard for me to imagine the above in conjunction with an actualist position.

I misunderstood this passage from your BSL paper.  There’s often a kind of translation problem when philosophical terms get used and we professional philosophers assume they’re intended in more or less the same sense as we use them.  In this case, the troublesome term is ‘ontological’.

I took you to be denying that the subject matter of set theory is an objectively existing abstract realm, not to be saying anything at all about how we should best think of V (as completed or potential) while doing set theory.

So now we have to understand better what you are out to do, which means understanding better what these ‘thickenings’ are supposed to be.

I realize that you and Hugh have been discussing this at a very high level, but at some point could you give one explicit HP-generated mathematical principle that you endorse, and explain its significance to us philosophers?

All best,
Pen

# Re: Paper and slides on indefiniteness of CH

Pen,

You wrote to Sy,

“OK!  So we have an answer to the question Peter has been asking:  are you an actualist or a potentialist?  Answer:  a potentialist.  So you aren’t really out to settle CH in the ordinary way people think of that project; you aren’t out to discover new things about V (because there is no V).”

But here, ‘V’ can be replaced by “any standard initial segment of a (not “the”!) cumulative hierarchy of sets with full power sets up to and including rank $\omega+2$, for CH is determinate (semantically) there. And to accommodate the proof theory of “the ordinary way people think of that project”, one can respect that by replacing “new things about V” with “new things about any standard (well-founded with full power sets) model of T, where T is the relevant extension of ZFC for the result in question. This seems to be at least one way in which a potentialist can respect mathematical practice of higher set theory.

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

But haven’t you softened your position as well? You went from:

“I don’t see why anyone has reason to sign onto this project, or to care about it one way or the other, unless it reveals some mathematical interest despite you.”

to:

“my strongest protest — why should mathematicians care about a program that doesn’t even aim to produce any good mathematics? — is no longer valid.  And my interest revives …”

You changed the word “anyone” to “mathematicians”! Of course I agree that mathematicians shouldn’t care unless there is “good math” coming out! My disappointment was with your earlier version, which implied that not even *philosophers* should care unless there is “good math” coming out! That I took as a denunciation of the HP on purely philosophical grounds.

As it happens, I had typed ‘anyone’ in my recent message, then changed it to ‘mathematicians’ in hope of sharpening the point!  Anyway, your suggestion is that some one might be interested in the program as philosophy, not as mathematics.  Well, I suppose that’s possible.  But it looks to me as if the kind of philosophy in question would have to be a kind of pure philosophy, like the study of radical skepticism or analytic metaphysics.  It wouldn’t be philosophy of mathematics, because philosophy of mathematics is about what’s doneas mathematics.  To make an extreme comparison (swiped from Wittgenstein), some one might be interested in the program because it generates such attractive arrays of symbols, so attractive that they make nice wallpaper patterns.  I was assuming your program is a mathematical one, not an exercise in pure philosophy or interior decoration.

So here I was intrigued by your exchange with Peter.  But I blush to admit that I still haven’t grasped your answer to the flat-footed question:  if there is no actual V, in width or height, what are we asking about when we ask about CH?

Finally you ask an easy question! (Your other questions were all very challenging.)

Answer to this question: We have many pictures of V. Through a process of comparison we isolate those pictures which best exhibit the feature of Maximality, the “optimal” pictures. Then we have 3 possibilities:

a. Does CH hold in all of the optimal pictures?
b. Does CH fail in all of the optimal pictures?
c. Otherwise

In Case a, we have inferred CH from Maximality, in Case b we have inferrred -CH from Maximality and in Case c we come to no definitive conclusion about CH on the basis of Maximality.

OK!  So we have an answer to the question Peter has been asking:  are you an actualist or a potentialist?  Answer:  a potentialist.  So you aren’t really out to settle CH in the ordinary way people think of that project; you aren’t out to discover new things about V (because there is no V).

So now we have to understand better what you are out to do, which means understanding better what these ‘thickenings’ are supposed to be.  Is this what Harvey has been trying to pin down?

(Let me again suggest that it might be worth reconstituting the list so that people can remove themselves in private.)

I still think that this discussion will soon “fizzle out” but I thought that before and was proved wrong. So I think you have a good idea. Do you know how to do that? I am not clever with “e-mail news groups”; can someone volunteer to set things up so that each of us can subscribe or unsubscribe freely? As it currently stands, someone will write to me occasionally and ask to be left out, but then I have to police the situation to make sure that future mails don’t go to that person; not an ideal setup.

This discussion will eventually fizzle out, but it occurs to me that there may be (and indeed have been) other questions of interest to this group that could benefit from discussion in this forum.  But maybe not.

In any case, no, I don’t know how to set these things up.

All best,
Pen

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

Fair question! Why should I say there are still things to discuss when I’ve also said our discussion had reached a natural conclusion? Let me try to explain (not excuse) myself…

1. I wasn’t so much saying that there is more for you and me to discuss. It looks as if there are several further topics that are ripe for discussion by others, and that are being discussed — topics of interest to many of us as bystanders.
2. In any case, it does seem to me that the discussion between you and me about whether the HP gives access to the uniquely correct type of set-theoretic truth has been resolved. In your recent summary, you acknowledged that this is one kind of set-theoretic truth, but there are other legitimate forms. (The part of our discussion that may not be resolved is whether your type of set-theoretic truth is intended to be intrinsic to the familiar iterative conception or to some new conception of your own (see (4) below).)
3. I also thought I’d detected a softening of your strong stance against the relevance of the quality of the mathematics the HP produces. Maybe this is just projection, but that’s how I read this:

I now acknowledge two goals for the programme. One is as before, to gain new understanding of set-theoretic truth based on maximality as an intrinsic feature of the set-concept. The other is to produce a new genre of set-theoretic principles which are demanding of new set-theoretic methods and which interact in interesting ways with current themes in the practice of set theory. (9/18)

and this:

Finally, I acknowledge that the legitimacy of this approach to discovering new first-order statements based on intrinsic evidence hinges on the way this approach treats Maximality via depicted universes and formulates mathematical criteria for Maximality. If these are regarded as illegitimate then I still maintain that the programme is worthwhile for purely mathematical reasons, as it generates set-theoretic properties that would not have otherwise been explored and which demand the development of new set-theoretic methods as well as new uses for known methods. (9/20)

If I’m right about this softening, then my strongest protest — why should mathematicians care about a program that doesn’t even aim to produce any good mathematics? — is no longer valid. And my interest revives, as I said, though I’m not in a position to contribute much to that discussion myself!

1. About that issue left dangling in (1). I was taking it to have been resolved that you were no longer claiming that your intrinsic justification are based in the familiar iterative conception, but I was ignoring this from you:

No, this is not correct … The distinction between actualism and potentialism is not about two different concepts of set. They are just different ways of treating the maximal iterative conception. Radical potentialism is (naturally) a form of potentialism, nothing more.

You’re quite right that I should have acknowledged and responded to this. My apologies.

I don’t think potentialism is ‘implicit in the iterative conception’, but it is one natural way of thinking, so it would be of interest if the HP could be based on some such picture (with potentialism for width as well as height). Now that the quality of the mathematical consequences is in play (I hope!), it’s worth trying to see if a workable intuitive picture to generate those consequences can be formulated. So here I was intrigued by your exchange with Peter. But I blush to admit that I still haven’t grasped your answer to the flat-footed question: if there is no actual V, in width or height, what are we asking about when we ask about CH?

(Let me again suggest that it might be worth reconstituting the list so that people can remove themselves in private.)

All best,
Pen