Re: Paper and slides on indefiniteness of CH

Dear Claudio and Sy,

1. I was trying to figure out whether the HP aims to come up with a single, accepted theory. I asked whether:

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 replied:

HP implies an irreversible departing from the idea of finding a single, unified body of set-theoretic truths.

Sy replied:

“Unify” plays a huge role in the Hyperuniverse analysis. I called it “synthesis” before. It is only with “Unification” that one gets convergence towards a single theory of truth in the HP.

2. I was trying to figure out whether the hyperuniverse a collection of ctms inside V:

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.

Sy replied:

Yes, but what is new is to use a multiverse as a tool to gain knowledge about V.

Claudio replied:

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’m at a loss.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

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 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?

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?

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

I very much appreciate your efforts, Claudio but the picture still isn’t clear to me. A simple, readily understandable intuitive picture can be an immensely fruitful tool, as the iterative conception has amply demonstrated, but this one, the intuitive picture behind the HP, continues to elude me.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I think it’s fair to say that contemporary set theory also has the goal of resolving CH somehow.

No, Type 1 considerations (ST as a branch of math) are not concerned with resolving CH, that is just something that a handful of set-theorists talk about. The rest are busy developing set theory, independent of philosophical concerns. Both Hugh and I do lots of ST for the sake of the development of ST, without thinking about this philosophical stuff. Philosophers naturally only see a small fraction of what is going on in ST, for the simple reason that 90% of what’s going on does not appear to have much philosophical significance (e.g. forcing axioms).

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.

Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously.

Surely doing serious set-theoretic mathematics with the hope of resolving CH isn’t a mere ‘philosophical discussion’!

In any case, for the record, only the foundational goal figured in my case for the methodological principles of maximize and unify. The goal of resolving CH was included to illustrate that I wasn’t at all claiming that this is the only goal of set theory.  Your further examples will serve that purpose just as well:

The goals I’m aware of that ST-ists seem to really care about are much more mathematical and specific, such as a thorough understanding of what can be done with the forcing method.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

For present purposes, what matters is that set theory has, as one of its goals, the kind of thing Zermelo identifies. This is part of the goal of providing the sort of foundation that Claudio and I were talking about (a kind of certification and a shared arena).

I interpreted the Zermelo quote to mean that ST’s task is to provide a useful foundation for mathematics through a mathematical clarification of ‘number’, ‘order’ and ‘function’, Is that correct? This goal is then Type 2, i.e. concerned with ST’s role as a foundation for mathematics.

Yes, in your classification (if I’m remembering it correctly), this would be a Type 2 goal, that is, a goal having to do with the relations of set theory to the rest of mathematics. (My recollection is that a Type 1 goal is a goal within set theory itself, as a branch of mathematics, and Type 3 is the goal of spelling out the concept of set, regardless of its relations to mathematics of either sort, as a matter of pure philosophy.)

I don’t see that it’s being Type 2 in any way disqualifies it as a goal of set theory, with attendant methodological consequences. It’s true that set theory has been so successful in this role and is now so entrenched that it’s become nearly invisible, and neither set theorists nor mathematicians generally give it much thought anymore, but it was explicit early on and it remains in force today (as that recent quotation from Voevodsky indicates).

I think it’s fair to say that contemporary set theory also has the goal of resolving CH somehow.

No, Type 1 considerations (ST as a branch of math) are not concerned with resolving CH, that is just something that a handful of set-theorists talk about. The rest are busy developing set theory, independent of philosophical concerns. Both Hugh and I do lots of ST for the sake of the development of ST, without thinking about this philosophical stuff. Philosophers naturally only see a small fraction of what is going on in ST, for the simple reason that 90% of what’s going on does not appear to have much philosophical significance (e.g. forcing axioms).

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject. To say this is in no sense to deny that you and Hugh and other set theorists have many other goals besides. (Incidentally, I don’t see why you think forcing axioms are of no interest to philosophers, but let that pass.)

There are others.

Such as? I think that just as the judgments about “good” or “deep” ST must be left to the set-theorists, perhaps with a little help from the philosophers, so must judgments about “the goals of set theory”.

I haven’t attempted to list other goals because, as a philosopher, I’m not well-placed to do so (as you point out).

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

With Claudio’s permission I will reply to your latest message (and also take the opportunity to make some further remarks).

I took Claudio to be proposing an interpretation of the HP program different from yours, one that’s an explicit multiverse program. No?  If not, then I guess we return to the place where I still don’t quite grasp what the HPer is up to.  The universer is studying V. The HPer is studying …

And indeed in the HP we have a “single-universe view” (the “ideal V”) which is analysed via a multiverse construct (the Hyperuniverse). It is a hybrid, and that may have caused confusion, for which I apologise. Another added ingredient is the consideration of “thickenings” in quotes, in addition to the lengthenings of the height potentialist. Both are reduced to the mathematical study of the Hyperuniverse (the reduction to the Hyperuniverse”, a bit more about that below).

… an ‘ideal V’?  What is that?

The Hyperuniverse is also an ideal construct, it is defined within the ideal V. Conversely, truth in the ideal V is clarified through an analysis of its associated (ideal) Hyperuniverse. There is a dynamic interplay, a “dualism” as Claudio explained. There is no thick ontology for the ideal Hyperuniverse just as there is none for the ideal V.

The hyperuniverse is also ideal:  the collection of ctms inside ‘ideal V’, right?   What work is ‘ideal’ doing here?  Why not just V and the collection of ctms in V?

1. I can acknowledge that a single-universe view (with no multiverse considerations) is the most obvious view for the foundations of Set Theory. But I don’t think it has better prospects than a multiverse view (or “hybrid” view as in the HP), since it has been quite obvious for a long time that independence is extensive in ST (Set Theory) and in my view this makes any single universe view that exceeds TR (Thin Realism) quite useless. What good is a single universe if we don’t know what form it takes and can only imagine a wide range of possibilities for that form?

Sure, the ordinary universer has a long way to go in figuring out the features of V.  But the aim remains a single theory of sets, which is the most natural way of satisfying the foundational goal. My point is just that this goal generates a methodological maxim I once called ‘Unifiy':  go for one accepted theory of sets if at all possible.  (It might turn out to be impossible, given other goals, but it doesn’t seem to me that that has happened yet.)

What Claudio and I established was that, on his understanding of the HP, the HP has embraced the methodological maxim of Unify — not in the universer’s way, in a different way — so I was ready to move on to the next question:  how to understand the ontology of the HP.

As for the Thin Realist, she’s beholden to those extrinsic payoffs. Until they abound, she sticks to her simple universe understanding.

“HP identifies a core model-theoretic construct, that is, c.t.m., as the only constituent of multiverse [thin] 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.”

I hope that this point has finally come across. Pen, Hugh and Harvey have each asked how I make the reduction to ctm’s and I can only ask them to please re-read what I have said about this at great length and through great effort in this exchange. The HP analyses Maximality through the study of certain very particular properties of ctm’s, but unlike what both Hugh and Harvey have tried to claim, it is much more than the study of ctm’s. The ctm’s are just the mathematical tool needed.

I realize this is frustrating for you, but what you’ve said so far about the ‘reduction to ctms’ hasn’t yet produced understanding (for me) or conviction (for Hugh or Harvey).  I’m still stuck, as above, on figuring out where the ctms live, what makes everything ‘ideal’, and so on. They have other concerns.

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I agree completely with Pen, but would like a clarification of just one thing: What do you mean by “the goals of set theory”? You have used that phrase before and I think it could very easily be misinterpreted. Do you just mean what you attribute to Zermelo above, or something more?

For present purposes, what matters is that set theory has, as one of its goals, the kind of thing Zermelo identifies.  This is part of the goal of providing the sort of foundation that Claudio and I were talking about (a kind of certification and a shared arena).  I think it’s fair to say that contemporary set theory also has the goal of resolving CH somehow.  There are others.

You’re probably wondering:  what makes a goal legitimate?  Could we just set up any old goal and justify whatever we want to do that way?   Perhaps my answer is predictable by now:   goals are legitimate insofar as they generate ‘good’ (‘deep’) mathematics.  We have pretty good evidence that the sort of foundational goal in play here has been immensely productive.  Harvey thinks the goal of resolving CH is unlikely to be legitimate in this sense, but others (obviously) disagree.  Time will tell.
All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Claudio,

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.

OK. I don’t think we need to resolve the terminological question whether an ordinary potentialist (one who says ‘the universe is indefinitely extendible’, or something like that) counts as a multiverser. I take it your multiverser isn’t the ordinary potentialist; your multiverser has a ‘distinctive framework’. That’s all I wanted to be sure of.

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).

I’m sorry to be dim about this, but where do these ctms live? The universer can study the ctms in V, but your multiverser can’t mean that (because for him there is no V).

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Claudio,

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

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I believe that the usual discussion of “extrinsic reasons for set theory” is deeply flawed and represents a lack of acknowledgement of many key features in mathematics generally, and many key attitudes of mathematicians. Specifically, there is a kind of RESTRICT! or DON’T MAXIMIZE! that is going on pervasively – at some important level – all through mathematics and with mathematicians generally. The “extrinsic/maximize” proponents (including Pen and Peter) surely have a defense against this loud attack. They can try to draw a distinction between what mathematics and mathematicians want to RESTRICT! and what would amount to restriction in set theory such as restricting to L. And then the debate goes on, with deeper issues as to the very point of what mathematics is and what higher set theory is, taking front and center. I am up for this debate, but I have as of yet no indication that Pen and Peter and others are up for this debate.

I’m perfectly happy to engage in this debate, Harvey. As you suggest, my position is that different parts of mathematics undertake different jobs, pursue different goals, and this is as it should be. (E.g., number theorists want to understand the particular structure of the standard model of arithmetic; algebraists want to isolate mathematically important features shared by many different structures.) To take an important and well-studied example: many mathematicians, for very good reasons, want to restrict their attention to continuous functions, or differentiable functions, or smooth functions, or whatever, but when it comes to setting out the notion of function itself, it turns out to be best to use the most general, least restrictive concept. One of set theory’s jobs is to formulate those most general notions. (Zermelo writes, ‘Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions ‘number’, ‘order’ and ‘function’, taking them in their pristine, simple form.’) For that matter even set theorists study narrower classes of functions, e.g., in descriptive set theory, but it’s still best that the underlying notion be unfettered. (I hope I don’t have to make that case. It goes back to Euler, D’Alembert and the vibrating string.)

So I don’t see any conflict between saying, on the one hand, that set theory, given its goals, should be unrestricted, and on the other, that it makes good sense for many mathematics to restrict what they’re studying. (This might be called a kind of maximization principle for set theory, but one that traces to the goals of set theory and the practice of mathematics — not one that’s derived as an intrinsic consequence of the iterative conception.)

All best,
Pen

Re: Paper and slides on indefiniteness of CH

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’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 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?

All best,
Pen