# Re: Paper and slides on indefiniteness of CH

That doesn’t answer the question: If you assert that we will know the truth value of CH, how do you account for the fact that we have many different forms of set-theoretic practice? Do you really think that one form (Ultimate L perhaps) will “have virtues that will swamp all the others”, as Pen suggested?

Look, as I have stated repeatedly I see the subject of the model theory of ctm’s as separate from the study of V (but this is not to say that theorems in the mathematical study of ctm’s cannot have significant consequences for the study of V). I see nothing wrong with this view or the view that the practice you cite is really in the subject of ctm’s, however it is presented.

??? My question has nothing to do with ctm’s! It has nothing to do with the HP either (which I repeat can proceed perfectly well without discussing ctm’s anyway). I was referring to the many different forms of set-theoretic practice which disagree with each other on basic questions like CH. How do you assign a truth value to CH in light of this fact?

For your second question, If the tests are passed, then yes I do think that V = Ulitmate-L will “swamp all the others” but only in regard to a conception of V, not with regard to the mathematics of ctm’s. There are a number of conjectures already which I think would argue for this. But we shall see (hopefully sooner rather than later).

Here come the irrelevant ctm’s again. But you do say that V = Ultimate L will “swamp all the others”, so perhaps that is your answer to my question. Now do you really believe that? You suggested that Forcing Axioms can somehow be “part of the picture” even under V = Ultimate L, but that surely doesn’t mean that Forcing Axioms are false and Ultimate L is true.

Pen and Peter, can you please help here? Pen hit me very hard for developing what could be regarded as “Sy’s personal theory of truth” and it seems to me that we now have “Hugh’s personal theory of truth”, i.e., when Hugh develops a powerful piece of set theory he wants to declare it as “true” and wants us all to believe that. This goes far beyond Thin Realism, it goes to what Hugh calls a “conception of V” which far exceeds what you can read off from set-theoretic practice in its many different forms. Another example of this is Hugh’s claim that large cardinal existence is justified by large cardinal consistency; what notion of “truth” is this, if not “Hugh’s personal theory of truth”?

Pen’s Thin Realism provides a grounding for Type 1 truth. Mathematical practice outside of set theory provides a grounding for Type 2 truth. Out intuitions about the maximality of V in height and width provide a grounding for Type 3 truth. How is Hugh’s personal theory of truth grounded?

Look: There is a rich theory about the projective sets in the context of not-PD (you yourself have proved difficult theorems in this area). There are a number of questions which remain open about the projective sets in the context of not-PD which seem very interesting and extremely difficult. But this does not argue against PD. PD is true.

I want to know what you mean when you say “PD is true”. Is it true because you want it to be true? Is it true because ALL forms of good set theory imply PD? I have already challenged, in my view successfully, the claim that all sufficiently strong natural theories imply it; so what is the basis for saying that PD is true?

If the Ultimate-L Conjecture is false then for me it is “back to square one” and I have no idea about an resolution to CH.

I see no virtue in “back to square one” conjectures. In the HP the whole point is to put out maximality criteria and test them; it is foolish to make conjectures without doing the mathematics. Why should your programme be required to make “make or break” conjectures, and what is so attractive about that? As I understand the way Pen would put it, it all comes down to “good set theory” for your programme, and for that we need only see what comes out of your programme and not subject it to “death-defying” tests.

One more question at this point: Suppose that Jack had succeeded in proving in ZFC that $0^\#$ does not exist. Would you infer from this that V = L is true? On what grounds? Your V = Ultimate L programme (apologies if I misunderstand it) sounds very much like saying that Ultimate L is provably close to V so we might as well just take V = Ultimate L to be true. If I haven’t misunderstood then I find this very dubious indeed. As Pen would say, axioms which restrict set-existence are never a good idea.

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Pen,

Please understand that when I wrote that I was being subjected to a “valuable ‘grilling'” by you the emphasis is on the word “valuable”. I appreciate this very much.

Prescript: Dear esteemed colleagues, are you tired of this discussion? If so, please send me a short e-mail to that effect and I will remove your name from my future mailings.

Thanks for this, Sy.  I’ve been uncomfortable imposing this discussion on so many innocent bystanders. (Maybe it would be more humane to go for opt-in rather than opt-out.)

For now I’ll leave it as opt-out as I think we may be approaching the end of this very stimulating discussion.

On truth –

3.  The HP and the current practice chug along side-by-side; they’re just ‘investigating different notions of truth’.

Yes, but in 3 I wouldn’t say “the current practice” but rather “the practice-based investigation of truth”. I think that doing set theory and investigating set-theoretic truth based on practice are different things.

Now you’ve got me confused.  Here’s the original question you raised to Sol:

So what is the relationship between truth and practice? If there are compelling arguments that the continuum is large and measurable cardinals exist only in inner models but not in V will this or should this have an effect on the development of set theory? Conversely, should the very same compelling arguments be rejected because their consequences appear to be in conflict with current set-theoretic practice?

This is a question about the relations between the HP and current practice, isn’t it?  And here again are the three options I thought we’d settled on:

1. The current practice has ‘veto power’. That is, if the HP followers come up with a principle that contradicts the existence of MCs, they say, ‘oops, back to the drawing board’.
2. The HP has veto power.  That is, if the HP followers come up with a principle that contradicts the existence of MCs, they say to the community, ‘terribly sorry, but you’ll have to give that up’.
3. The HP and the current practice chug along side-by-side; they’re just ‘investigating different notions of truth’.

How did ‘the current practice’ drop out and ‘a practice-based investigation of truth’ slip in?  (If you want to insist on the later, I’m going to have to ask what it is and who’s doing it.  What I’ve been advocating on my own behalf is a move away from ‘truth’ as the relevant notion in any of this.)

You are right, I confused two different things:

1. The relationship between truth investigation and practice.
2. The relationship between two different truth investigations, one “practice-based” and the other not.

I didn’t realise that you view Hugh’s approach to truth as part of practice! I should have caught this earlier when you said:

3.  The HP and the current practice chug along side-by-side; they’re just ‘investigating different notions of truth’.

…The idea is that the HP and Hugh’s project are both fine, legitimate, worthy, whatever — they’re aiming at different sorts of truth — and you’re just pursuing yours.

I misread this to mean that Hugh and I are each pursuing truth, his more practice-based than mine, but neither approach is to actually be viewed as part of practice.

But this is all moot anyway, given what you say below and my response to it.

Hugh’s project is a trickier issue as it raises the following question: When is mathematics relevant to the investigation of truth and when is it just good mathematics? You may feel that this question doesn’t need answering, and we should welcome any investigation which a mathematician reassures us is relevant to the investigation of truth.

What I think is that doing good mathematics is the goal of mathematical practice, in set theory and elsewhere.

No debate there!

A person can call this the search for truth if he likes (as Hugh and my other figure, the Thin Realist, both do), but if so (I say) then the grounding of this truth is in the goodness of the mathematics.  (So I guess you might say that my other figure, the Thin Realist, is in pursuit of ‘practice-based truth’, but if so, she conducts this pursuit just by doing set theory.)

Oh. My. God. I see that you have no concept of truth independent of practice anyway! To be fair I was forewarned; with regard to the HP and Hugh’s project you wrote:

I don’t think the use of ‘truth’ or ‘different notions of truth’ here is appropriate or helpful. In my mind, we’re just considering two different set-theoretic projects.

And when I referred to the misuse of good mathematics to make unjustified claims about truth you wrote:

…this is a pointless squabble over window-dressing, over an empty honorific ‘true’. The real debate is between the two set-theoretic projects and their fruits.

Also:

My take is that your so-called ‘intrinsic justifications’ are actually functioning as heuristics that are helping you get to some interesting ideas, but that the justification will come from the extrinsic success of those ideas.

I feel even dumber now, because I have several times referenced a poignant quote of yours in my lectures:

What, then, does naturalism suggest for the case of the CH? First, that we needn’t concern ourselves with whether or not the CH has a determinate truth value … Instead, we need to assess the prospects of finding a new axiom that is well-suited to the goals of set theory and also settles CH.’

But what are “the goals of set theory”? Who chooses those? I can imagine set-theorists looking back and aseessing the most important achievements of the past, but surely they will be wrong when they guess about the most important achievements of the future. If some good mathematical work in set theory is not well-suited to the “goals of set theory” is it to be “jettisoned”, exactly what you objected to earlier?

Can you imagine at all that there might be some features of the set concept that are “intrinsic” and therefore of lasting importance, independent of the mathematical development of the subject? Does any statement of set theory at all have a truth value? My use of “intrinsic” is intended to extend its use to justify the axioms of ZFC. In your view are the axioms of ZFC true, or should we not be concerned about that because they are “well-suited to the goals of set theory” (whatever those are)?

I now have the impression that you are not really interested in truth at all, perhaps only in what others have to say about it!

So please tell me:

1. Do you see any difference between “standard” work in set theory (i.e. what set-theorists typically do on a daily basis) and what I am trying to do in the HP? Or is the HP just another example of set-theoretic practice where “truth” is being used as an excuse to develop some new mathematics?
2. Do you see any difference between “Hugh’s project” (I won’t be more specific because I think you are happy with that phrase) and “standard” work in set theory? Or is Hugh also just using “the search to resolve CH” just as an excuse to develop some new mathematics?
3. When Hugh says that “PD is true” how do you interpret that? Could you ever believe him?
4. This all began with Sol’s suggestion that CH is not a definite logical problem. Does this suggestion have any meaning for you, and if so, what? In your view, could Sol’s suggestion ever be refuted?

If you feel that there is only set-theoretic practice and nothing else then for you most of what I have said in our discussion is simply meaningless. I wonder why you put so much time into it (in any case I am very grateful for that).

But surely if the conclusions of such an investigation are interesting, such as a solution to CH, we would want to verify that the arguments which led there were well-grounded philosophically and that there were not mathematical choices made along the way just to make things work.

I don’t see anything at all wrong with ‘mathematical choices made along the way just to make things work’ — or as I might phrase it more generously, ‘mathematical choices made along the way in order to uncover good mathematics’.  This is how we form the various central concepts of mathematics (e.g., group) and I would say it’s how we chose (or ought to choose) new set-theoretic axioms.

Specifically with regard to Hugh’s projects, it is worrisome that huge mathematical prerequisites are required to understand even the statements of, let alone the motivation for, what Hugh presents as his key conjectures. As a mathematician I find this difficult, it must be even more difficult for the philosopher.

This is an entirely different matter.  Hugh’s mathematics is very difficult, largely inaccessible to many of us. This makes it hard for the community to come to informed judgments about its ‘goodness’ or ‘depth’.  But there’s no reason at this point not to applaud his efforts, and to wait for the inevitable progress of mathematics to better digest what he’s doing and for the inevitable judgment of history to determine its value.

Fine, but as I understand it in your view there is no possibility whatsoever that Hugh or I will “solve the continuum problem” during our lifetimes because for you the truth of CH is either just meaningless or could only be resolved by developing mathematics of time-tested importance that proves or refutes it, and given that Hugh and I are about 60 years old there just ain’t enough time.

On concepts  –

First question:  Is this your personal picture or one you share with others?

I don’t know, but maybe I have persuaded some subset of Carolin Antos, Tatiana Arrigoni, Radek Honzik and Claudio Ternullo (HP collaborators) to have the same picture; we could ask them.

Why do you ask? Unless someone can refute my picture then I’m willing to be the only “weirdo” who has it.

Now here you surprise me, Sy!  Most people who go in for conceptualism of some brand or other take the relevant concepts to be shared by the community — ‘we’re all out to investigate the concept of set (or set-theoretic universe)’, or something like that.  I thought you might hesitate to claim that set-theoretic truth is determined by a picture in Sy Friedman’s head (though others are welcome to be instructed by him on its contours).  No?

You got this wrong. I indeed expect that others have similar pictures in their heads but can’t assume that they have the same picture. There is Sy’s picture but also Carolin’s picture, Tatiana’s picture … Set-theoretic truth is indeed about what is common to these pictures after an exchange of ideas. My ideas result from intrinsic considerations as refined through the HP.

Here you seem to say the same thing:

(Does the phrase ‘refinement of what we take as true’ trouble you at all? Don’t ‘true’ and ‘what we take as true’ at least potentially diverge?)

I have no concept of “true” other than “what I take as true based on my picture of V”, which is constantly being refined.

What’s true in set theory is what Sy Friedman takes to be true based on his picture of V, which he constantly refines as he sees fit?

No, I am influenced by the exchange of ideas with others. But what influences me the most are ideas that are intrinsically-based as in the HP.

On my second question –

In the second kind of case, what grounds those refinements?  It can’t be that they’re faithful to the concepts, so what is it?

They are faithful to the motivating intrinsic philosophical principles such as maximality.

But those motivating intrinsic principles are supposed to be implicit in the concept, aren’t they?  If not, where do they come from?

In general –

At this point, it sounds as if the HP works like this.  SF has a picture, he refines the picture.  He eschews any extrinsic standard (now removing even the clause about principles being tested ‘by their ability to settle independent questions’).  He’s willing to follow this notion of set-theoretic truth even if the mathematics generated is trivial and boring (‘a risk I have to take’).   He assures us that ‘the mathematics is secondary’.

Now the question of why we should care becomes acute.  Why should someone want to learn your concept and help develop it if it doesn’t produce good mathematics?  If the goal isn’t to produce good mathematics, if it’s not to be judged by shared mathematical standards, in what sense is it even mathematics?

You lose me here. We know what mathematics is when we see it. It is obvious that part of the HP work is mathematics. And part of it has to do with formulating intrinsically-based criteria for the choice of preferred universes, which I regard as a mix of mathematics and philosophy. The goal is to gain a better understanding of truth in set theory and as I said before. that is a worthy goal, whether or not good mathematics comes out. It seems obvious that any philosopher who assigns meaning to set-theoretic truth would find this a worthy goal. I have to conclude that you do not assign meaning to set-theoretic truth. How unexpected, given that you are the philosopher and I am the mathematician!

For me the practical point is this:  even if you don’t give a hoot about extrinsic success, it doesn’t follow that you aren’t, in fact, generating some good mathematics, despite yourself so to speak.  It doesn’t matter if Newton thought he was writing down the thoughts in the mind of God,

Doesn’t matter to you, but mattered to him. Are you deciding for Newton what was important about his work?

what he actually did was science of the highest order.  This is what I meant a while back by saying I thought that your analysis of your concepts was actually functioning as a sort of heuristic for generating ideas that would then be judged extrinsically.

No objection there, I don’t object to extrinsic judgment but I am also not motivated by it. Perhaps my non-HP mathematics will be better than my HP-mathematics and that’s OK.

But of course this means I don’t see how you can lay special claim to some privileged notion of set-theoretic truth.

It comes down to the difference intrinsically-based and practice-based approaches to truth. It seems that you regard the former (and even the latter!) as just excuses to discover new mathematics, but you are wrong.

I have no objection to other investigations of set-theoretic truth, but I do think that we need philosophers to play a role in deciding what qualifies as an investigation of truth and what is just good mathematics.I can tell you as a mathematician that it is not hard to deceive oneself into thinking that one’s exciting new results have important implications for truth in set theory. That is why we need philosophers to police the situation. Tatiana, Claudio and other philosophers have helped to keep me honest. And aren’t I being currently subjected to a valuable “grilling” by an expert in the philosophy of mathematics (you)? I think that any mathematician who claims to investigate truth should be subjected to such a “grilling”. Philosophers of mathematics: We need you!

By now it should be clear that this philosopher has no interest whatsoever in distinguishing between ‘an investigation of set-theoretic truth’ and ‘the pursuit of good mathematics’ — or for that matter in ‘policing’ anyone.  (If extrinsic considerations are the proper measure, as I claim, then these matters are to be judged on mathematical, not philosophical grounds.  While you’re right that it’s often hard for mathematicians to tell immediately what’s good and what’s not, this is no reason at all to defer to philosophers, who are much more poorly placed to make that call.)

Yes, if only practice matters then leave the quality judgments to the mathematicians, But I claim, and you disagree, that there is more to truth than simply what makes for good math, and to carry out a good investigation of truth one needs to not only know math but also to make sound philosophical judgments.

Philosophers out there: Help! Pen says you are useless for the investigation of truth, but I need you!

I also haven’t intended to ‘grill’ you, and apologize that it came across that way.

No apology please! It is a privilege for me that you have taken so much time to teach me so much about concepts, truth, the extrinsic-intrinsic distinction and more.

I have been trying to figure out precisely what your position is, and I have then pointed to some areas where the answers seem to me to be problematic. But as I’ve said before, one person’s reductio is another’s revolution!

Indeed. I do think that “one more round” would be of value to me, if you are willing (I’m very interested to know the answers to the questions I posed). But perhaps you agree that we are near the end of the discussion (for now), which as I say has been very enjoyable and instructive for me.

All the best,
Sy