Tag Archives: Concept of set

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

Re: Paper and slides on indefiniteness of CH

Dear Sy,

My apologies!  After sending my last message, I began to worry that I’d gotten you wrong, and I had.  We have on the table three possible attitudes on the relations between HP and the current practice of set theory:

  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’.

When you backed off of (1), I was so worried about (2) that I didn’t fully appreciate that you were advocating (3).  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.  (If I may for a moment insert my own views here, 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, but I’ll suppress that for the sake of discussion.)

About concepts –

OK, so what grounds your ‘notion of truth’?  (We’ll leave Hugh aside.)  Answer:  it’s faithful to intrinsic considerations, to the concepts of set and set-theoretic universe.  And what are these concepts?  You say,

I have a mental picture of sets and of the universe of sets and my concepts of set and universe coincide with these mental pictures. My claim is that one can indeed make new discoveries that refine these pictures through the Hyperuniverse Programme. With my dynamic and epistemic approach to truth there can be no presumption of a “fact of the matter” regarding truth but only a process of refinement of what we take as true through the continued exploration of intrinsically-motivated criteria for the choice of preferred pictures of the universe of sets.

First question:  Is this your personal picture or one you share with others?  Does Hugh have this picture, too, and just ignore it?

Second question:  It sounds as if you’re saying that sometimes what happens in the HP is that we make new discoveries about the relevant concepts, and sometimes we ‘refine’ those concepts in ways that aren’t predetermined by the concepts.  (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?)  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?

You will deny that anything extrinsic is involved here, that any consideration of how set theory might be fruitfully guided comes into it.  But our agreed-upon summary of your position includes the claim that HP-generated principles are tested by ‘their ability to settle independent questions’.  This is at least a minimally extrinsic criterion.  And I doubt you would want to say that all ways of settling the independent questions are equally good.  I also doubt that you’d continue to pursue a principle suggested by your HP project if it only led to trivial and boring mathematics.

About truth –

My “gripe” is when good mathematics gets promoted to the status of “discovery about truth” without adequate justification. I think that Hugh, for example, has done this. I would like philosophers to take a more active role in preventing this from happening. If there is something that I would like to discard it is not good mathematics but the misuse of good mathematics to make unjustified claims about set-theoretic truth.

This doesn’t sound quite as (3)-like as I would have expected.  It does sound as if you think you have a special right to the word ‘true’, and that Hugh hasn’t properly earned his use of it.  (Again, if you ask this philosopher to take a more active role, I’d say that 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.)

A clarification –

(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.)

That is unfair. I am not just looking for ways of generating interesting new mathematics (although that >has certainly occurred) but am motivated by the desire to come as close as I can to “the right picture >of V”. My ideas are in no need of extrinsic justification (which of course cannot be ruled out).

In that parenthetical passage, I allowed myself to step outside the project of characterizing your position and to explain how I see the situation in my own voice — as I’ve allowed myself to do a couple of times above.  I’m sorry it came across instead as an unfair reading of you.  It certainly would be unfair as a reading of you!

All best,
Pen

Re: Paper and slides on indefiniteness of CH

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. I will not be offended! This may go on for a while, as I continue to learn a lot from Pen.

Dear Pen,

On Sat, 9 Aug 2014, Penelope Maddy wrote:

Dear Sy,

My apologies!  After sending my last message, I began to worry that I’d gotten you wrong, and I had.  We have on the table three possible attitudes on the relations between HP and the current practice of set theory:

  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’.

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.

When you backed off of (1), I was so worried about (2) that I didn’t fully appreciate that you were advocating (3).  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 wasn’t thinking of Hugh but of you. Indeed I no longer feel that the HP and the practice-based investigation of truth are in conflict (although they may reach different conclusions!).

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. 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. 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

(If I may for a moment insert my own views here, 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, but I’ll suppress that for the sake of discussion.)

(Here is my matching parenthetical remark: Clearly some projects are just mathematical and others are foundational; we should not ignore the distinction. Tomorrow I will discuss a project concerning ‘cardinal characteristics of the generalised continuum’ with one of my doctoral students; this will feel very different than discussing the HP.)

About concepts –

OK, so what grounds your ‘notion of truth’?  (We’ll leave Hugh aside.) Answer:  it’s faithful to intrinsic considerations, to the concepts of set and set-theoretic universe.  And what are these concepts?  You say,

I have a mental picture of sets and of the universe of sets and my concepts of set and universe coincide with these mental pictures. My claim is that one can indeed make new discoveries that refine these pictures through the Hyperuniverse Programme. With my dynamic and epistemic approach to truth there can be no presumption of a “fact of the matter” regarding truth but only a process of refinement of what we take as true through the continued exploration of intrinsically-motivated criteria for the choice of preferred pictures of the universe of sets.

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.

Does Hugh have this picture, too, and just ignore it?

Only Hugh knows if he has this picture so he’s the one to ask!

Second question:  It sounds as if you’re saying that sometimes what happens in the HP is that we make new discoveries about the relevant concepts, and sometimes we ‘refine’ those concepts in ways that aren’t predetermined by the concepts.

Yes.

(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.

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.

You will deny that anything extrinsic is involved here, that any consideration of how set theory might be fruitfully guided comes into it. But our agreed-upon summary of your position includes the claim that HP-generated principles are tested by ‘their ability to settle independent questions’.  This is at least a minimally extrinsic criterion.

Sorry, I wasn’t being sufficiently attentive earlier. Please delete the bit “They’re further tested by their ability to settle independent questions.” Remember that I expect my continually refined picture to produce a converging set of first-order truths, as I am constantly “synthesising” the different criteria. At least at the moment I’m not worried that it will produce only boring such truths.

And I doubt you would want to say that all ways of settling the independent questions are equally good.

I’ve lost you here. My optimistic view is what I discussed with Jan Mycielski: It is a process of exploring the tree of possibilities with the aim of isolating the “ideal” branch. Of course Jan and I were discussing the tree of possible first-order theories, I am now talking about the “tree” of criteria for preferred universes. (It’s not really a tree, but let’s not get technical.)

I also doubt that you’d continue to pursue a principle suggested by your HP project if it only led to trivial and boring mathematics.

That is a risk I have to take. For example the original IMH has very powerful consequences. But when I later synthesised it with #-generation (ordinal maximality) that power was lost. Currently I am bringing omniscience into the picture and we’ll see whether the strength is restored. There may be a principle of “internal unreachability” inspired by a remark of Neil Barton at Chiemsee to add to the picture which could lead to very interesting criteria. I am greedy: at any point I want to synthesise all of these criteria together into a master criterion. I would love to get large large cardinals into the picture (and therefore Hugh’s beloved PD) but don’t yet see how to do that.

But please believe me: Even though the mathematics behind all of this is very difficult and interesting (Radek and I are developing a new type of iterated class forcing to understand omniscience, for example) the mathematics is secondary. My sole aim is to approximate as closely as possible my “ideal picture of V” based on philosophically well-motivated principles. I expect that doing so will lead to the resolution of interesting first-order problems but there is no guarantee.

About truth –

My “gripe” is when good mathematics gets promoted to the status of “discovery about truth” without adequate justification. I think that Hugh, for example, has done this. I would like philosophers to take a more active role in preventing this from happening. If there is something that I would like to discard it is not good mathematics but the misuse of good mathematics to make unjustified claims about set-theoretic truth.

This doesn’t sound quite as (3)-like as I would have expected.  It does sound as if you think you have a special right to the word ‘true’, and that Hugh hasn’t properly earned his use of it.  (Again, if you ask this philosopher to take a more active role, I’d say that 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.)

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!

A clarification –

(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.)

That is unfair. I am not just looking for ways of generating interesting new mathematics (although that >has certainly occurred) but am motivated by the desire to come as close as I can to “the right picture of V”. My ideas are in no need of extrinsic justification (which of course cannot be ruled out).

In that parenthetical passage, I allowed myself to step outside the project of characterizing your position and to explain how I see the situation in my own voice — as I’ve allowed myself to do a couple of times above. I’m sorry it came across instead as an unfair reading of you.  It certainly would be unfair as a reading of you!

OK, but I now see that you have doubts about my use of the word “intrinsic” and you feel that the approach is in need of some kind of “justification” which can only come from what you call “extrinsic success”. I understand neither of these points but they can be left for a future discussion.

Thanks again!
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I think we’ve reached the crux, but let me try one more time to summarize your position accurately:

We reject any ‘external’ truth to which we must be faithful, either in the form of a platonist ontology or some form of truth-value realism, but we also deny that the resulting ‘true-in-V’ arises strictly out of the practice (as my Arealist would have it).  One key is that ‘true-in-V’ is answerable to various intrinsic considerations.  The other key is that it’s also answerable to some set-theoretic claims, namely ZFC and the consistency of LCs.

The intrinsic constraints aren’t limited to items that are implicit in the concept of set.  One of the items present in this concept is a notion of maximality.  The new intrinsic considerations arise when we begin to consider, in addition, the concept of a universe of sets.  We investigate this new concept with the help of a mathematical construct, the hyperuniverse.  This analysis reveals a new notion of maximality that’s implicit in the concept of a universe of sets and that generates the schema of Logical Maximality and its various instances (and more, set aside for now).

At this point, we have ZFC+the consistency of LCs and various maximality principles.  If the consequences of the maximality principles conflict with ZFC+the consistency of LCs, they’re subject to serious question.  They’re further tested by their ability to settle independent questions.  Once we’ve settled on a principle, we use it to define ‘preferred universe’ and count as ‘true-in-V’ anything that’s true in all preferred universes.

Now two remarks (not ‘attacks’ for goodness sake!):

1.  About ‘external’ and the ‘concept':

We reject any ‘external’ truth to which we must be faithful, but we also deny that the
resulting ‘true-in-V’ arises strictly out of the practice (as my Arealist would have it).

I think that I agree but am not entirely clear about your use of the term “external truth”. For example, I remain faithful to the extrinsically-confirmed fact that large cardinal axioms are consistent. Is that part of what you mean by “external truth”? With this one exception, my concept of truth is entirely based on intrinsic evidence.

Actually, I adopted the word ‘external’ from you, Sy:  ‘No “external” constraint is imposed … such as an already existing reality to which one must be faithful’ (BSL, p. 80).  Later, in these exchanges, you also distanced yourself from truth-value realism:

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

So my concern was just that you’d need an account of what your ‘concepts’ are that doesn’t end you up with something you find uncomfortably close to these things you reject.  In response, you write:

The concept of set is clear enough in the discussion, I have not proposed any change to its usual meaning.

Maybe so, but what is a concept?  An abstract item (a property? a universal? a meaning?)  Something mental?  Just a fancy way of talking about our shared practices in using a particular word?  If we’re to understand what your intrinsic justifications come to, we have to know what grounds them, and for that we have to know what a concept is.  My guess is that your aversion to abstract ontology and truth-value realism would lead you to rule out the first.  Would you be happy with a view of concepts as some kind of mental construction, of an individual or of a group (however that would work)?  Would it be possible for these mental concepts to have features that we don’t now know about but can somehow discover?  (Can we discover how long Sherlock Holmes’s nose is?)  Wanting there to be a fact-of-the-matter we’re out to discover is what pushes many people in the direction of the first option (some kind of abstract item).  I don’t have any horse in this race — I’m a bit of a concept&meaning-phobe myself — I’m just pointing out that you need a notion of concept that does all the things you want it to do and doesn’t land you in a place you don’t like.

2.  About truth.

Given the general tenor of your position, this sounds to me like the right move for you to make:

I think that I just fell over the edge and am ready to revoke my generous offer of “veto power” to the working set-theorist. Doing so takes the thrust out of intrinsically based discoveries about truth. You are absolutely right, “veto power” would constrain the necessary freedom one needs in the investigation of intrinsically-based criteria for the choice of preferred universes.

Then I come back to my original concern about intrinsic justifications in general:

The challenge we friends of extrinsic justifications like to put to defenders of intrinsic justifications is this:  suppose some candidate principle generates a lot of deep-looking mathematics, but conflicts with intrinsically generated principles; would you really want to say ‘gee, that’s too bad, but we have to jettison that deep-looking mathematics’?

You seem perfectly prepared to say just that:

The only way to avoid that would be to hoard together a group of brilliant young set-theorists whose minds have not yet been influenced (polluted?) by set-theoretic practice, deny them access to the latest results in set theory and set them to work on the HP in isolation. From time to time somebody would have to drop by and provide them with the mathematical methods they need to create preferred universes. Then after a good amount of time we could see what conclusions they reach! LC? PD? CH? What?

Whatever they come up with wins, even if it means jettisoning what looks like deep and important mathematics.

So that’s the crux.  To me this sounds like a reductio.  To you it sounds revolutionary.  Here’s your defense:

I would like to have a notion of truth in set theory that is immune to the influence of fads, forceful personalities, available grant money, … I really am not confident that what we now consider to be important in the subject will be important in the future; I am more confident about the “stability” of Sy truth. Second, and this may appeal to you more, it is already clear that the new approach taken by the HP has generated new mathematical ideas that never would have been generated through the usual practice. Taking a practice-independent look at set-theoretic truth generates new forms of set-theoretic practice. And I do like the practice of set theory, even though I don’t want it to dictate investigations of truth! It is valuable to develop set theory in new directions.

I sympathize with your desire to know, right now, what the good math is and what the lousy math is, but the history of the subject seems to me to show that we can’t know this for sure right now, that it can take decades, or longer, for matters to sort themselves out.  Of course there are nihilists who think that there’s really no difference between good and lousy, that it’s all just fads, personalities, politics, etc., but despite the undeniable fact that factors like these are always in play, again the history of the subject makes me hopeful that we do, eventually, attain a fair view of the terrain.

As for your second point, yes, I do like it!  I have no more desire to curtail the HP program than to curtail any other promising mathematical avenue.  (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.)  My gripe only comes in when you lay claim to an intrinsic way of limiting everyone else.  In the end, of course, I hope that whatever good comes out of your program and out of Hugh’s program and out of other programs, can be combined into one overarching subject it seems natural to continue to call ‘set theory’, but if not, well, we’ll face that when it comes.  But here’s another of my wagers:  however we do it, we won’t decide to throw out any good mathematics.

All best,
Pen

PS:  You asked about my notion of truth.  I haven’t been out to expound my own position here; I only threw in the Arealist because Sol suggested that something of mine might help clarify your views.  For what it’s worth, my Arealist doesn’t think that truth is what we’re after in doing set theory; rather, we’re doing our best to devise an effective theory to do certain mathematical jobs.  But while he’s doing set theory, the Arealist is happy to use the word ‘true’ in conventional ways:  for example, if he accepts, works in, a theory with lots of LCs, he’s happy to say things like ” ‘MCs exist’ is true in V, but not in L’.  My own position is close to the Arealist’s but not identical.  I realize it’s tedious of me to have written a book, let alone books, but if you’re ever curious, the one to read is Defending the Axioms.  It’s very short!

Re: Paper and slides on indefiniteness of CH

Dear Penny (or do you prefer Pen or Penelope?),

Thanks for these clarifications and amendments!  (The only changes of mind that strike me as objectionable are the ones where the person pretends it hasn’t happened.)

I am relieved to hear that.

I’m still keen to formulate a nice tight summary of your approach, and then to raise a couple of questions, so let me take another shot at it. Here’s a revised version of the previous summary:

Many thanks for doing this. But you may lose patience with me, as I am still going to be difficult and fine-tune your description even further! My apologies in advance.

We reject any ‘external’ truth to which we must be faithful, but we also deny that the resulting ‘true-in-V’ arises strictly out of the practice (as my Arealist would have it).

I think that I agree but am not entirely clear about your use of the term “external truth”. For example, I remain faithful to the extrinsically-confirmed fact that large cardinal axioms are consistent. Is that part of what you mean by “external truth”? With this one exception, my concept of truth is entirely based on intrinsic evidence.

One key is that ‘true-in-V’ is answerable, not to a realist ontology or some sort of ‘truth value realism’, but to various intrinsic considerations.  The other key is that it’s also answerable to a certain restricted portion of the practice, the de facto set-theoretic claims.  These are the ones that need to be be taken seriously as we evaluate any candidate for a new set theoretic axiom or principle.  They include ZFC and the consistency of LCs.

Tatiana and I talked about “de facto” truth in our BSL paper, but I no longer think that this is necessary. The only “de facto” claims other than the consistency of large cardinals that must be strictly respected are intrinsic (the axioms of ZFC, for example).

The intrinsic constraints aren’t limited to items that are implicit in the concept of set.  One of the items present in this concept is a notion of maximality.  The new intrinsic considerations arise when we begin to consider, in addition, the concept of the hyperuniverse.

Sorry to be difficult, but the 2nd concept is not “hyperuniverse” but “universe of sets”. The hyperuniverse is just a mathematical construct that facilitates the investigation and clarification of the concept of “universe of sets”. This approach is so thoroughly anti-platonistic and epistemic that all one has is an idea of “V = the universe of sets” that is expressed through a wide spectrum of different “pictures of V”. The hyperuniverse allows one to capture this idea of “picture of V” in a sufficiently precise way to enable the logico-mathematical examination and comparison of the different pictures.

One of the items present in this concept is a new notion of maximality, building on the old, that generates the schema of Logical Maximality and its various instances (and more, set aside for now).

Yes, but I would not say that it “builds on the old” if this refers just to the maximal iterative conception. The new maximality principles are different from that conception not just because they deal with external features of universes and are logical in nature, but also because they typically emphasize not the the length of the ordinal sequence (vertical maximality) but the strength of the powerset operation (horizontal maximality). But maybe by the “old notion of maximality” you were referring to more than just the maximal iterative conception. In any case, this is not a major issue as one could just drop the phrase “building on the old”.

At this point, we have the de facto part of practice and various maximality principles.  If the principles conflict with the de facto part, they’re subject to serious question (see below).  They’re further tested by their ability to settle independent questions.  Once we’re settled on a principle, we use it to define ‘preferred universe’ and count as ‘true-in-V’ anything that’s true in all preferred universes.

Yes, but as I said above I’m not too generous about the meaning of “de facto”. Surely many of my colleagues would insist that the existence of large large cardinals is a de facto truth to be respected, but I do not. Second (minor point), the potential conflict with de facto truth is not revealed by the criteria (what you call principles) themselves, but by their first-order consequences, which may take time to extract. Third, notice that there is no static or definitive aspect to this kind of truth. It is a process of dynamic investigation and discovery, because neither the motivating philosophical principles (maximality, omniscience, internal unreachability, …) nor the choice of logico-mathematical criteria instantiating them is fixed; they are subject to enrichment and improvement with the belief that as things progress one is converging towards a coherent and well-justified interpretation of set-theoretic truth.

I hope this has inched a bit closer!  Assuming so, here are the two questions I wanted to pose to you:

Despite all of my complaining above, I do think that your summary is accurate enough that you can now fairly start to attack the programme (!). Just one more thing before I respond to your points below: My collaborator Claudio Ternullo (co-author of “Believing the New Axioms”, under review, on my webpage, the source of much of what I have said in these mails) suggested that I might clarify two more points. First: The criteria for the choice of preferred universes (Claudio has named them “H-axioms”) should be viewed as expressing “higher-order” features of V. This may be reminiscent of Zermelo’s use of full 2nd order set theory, but in fact it is quite different as Zermelo’s very strong criteria leads to a very scanty family of universes, which can be used to reveal nothing about V other than reflection principles (I made this point to Hugh). In fact nearly all of our H-axioms are expressible in a very restricted fragment of 2nd order set theory (using Barwise’s work in infinitary logic they are first-order over the least universe containing the given universe as an element). So a key feature of the programme is to use intrinsic higher-order features of V to make new discoveries about first-order truth which cannot be seen as intrinsic without the use of higher-ordrer ideas. This fits well with my claim that intrinsic first-order evidence is too limited in its power. Second: I should clarify that when I refer to “maximality” of universes I have in mind a logical notion of maximality, i.e., we are not maximising the fanily of sets based on some concept of the “absolute infinite” but rather maximising a family of logical properties; if a logical property occurs externally then it also occurs internally.

Having said that I turn now to your questions.

1.  What is the status of ‘the concept of set’ and ‘the concept of set-theoretic universe’?

This might sound like a nicety of interest only to philosophers, but there’s a real danger of falling into something ‘external’, something too close for comfort to an ontology of abstracta or a form of truth-value realism.

The concept of set is clear enough in the discussion, I have not proposed any change to its usual meaning.

Let me illustrate the concept of universe using maximality. Indeed the word “external” does enter the discussion but only in a very indirect and limited way.

Internal maximality is the usual form, an example is given by the maximal iterative conception. It is an assertion about the freedom to generate new sets through methods of construction and iteration.

External maximality is only conceivable in the context of a non-platonistic conception of V. As there is no fixed ontology, we can easily imagine that there might be a broader interpretation of V, as a universe containing more sets than does our initial interpretation. However we have no clear mechanism for producing such a broadening, there remains only the anti-ontological idea that this should be possible. External maximality asserts that nothing meaningful would be gained by such a broadening. However as we have no clear mechanism for producing such broadenings we can only work indirectly with small pictures of V, elements of the hyperuniverse, where such broadenings are indeed possible and the concept of broadening takes on a precise meaning (a broadening is simply another picture of V which contains the given picture as a subset without changing its ordinals). A worry is that by substituting V with a small picture we have not been faithful to our conception of V; this is true (for example V is not countable but the pictures of V as provided by the hyperuniverse are), however our pictures of V do retain first-order properties of V and that’s all we are interested in anyway. Thus we can fairly capture the first-order consequences of the external maximality of V using pictures of V that are genuinely maximal in a context where we really do have a mechanism for broadening to larger (pictures of) universes.

So my point is that there is no need in the HP to embed V itself into a multiverse and no danger of falling victim to a Balaguerian full-blooded Platonism. The only multiverse needed is the hyperuniverse of (countable) pictures of V.

Thus the concept of universe refers to possible interpretations of V and we can study these universes through pictures of them provided by the hyperuniverse. This is not to say however that the hyperuniverse is fixed! It is just as epistemically conceived as is V. Indeed each interpretation of V gives rise to an interpretation of the hyperuniverse H.

2.  The challenge we friends of extrinsic justifications like to put to defenders of intrinsic justifications is this: suppose some candidate principle generates a lot of deep-looking mathematics, but conflicts with intrinsically generated principles; would you really want to say ‘gee, that’s too bad, but we have to jettison that deep-looking mathematics’?  (I’d argue that this isn’t entirely hypothetical.  Choice was initially controversial largely because it conflicted with one strong theme in the contemporary concept of set, namely, the idea that a set is determined by a property.  The mathematics generated by Choice was so irresistible that (much of the) mathematical community switched to the iterative conception. Trying to shut down attractive mathematical avenues has been a fool’s errand in the history of mathematics.)

You’ve had some pretty interesting things to say about this!  This remark to Hugh, which you repeat, was what made me realize I’d so badly misunderstood you the first time around:

The basic problem with what you are saying is that you are letting set-theoretic practice dictate the investigation of set-theoretic truth!

And these remarks to Sol also jumped out:

Another very interesting question concerns the relationship between truth and practice. It is perfectly possible to develop the mathematics of set theory without consideration of set-theoretic truth. Indeed Saharon has suggested that ZFC exhausts what we can say regarding truth but of course that does not force him to work just in ZFC. Conversely, the HP makes it clear that one can investigate truth in set theory quite independently from set-theoretic practice; indeed the IMH arose from such an investigation and some would argue that it conflicts with set-theoretic practice (as it denies the existence of inaccessibles). 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?

And today, to me, you add:

I see that the HP is the correct source for axiom *candidates* which must then be tested against current set-theoretic practice. There is no naturalist leaning here, as I am in no way allowing set-theoretic practice to influence the choice of axiom-candidates; I am only allowing a certain veto power by the mathematical community. The ideal situation is if an (intrinsically-based) axiom candidate is also evidenced by set-theoretic practice; then a strong case can be made for its truth.
But I am very close to dropping this last “veto power” idea in favour of the following (which I already mentioned to Sol in an earlier mail): Perhaps we should accept the fact that set-theoretic truth and set-theoretic practice are quite independent of each other and not worry when we see conflicts between them. Maybe the existence of measurable cardinals is not “true” but set theory can proceed perfectly well without taking this into consideration.

Let me just make two remarks on all this.  First, if you allow the practice to have ‘veto power’, I don’t see how you aren’t allowing it to influence the choice of principles.  Second, if you don’t allow the practice to have ‘veto power’, but you also don’t demand that the practice conform to truth (as I was imagining in my generic challenge to intrinsic justification given above), then — to put it bluntly — who cares about truth?  I thought the whole project was to gain useful guidance for the future development of set theory.

It seems that you have now isolated the key point in this discussion: what is the point of trying to clarify truth in set theory? I never imagined that it was to guide the future development of set theory! (I’ll say below what I thought the point was.)

But I think that I just fell over the edge and am ready to revoke my generous offer of “veto power” to the working set-theorist. Doing so takes the thrust out of intrinsically based discoveries about truth. You are absolutely right, “veto power” would constrain the necessary freedom one needs in the investigation of intrinsically-based criteria for the choice of preferred universes. The only way to avoid that would be to hoard together a group of brilliant young set-theorists whose minds have not yet been influenced (polluted?) by set-theoretic practice, deny them access to the latest results in set theory and set them to work on the HP in isolation. From time to time somebody would have to drop by and provide them with the mathematical methods they need to create preferred universes. Then after a good amount of time we could see what conclusions they reach! LC? PD? CH? What?

Obviously my plan is unrealistic. So forget the “veto power”. Now what? Well, I guess we have a bifurcation:

Penny truth: Truth derived from and intended for the guidance of the development of set theory.

Sy truth: Truth resulting purely from an investigation of intrinsically-based criteria for the choice of preferred pictures of the universe of sets. The only deference paid to set-theoretic practice is to respect the consistency of large cardinals.

There is no a priori reason to think that these two forms of truth will be compatible with each other.

I owe you an answer to the question: Why study Sy truth?

Aside from the obvious appeal on purely philosophical grounds of understanding what is intrinsic about the fundamental and important concept of set, I would like to have a notion of truth in set theory that is immune to the influence of fads, forceful personalities, available grant money, … I really am not confident that what we now consider to be important in the subject will be important in the future; I am more confident about the “stability” of Sy truth. Second, and this may appeal to you more, it is already clear that the new approach taken by the HP has generated new mathematical ideas that never would have been generated through the usual practice. Taking a practice-independent look at set-theoretic truth generates new forms of set-theoretic practice. And I do like the practice of set theory, even though I don’t want it to dictate investigations of truth! It is valuable to develop set theory in new directions.

Now by bifurcating truth into Penny truth and Sy truth does one in fact eliminate some of the conflicts that you have seen arise in my exchanges with Hugh? I imagine that PD is Penny-true but whether it is Sy-true or not is still open.

I end this mail with a question for you: How does what I call Penny-truth work (it’s OK to just tell me to read your books)? And do you think that it has succeeded or will succeed in guiding the development of set theory? Is there a danger that it will guide it away from areas that should have been investigated?

All the best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

Thanks for these clarifications and amendments!  (The only changes of mind that strike me as objectionable are the ones where the person pretends it hasn’t happened.)  I’m still keen to formulate a nice tight summary of your approach, and then to raise a couple of questions, so let me take another shot at it. Here’s a revised version of the previous summary:

We reject any ‘external’ truth to which we must be faithful, but we also deny that the resulting ‘true-in-V’ arises strictly out of the practice (as my Arealist would have it).  One key is that ‘true-in-V’ is answerable, not to a realist ontology or some sort of ‘truth value realism’, but to various intrinsic considerations.  The other key is that it’s also answerable to a certain restricted portion of the practice, the de facto set-theoretic claims.  These are the ones that need to be be taken seriously as we evaluate any candidate for a new set theoretic axiom or principle. They include ZFC and the consistency of LCs.

The intrinsic constraints aren’t limited to items that are implicit in the concept of set.  One of the items present in this concept is a notion of maximality.  The new intrinsic considerations arise when we begin to consider, in addition, the concept of the hyperuniverse. One of the items present in this concept is a new notion of maximality, building on the old, that generates the schema of Logical Maximality and its various instances (and more, set aside for now).

At this point, we have the de facto part of practice and various maximality principles.  If the principles conflict with the de facto part, they’re subject to serious question (see below).  They’re further tested by their ability to settle independent questions.  Once we’re settled on a principle, we use it to define ‘preferred universe’ and count as ‘true-in-V’ anything that’s true in all preferred universes.

I hope this has inched a bit closer!  Assuming so, here are the two questions I wanted to pose to you:

  • What is the status of ‘the concept of set’ and ‘the concept of set-theoretic universe’?

This might sound like a nicety of interest only to philosophers, but there’s a real danger of falling into something ‘external’, something too close for comfort to an ontology of abstracta or a form of truth-value realism.

  • The challenge we friends of extrinsic justifications like to put to defenders of intrinsic justifications is this: suppose some candidate principle generates a lot of deep-looking mathematics, but conflicts with intrinsically generated principles; would you really want to say ‘gee, that’s too bad, but we have to jettison that deep-looking mathematics’?  (I’d argue that this isn’t entirely hypothetical.  Choice was initially controversial largely because it conflicted with one strong theme in the contemporary concept of set, namely, the idea that a set is determined by a property.  The mathematics generated by Choice was so irresistible that (much of the) mathematical community switched to the iterative conception. Trying to shut down attractive mathematical avenues has been a fool’s errand in the history of mathematics.)

You’ve had some pretty interesting things to say about this!  This remark to Hugh, which you repeat, was what made me realize I’d so badly misunderstood you the first time around:

The basic problem with what you are saying is that you are letting set-theoretic practice dictate the investigation of set-theoretic truth!

And these remarks to Sol also jumped out:

Another very interesting question concerns the relationship between truth and practice. It is perfectly possible to develop the mathematics of set theory without consideration of set-theoretic truth. Indeed Saharon has suggested that ZFC exhausts what we can say regarding truth but of course that does not force him to work just in ZFC. Conversely, the HP makes it clear that one can investigate truth in set theory quite independently from set-theoretic practice; indeed the IMH arose from such an investigation and some would argue that it conflicts with set-theoretic practice (as it denies the existence of inaccessibles). 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?

And today, to me, you add:

I see that the HP is the correct source for axiom *candidates* which must then be tested against current set-theoretic practice. There is no naturalist leaning here, as I am in no way allowing set-theoretic practice to influence the choice of axiom-candidates; I am only allowing a certain veto power by the mathematical community. The ideal situation is if an (intrinsically-based) axiom candidate is also evidenced by set-theoretic practice; then a strong case can be made for its truth.

But I am very close to dropping this last “veto power” idea in favour of the following (which I already mentioned to Sol in an earlier mail): Perhaps we should accept the fact that set-theoretic truth and set-theoretic practice are quite independent of each other and not worry when we see conflicts between them. Maybe the existence of measurable cardinals is not “true” but set theory can proceed perfectly well without taking this into consideration.

Let me just make two remarks on all this.  First, if you allow the practice to have ‘veto power’, I don’t see how you aren’t allowing it to influence the choice of principles.  Second, if you don’t allow the practice to have ‘veto power’, but you also don’t demand that the practice conform to truth (as I was imagining in my generic challenge to intrinsic justification given above), then — to put it bluntly — who cares about truth?  I thought the whole project was to gain useful guidance for the future development of set theory.

All best,
Pen