I’m pretty sure Hugh would disagree with what I’m about to say, which naturally gives me pause. With that understood, I confess that from where I sit as a relatively untutored observer, it looks as if the evidence Hugh is offering is overwhelming of your Type 1 (involving the mathematical virtues of the attendant set theory).
Let me give you a counterexample.
Maximality Criterion. For each infinite cardinal , of is less than .
Both Hugh and I feel that this Criterion violates the existence of certain large cardinals. If that is confirmed, then I will (tentatively) conclude that Maximality contradicts the existence of large cardinals. Hugh will conclude that there is something wrong with the above Maximality Criterion and it therefore should be rejected.
My point is that Hugh considers large cardinal existence to be part of set-theoretic truth. Why? I have yet to see an argument that large cardinal existence is needed for “good set theory”, so it does not follow from Type 1 evidence. That is why I think that large cardinal existence is part of Hugh’s personal theory of truth.
My guess is he’d also consider type 2 evidence (involving the relations of set theory to the rest of mathematics) if there were some ready to hand.
There is some ready to hand: At present, Type 2 evidence points towards Forcing Axioms, and these contradict CH and therefore contradict Ultimate L.
That’s a lot to put in Hugh’s mouth. Probably we should invite Hugh to confirm what you say above.
As with your description above, the “picture” as you call it keeps changing, even with the HP. Recall that the programme began solely with the IMH. At that time the “picture” of V was very short and fat: No inaccessibles but lots of inner models for measurable cardinals. Then came #-generation and the ; a taller, handsomer universe, still with a substantial waistline. As we learn more about maximality, we refine this “picture”.
the only legitimate evidence is Type 3. As we’ve determined over the months, in this case the picture involved has to be shared, so that it won’t degenerate into ‘Sy’s truth’. So far, to be honest, I’m still not clear on the HP picture, either in its height potentialist/width actualist form or its full multiverse form. Maybe Peter is doing better than I am on that.
I have offered to work with the height potentialist/width actualist form, and even drop the reduction to ctm’s, to make people happy (this doesn’t affect the mathematical conclusions of the programme). Regarding Peter: Unless he chooses to be more open-minded, what I hear from him is a premature pessimism about the HP based on a claim that there will be “no convergence regarding what can be inferred from the maximal iterative conception”. To be honest, I find it quite odd that (excluding my coworkers Claudio and Radek) I have received the most encouragement from Hugh, who seems open-minded and interested in seeing what comes out of the HP, just as we all want to see what comes out of Ultimate L (my criticisms long ago had nothing to do with the programme itself, only with the way it had been presented).
Pen, I know that you have said that in any event you will encourage the “good set theory” that comes out of the HP. But the persistent criticism (not just from you) of the conceptual approach, aside from the math, while initially of extraordinary value to help me clarify the approach (I am grateful to you for that), is now becoming somewhat tiresome. I have written dozens of e-mails to explain what I am doing and I take it as a good sign that I am still standing, having responded consistently to each point. If there is something genuinely new to be said, fine, I will respond to it, but as I see it now we have covered everything: The HP is simply a focused investigation of mathematical criteria for the maximality of V in height and width, with the aim of convergence towards an optimal such criterion. The success of the programme will be judged by the extent to which it achieves that goal. Interesting math has already come out of the programme and will continue to come out of it. I am glad that at least Hugh has offered a bit of encouragement to me to get to work on it.
How good of you to read the book! I’m glad it wasn’t too unpleasant an experience.
Don’t you find that a bit harsh for the concept of “intrinsic”?
Maybe you haven’t yet reached the punch lines in the final chapter. There I lay out the view of intrinsic justifications we’ve been talking about here: they’re very valuable, but only as means toward outcomes with extrinsic support.
You probably do: my answer is yes.
As I said the last time you asked this question, I don’t think we can decide this in advance of seeing the actual theories.
It occurs to me that some of my disagreements with Pen and Sol could be resolved just by being clear about how the term “Set Theory” is used.
It is a branch of mathematics.
It is a foundation for mathematics.
It is the study of the concept of set.
Regarding 3: It is plain as pie that there is indeed a “concept of set”, familiar to schoolchildren who are victims of the “new math” (Venn diagrams, essentially). Even kids understand basic set-theoretic operations; probably once they are out of short pants they understand what we mean by powerset.
As Boolos clarified in his paper on the iterative conception (IC), the axioms of Zermelo set theory are derivable from the concept of set as expressed by that conception. Replacement is not derivable from the IC, but it easily follows once we invoke Maximality, i.e. we strengthen the IC to the MIC (maximal iterative conception), also part of the concept of set.
As Pen has clearly expressed, the Axiom of Choice is a different matter: It does not follow from the MIC, but it does follow from the role of Set Theory as a foundation for mathematics. She can say this better than I, but the idea is that mathematics did much better once the old restrictive idea of set given by a rule was liberated through AC.
Now here we come to an important distinction that is ignored in discussions of Thin Realism: The Axiom of Choice didn’t get elected to the club because it is beneficial to the development of Set Theory! It got elected only because of its broader value for the development of mathematics outside of Set Theory, for the way it strengthens Set Theory as a foundation of mathematics. It is much more impressive for a statement of Set Theory to be valuable for the foundations of mathematics than it is for it to be valuable for the foundations of just Set Theory itself!
In other words when a Thin Realist talks about some statement being true as a result of its role for producing “good mathematics” she almost surely means just “good Set Theory” and nothing more than that. In the case of AC it was much more than that.
True in the sense of Pen’s Thin Realist, i.e. a statement is true because of its importance for producing “good Set Theory”.
True in the sense assigned to AC, i.e., a statement is true based on Set Theory’s role as a foundation of mathematics, i.e. because it is important for the development of areas of mathematics outside of Set Theory.
True in the intrinsic sense, i.e., derivable from the maximal iterative conception of set.
Pen’s model Thin Realist John Steel will go for Hugh’s Ultimate-L axiom, assuming certain hard math gets taken care of. Will he then regard it as “true” based on its importance for producing “good Set Theory”? I assume so. If not, then maybe Pen will have to look for a new Thin Realist.
Examples here are much harder to find! What have axioms beyond ZFC done for areas of math outside of Set Theory? Surely forcing axioms have had some dramatic combinatorial consequences, but large cardinals haven’t yet had a similar impact. Descriptive Set Theory has had recent and major implications for functional analysis, but the DST being used is just part of good old ZFC. To understand this situation better I think it’s time for set-theorists to stop being so self-centered and to take a close look at independence outside of set theory, with the aim of seeing which axioms beyond ZFC are the most fruitful for resolving those cases of independence (I’m happy to lead the charge!).
Small large cardinals come easily out of the MIC. Precisely what I am doing with the HP is to derive further consequences. Maybe the negation of CH! Work in progress.
Now I see absolutely no argument for rejecting any of these three notions of Truth in Set Theory. Nor do I see an argument that they should reach common conclusions! Maybe you’ll find this to be excessively diplomatic, taking the heat and excitement out of the Great Set Theory Truth Debate, but I’m sure that even if we agree to this proposed Grand Truce, we’ll still find interesting things to argue about.
As I understand it (I am happy to be corrected), Pen is no fan of Type 3 truth and Sol is no fan of Type 1 truth. OK, I have nothing against aesthetic preferences. But to say that an answer to the Continuum Problem based on one of these three takes on Truth is “illegitimate” is going too far. If someone is going to say that CH is true (or false) then she has to say what notion of Truth is being referenced. Indeed, maybe CH is Type 2 true but Type 3 false!
In any case, it is clearly very hard (but in my view possible) to come to conclusions about what is true in any of these senses. As I have emphasized in the HP (Type 3 truth), for me to make a verdict about CH I will have to first produce “optimal” maximality criteria and show that CH is decided in the same way by those criteria. That is very hard work. For Type 2 truth one would similarly have to show that the statements of Set Theory which are most fruitful for the further development of Set Theory as a foundation for mathematics converge on a theory which settles CH. We have barely begun an investigation of the class of such statements!
I am most pessimistic about Type 1 truth (Thin Realism). To get any useful conclusions here one would not only have to talk about “good Set Theory” but about “the Best Set Theory”, or at least show that all forms of “good Set Theory” reach the same conclusion about something like CH. Can we really expect to ever do that? To be specific: We’ve got an axiom proposed by Hugh which, if things work out nicely, implies CH. But then at the same time we have all of the “very good Set Theory” that comes out of forcing axioms, which have enormous combinatorial power, many applications and imply not CH. So it seems that if Type 1 truth will ever have a chance of resolving CH one would have to either shoot down Ultimate L, shoot down forcing axioms or argue that one of these is not “good Set Theory”. Pen, how do you propose to do that? Forcing axioms are here to stay as “good Set Theory”, they can’t be “shot down”. And even if Ultimate L dies, there will very likely be something to replace it. Why should we expect this replacement for Ultimate L to come to the same conclusion about CH that forcing axioms reach (i.e. that CH is false)?
Nevertheless, as a stubborn optimist I do still expect that at least one of these forms of truth will generate some useful conclusions. But I have given up on the idea that there is a unique, supreme notion of truth in Set Theory that overrides all others; there are at least three distinct and legitimate forms to be taken seriously (despite my pessimism about Thin Realism). And maybe there is even yet a another form of set-theoretic truth that I have overlooked.
You proposed “strong unreachability” as intrinsically justified on the basis of the maximal iterative conception of set, writing: “It is compelling that unreachability (and strong unreachability) with reflection is faithful to maximality but these criteria have not yet been systematically investigated”. Now, we know a bit more, in light of Hugh’s result: If you accept strong unreachability then you have to accept either V=HOD or PD.
[It] cannot be “true” because it violates the maximality of the universe of sets. Recall Sol’s comment about “sharpenings” of the set concept that violate what the set concept is supposed to be about. Maximality implies that there are sets (even reals) which are not ordinal-definable.
So what now? Do you accept PD? Do you claim that we now know that PD is intrinsically justified on the basis of the maximal iterative conception of set?
Or do you retract one of the above claims about what is intrinsically justified on the basis of the maximal iterative conception of set? And if “maximality” keeps suggesting principles that conflict and must be either revised or rejected, does that not indicate that we are not here dealing with a robust notion? Or do you see enough convergence and underlying unity to allay this worry? And, if so, can you, in hindsight, explain what went wrong in this case?
The relevant concept is the familiar iterative conception, which includes a rough idea of maximality in ‘height’ and ‘width’.
To give an intrinsic justification or intrinsic evidence for a set-theoretic principle is to show that it is implicit in the concept in (1).
The HP is a method for extracting more of the implicit content of the concept in (1) than has heretofore been possible.