I think we should all be grateful to you for this eloquent description of how we gather evidence for new axioms based on the development of set theory. The first two examples (and possibly the third) that you present are beautiful cases of how a body of ideas converges on the formulation of a principle or principles with great explanatory power for topics which lie at the heart of the subject. Surely we have to congratulate those who have facilitated the results on determinacy and forcing axioms (and perhaps in time Hugh for his work on Ultimate L) for making this possible. Further, the examples mentioned meet your high standard for any such programme, which is that it “makes predictions which are later verified”.
I cannot imagine a more powerful statement of how Type 1 evidence for the truth of new axioms works, where again by “Type 1″ I refer to set theory’s role as a field of mathematics and therefore by “Type 1 evidence” I mean evidence for the truth of a new axiom based on its importance for generating “good set theory”, in the sense that Pen has repeatedly emphasized.
But I do think that what you present is only part of the picture. Set theory is surely a field of mathematics that has its own key questions and as it evolves new ideas are introduced which clarify those questions. But surely other areas of mathematics share that feature, even if they are free of questions of independence; they can have analogous debates about which developments are most important for the field, just as in set theory. So what you describe could be analagously described in other areas of mathematics, where “predictions” are made about how certain approaches will lead to the solution of central open problems. Briefly put: In your description of programmes for set theory, you treat set theory in the same way as one would treat any field of mathematics.
But set theory is much more that. Before I discuss this key point, let me interrupt myself with a brief reference to where this whole e-mail thread began, Sol’s comments about the indefiniteness of CH. As I have emphasized, there is no evidence that the pursuit of programmes like the ones you describe will agree on CH. Look at your 3 examples: The first has no opinion on CH, the second denies it and the third confirms it! I see set theory as a rich and developing subject, constantly transforming itself with new ideas, and as a result of that I think it unreasonable based on past and current evidence to think that CH will be decided by the Type 1 evidence that you describe. Pen’s suggestion that perhaps there will be a theory “whose virtues swamp the rest” is wishful thinking. Thus if we take only Type 1 evidence for the truth of new axioms into account (Sol rightly pointed out the misuse of the term “axiom” and Shelah rightly suggested the better term “semi-axiom”), we will not resolve CH and I expect that we won’t resolve much at all. Something more is needed if your goal is to say something about truth in set theory. (Of coures it is fine to not have that goal, and only a handful of set-theorists have that goal.)
OK, back to the point that set theory is more than just a branch of mathematics. Set theory also has a role as a foundation for mathematics (Type 2). Can we really assume that Type 1 axioms like the ones you suggest in your three examples are the optimal ones for the role of set theory as a foundation? Do we really have a clear understanding of what axioms are optimal in this sense? I think it is clear that we do not.
The preliminary evidence would suggest that of the three examples you mention, the first and third are quite irrelevant to mathematics outside of set theory and the second (Forcing Axioms) is of great value to mathematics outside of set theory. Should we really ignore this in a discussion of set-theoretic truth? I mean set theory is a great branch of mathematics, rife with ideas, but can we really assert the “truth” of an axiom which serves set theory’s needs when other axioms that contradict it do a better job in providing other areas of mathematics what they need?
There is even more to the picture, beyond set theory as a branch of or a foundation for math. I am referring to its Type 3 role, as a study of the concept of set. There is widespread agreement that this concept entails the maximality of V in height and width. The challenge is to explain this feature in mathematical terms, the goal of the HP. There is no a priori reason whatsoever to assume that the mathematical consequences of maximality in this sense will conform to axioms which best serve the Type 1 or Type 2 needs of set theory (as a branch of or foundation for mathematics). Moreover, to pursue this programme requires a very different approach than what is familiar to the Type 1 set-theorist, perfectly described in your previous e-mail. I am asking you to please be open-minded about this, because the standards you set and the assumptions that you make when pursuing new axioms for “good set theory” do not apply when pursuing consequences of maximality in the HP. The HP is a very different kind of programme.
To illustrate this, let me begin with two quotes which illustrate the difference and set the tone for the HP:
I said to Hugh:
The basic problem with what you are saying is that you are letting set-theoretic practice dictate the investigation of set-theoretic truth!
In other words, my starting point is not what facilitates the “best set theory”, but what one can understand about maximality of V in height and width.
On a recent occasion, Hugh said to me:
[Yet] you propose to deduce the non existence of large cardinals at some level based on maximality considerations. I would do the reverse, revise maximality.
This second quote precisely indicates the difference in our points of view. The HP is intended to be an unbiased analysis of the maximality of V in height and width, grounded in our intuitions about this feature and limited by what is possible mathematically. These intuitions are indeed fairly robust, surely more so than our judgments about what is “good set theory”. I know of no persuasive argument that large cardinal existence (beyond what is compatible with V = L) follows from the maximlity of V in height and width. Indeed in the literature authors such as Gödel had doubts about this, whereas they have felt that inaccessible cardinals are derivable from maximality in height.
So the only reasonable interpretation of Hugh’s comment is that he feels that LC existence is necessary for “good set theory” and that such Type 1 evidence should override any investigation of the maximality of V in height and width. Pen and I discussed this (in what seems like) ages ago in the terminology of “veto power” and I came to the conclusion that it should not be the intention of the HP to have its choice of criteria dictated by what is good for the practice of set theory as mathematics.
To repeat, the HP works like this: We have an intuition about maximality (of V in height and width) which we can test out with various criteria. It is a lengthy process by which we formulate, investigate and compare different criteria. Sometimes we “unify” or “synthesise” two criteria into one, resulting in a new criterion that based on our intuitions about maximality does a better job of expressing this feature than did the individual criteria which were unified. And sometimes our criteria conflict with reality, namely they are shown to be inconsistent in ZFC. Here are some examples:
Synthesis: The IMH is the most obvious criterion for expressing the maximality of V in width. #-generation is the strongest criterion for expressing the maximality of V in height. If we unify these we get IMH#, which is consistent but behaves differently than either the IMH alone or #-generation alone. Our intuition says that the IMH# better expresses maximality than either the IMH alone or #-generation alone.
Inconsistency (examples with HOD): We can consistently assert the maximality principle . A natural strengthening is that of HOD is less than for all infinite cardinals . Still consistent. But then we go to the further natural strengthening of is less than for all subsets of (for all infinite cardinals ). This is inconsistent. So we back off to the latter but only for of cofinality . Now it is consistent for many such , not yet known to be consistent for all such . We continue to explore the limits of maximality in this way, in light of what is consistent with ZFC. A similar issue arises with the statement that is inaccessible in HOD for all infinite regular , which is not yet known to be consistent (my belief is that it is).
The process continues. There is a constant interplay betrween criteria suggested by our maximality intuitions and the mathematics behind these criteria. Obviously we have to modify what we are doing as we learn more of the mathematics. Indeed, as you pointed out in your more recent e-mail, there are maximality criteria which contradict ZFC; this has been obvious for a long time, in light of Vopenka’s theorem.
It may be too much to ask that your program at this stage make such predictions. But I hope that it aspires to that. For if it does not then, as I mentioned earlier, one has the suspicion that it is infinitely revisable and “not even wrong”.
Once again, the aim of the programme is to understand the consequences of the maximality of V in height and width. Your criterion of “making predictions” may be fine for your Type 1 programmes, which are grounded by nothing more than “good set theory”, but it is not appropriate for the HP. That is because the HP is grounded by an intrinsic feature of the set-concept, maximality, which will take a long time to understand. I see no basis for your suggestion that the programme is “infinitely revisable”, it simply requires a huge amount of mathematics to carry out. Already the synthesis of the IMH with #-generation is considerable progress, although to get a deeper understanding we’ll definitely have to deal with the and HOD-maximality.
If you insist on a “prediction” the best I can do is to say that the way things look now, at this very preliminary stage of the programme, I would guess that both not-CH and the nonexistence of supercompacts will come out. But that can’t be more than a guess at this point.
Now I ask you this: Suppose we have two Type 1 axioms, like the ones in your examples. Suppose that one is better than the other for Type 2 reasons, i.e., is more effective for mathematics outside of set theory. Does that tip the balance between those two Type 1 axioms in terms of which is closer to the truth? And I ask the same question for Type 3: Could you imagine joining forces and giving priority to axioms that both serve the needs of set theory as mathematics and are derivable from the maximality of V in height and width?
One additional worry is the vagueness of the idea of the ” ‘maximal’ iterative conception of set”. If there were a lot of convergence in what was being mined from this concept then one might think that it was clear after all. But I have not seen a lot of convergence. Moreover, while you first claimed to be getting “intrinsic justifications” (an epistemically secure sort of thing) now you are claiming to arrive only at “intrinsic heuristics” (a rather loose sort of thing). To be sure, a vague notion can lead to motivations that lead to a great deal of wonderful and intriguing mathematics. And this has clearly happened in your work. But to get more than interesting mathematical results — to make a case for for new axioms — at some stage one will have to do more than generate suggestions — one will have to start producing propositions which if proved would support the program and if refuted would weaken the program.
I imagine you agree and that that is the position that you ultimately want to be in.
No, the demands you want to make of a programme are appropriate for finding the right axioms for “good set theory” but not for an analysis of the maximality of V in height and width. For the latter it is more than sufficient to analyse the natural candidates for maximality criteria provided by our intuitions and achieve a synthesis. I predict that this will happen with striking consequences, but those consequences cannot be predicted without a lot of hard work.
PS: The above also addresses your more recent mail: I don’t reject a form of maximality just because it contradicts supercompacts (because I don’t see how supercompact existence is derivable form any form of maximality) and I don’t see any problem with rejecting maximality principles that contradict ZFC, simply because by convention ZFC is taken in the HP as the standard theory.
PPS: A somewhat weird but possibly interesting investigation would indeed be to drop the ZFC convention and examine criteria for the maximality of V in height and width over a weaker theory.