This message will try to say where we are in a generally understandable way.
1. Sol Feferman originally put out a request for comments on his paper and slides concerning the indefiniteness of CH. Sol maintains that CH = continuum hypothesis, is neither a mathematical NOR a logical problem. I think Sol is pretty definite that CH is never going to become a mathematical problem, and is definitely not currently a logical problem. And that there is some realistic possibility of it becoming a logical problem, provided some theory emerges of sufficiently widespread acceptance or interest in the set theory community, with the question of the status of CH within that theory.
NOTE: I have been wanting to turn to the Woodin program (Ultimate L) to see if this is clear, simple, and coherent enough to turn CH into a logical problem. I am planning to approach this from a high level generally understandable perspective shortly to see what shakes out. We have seen that the HP, at least in its present form, does not meet such standards.
2. My own position has a lot of overlap with Sol’s but differs in detail and emphasis. The bottom line is probably that both Sol and I agree that “the program of “settling CH” is not a relatively promising area of research in the foundations of mathematics”. In my foundational methodology, I never subscribe to philosophical views, but do think in terms of which side has the better arguments. I believe that arguments can always be strengthened, and attacks against arguments can also always be strengthened. I don’t come to a definite conclusion that “CH is not a mathematical problem”, even if prospects look poor right now for it being so. One thing is clear: it is not an important mathematical problem right now given the present research activities in mathematics today. It was at the time of Hilbert’s problem list. An interesting discussion is just why and how this changed, and whether this is simply due to Goedel/Cohen. I believe that the story goes well beyond Goedel/Cohen here, but this message is not about that. (Sol does deal with this, but I don’t think that it is the last word).
3. In particular, right now, the arguments against CH “having a definite truth value” are stronger than the arguments for CH “having a definite truth value”: but the argument against “having a definite truth value” being definite enough for most philosophically purposes, is, in my view, stronger than the argument for “having a definite truth value” being definite enough for most philosophical purposes. Incidentally, I was recently on the phone with a well known philosopher who strongly disagrees with me about this, and regards CH as (or equivalent to) a clearly stated problem in higher order logic, which he regards as automatically having a “definite truth value” in a “definite sense”.
4. There was little traffic in reaction to Sol’s original request for comments. Until Sy wrote forcefully and extensively about a “program” called HP = hyperuniverse program. Sy urged Sol to incorporate an account of HP in his paper(s). The headline statement of HP was very simple: study the countable transitive models of ZFC and their relationships, and this will reveal the “correct” or “best” axioms for set theory, including axioms that might well settle CH.
5. There were numerous attempts by Sy to justify the claim that such a study of countable transitive models of ZFC (dubbed the hyperuniverse) would provide the “correct” or “best” axioms for set theory, mostly under some form of “intrinsic maximality of the set theoretic universe”. Sy attempted to make this philosophically and foundationally coherent, but left a lot of objections by Pen and Peter unanswered (at least to the satisfaction of Pen and Peter and many others). One of his coworkers in HP answered Pen with answers in direct fundamental contradiction with those of Sy. No “reconciliation” between these very opposing views that Pen pointed out very clearly has been given.
6. On the philosophical side, Sy attempts to convince us that HP is a foundational program that responds to the “intrinsic maximality of the set theoretic universe”, without getting involved in the badly needed analysis of just what “intrinsic maximality of the set theoretic universe” means, or could mean, or should mean. Because of the lack of such a discussion – let alone creative ideas about it – it does not appear that anyone on this list, except a handful of HP coworkers, are being persuaded that HP is a legitimate foundational program.
7. In this connection, both Hugh and I believe that the HP is better viewed and better named as CTMP = countable transitive model program. There was an attempt by Sy to claim that ctms (countable transitive models) are of fundamental foundational importance for present purposes, based on the downward Skolem Lowenheim theorem. However, that is merely a technical point that comes after a framework for analyzing “intrinsic maximality of the set theoretic universe” is first accepted. In the absence of a careful and persuasive discussion of that framework, there is no relevant use of the Skolem Lowenheim theorem. Also, Hugh pointed out that in some of the recent HP proposals, the link between arbitrary sets and countable sets is broken. Another unanswered question of Hugh for Sy.
8. Another even more critical unanswered question of Hugh for Sy simply asks for clarity concerning what happens in CTMP (aka HP) after we see that the IMH contradicts the existence of inaccessible cardinals. IMH = inner model hypothesis, is the initial assertion coming out of the CTMP (aka HP). Hugh asked recently (and I think Hugh has been asking for months) for this clarity, and I have been expecting a response from Sy.
9. The issues of CTMP (aka HP) after IMH is critical. That IMH refutes the existence of an inaccessible cardinal should immediately make a thorough analysis of just what is meant by “intrinsic maximality of the set theoretic universe” urgent. Instead, Sy chose to reject IMH in favor of a number of “fixes”. Hugh has not gotten a satisfactory response as to what these “fixes” are. Furthermore, back channels affirm my suspicions that these “fixes” are ad hoc, taking the large cardinals as given, and then layering a kind of IMH on top of it. If this is the plan, then we really have, prima facie, a serious dose of philosophical incoherence.
10. With Hugh, Pen, Peter, Geoffrey, we see that Sy has to some extent lived up to his professional responsibility for interactive engagement (given that Sy has forcefully pushed the HP) – but only up to a point. There is still a significant degree of non responsiveness. If Sy would follow the principle of writing in generally understandable ways whenever practical, the non responsiveness and drawbacks would be apparent, and the discussion would be much more productive. However, with me, there is a complete refusal to engage. If he did engage in a professional manner, the prima facie emptiness of the HP would have gotten addressed months ago and either a new idea would have emerged from the interaction, or the HP would have simply morphed, rather pleasantly and uneventfully, into the not uninteresting CTMP.
11. Having been appalled at the utter waste of time for the overwhelming majority of people on this list, at least compared to what it could have been, I took the plunge and started a discussion of “intrinsic maximality in the set theoretic universe”. I was especially motivated by Sy saying that “intrinsic maximality of the set theoretic universe” doesn’t even generate AxC. This in the context of all sorts of pronouncements about what it does generate, of a comparatively technical nature. Sy’s coworker states unequivocally that “intrinsic maximality of the set theoretic unvierse” generates all of ZFC.
12. Already this has led to what appears to be an apparently new study of AxC and variants in the absolutely classically fundamental contexts of satisfiability of sentences in first order predicate calculus. But I am hoping to have a lot more to say about “intrinsic maximality” not only in the set theoretic universe, but much more generally. It appears that much of this thread is an important lesson in how NOT to do philosophy and foundations, and the fruits that come out if philosophy and foundations is done competently.