Many thanks for the interesting draft (I have studied only the first one so far). I would like to point out some recent developments, both in the direction of modern set theory and in approaches to problems like CH, that are relevant to the discussion. Overall I agree with much of what you say and can very well understand the conclusion you have reached regarding the intrinsic undecidability of the CH, given the assumptions that you make. Perhaps the strongest point of agreement that we have is that I do think that it will never be possible to achieve a resolution of CH based on intrinsic features of the concept of set. But that does not entail intrinsic undecidability, as I will explain below. Another point of agreement that we have is that I agree that the two programmes you outline, Woodin’s -logic and the Inner Model Programme, are inadequate to establish CH as a definite logical problem. Indeed regarding -logic I will go further and claim that it has little relevance for discussions of set-theoretic truth anyway.
Peter Koellner and I recently both presented tutorials on set-theoretic truth at the Chiemsee workshop hosted by Leitgeb-Petrakis-Schuster-Schwichtenberg. A key distinction that we both made was between intrinsic (a priori) and extrinsic (a posteriori) evidence in set theory. My position on such forms of evidence can be summarised as follows:
- Evidence based on intrinsic aspects of the set concept can lead no further than reflection principles. Reflection principles are weak, consistent with V = L. (Indeed may view is that they lead to any small large cardinal notion, i.e. any large cardinal notion consistent with V = L; see my paper with Honzik on this, on my webpage.) There is no hope of resolving a question like CH using them.
- Extrinsic evidence coming from set theory is also limited in its power. In particular, I do not find it sufficient to justify either the existence of large large cardinals or of PD (more on this below).
- Extrinsic evidence coming from outside set theory, either from other areas of logic (such as model theory, where non first-order questions demand more than ZFC) or from other areas of mathematics is not sufficiently explored to see what consequences this may have. This is an important gap in the literature that should be filled.
- There is a new source of intrinsic evidence in set theory that shows promise for resolving questions like CH, via my Hyperuniverse Programme (see my paper with Arrigoni on this, on my webpage, and the discussion below).
I’ll now fill in the above by referring explicitly to parts of your interesting paper.
CH has ceased to exist as a definite problem in the ordinary sense.
I agree, but keep in mind that there has been no systematic study of the axioms-candidates beyond ZFC that best suit the needs of mathematics or logic outside set theory. I cannot claim that there will be a consensus about the advisability or otherwise of CH among working mathematicians but this question has not been systematically explored.
…even its [CH's] status in the logical sense is seriously in question.
I will argue below via the HP (Hyperuniverse Programme) that there are serious reasons to possibly regard CH as a well-defined logical problem.
Since axioms for very large cardinals are taken for granted in current programs that aim to settle CH…
No. Large large cardinals are not taken for granted in my approach (more below).
But most importantly, as long as mathematicians think of mathematical problems as questions of truth or falsity, they do not regard problems in the logical sense relevant to their fundamental aims insofar as those are relative to some axioms or models of a formal language.
I mostly agree, but this may be changing. In particular, the remarkable combinatorial power of forcing axioms like PFA or MM which resolve such a wide array of questions (Farah, Moore, Todorcevic, …) may now be persuading mathematicians to use them in their work. This has already happened to some extent with MA.
The Continuum Hypothesis is perhaps unique in having originated as a problem in the ordinary sense and evolved into one in the logical sense.
Really? What about the Suslin Problem (of a similar flavour, but still different)? In fact couldn’t this characterisation apply to any problem of mathematics that was studied and later turned out to be independent of the axioms of ZFC?
Regarding Arnold’s collection: I do agree that mathematicians typically regard independent problems as not “real problems of mathematics”. But there is a new development in set theory, which I think is of the greatest importance, perhaps as profound as the discovery of independence: Unclassifiability results in descriptive set theory. The basic question is whether important classes of mathematical structures admit a desirable classification (nice invariants). For example, simple classes of countable structures have invariants which are real numbers but many interesting classes cannot be classified up to “equivalence” by countable structures of any kind. This is a dramatic development in set theory which has transformed areas of mathematics like operator algebras where good classifications were pursued and never found and now shown not to exist by descriptive set-theoretic methods. My point is that the next edition of Arnold’s collection may (perhaps should) take into account problems in this area, which have nothing to do with forcing or large cardinals, but are concerned with strictly mathematical issues regarding “Borel reducibility” (first discovered by Harvey, by the way, if I am not mistaken).
But in any case I fully agree that “CH can no longer be considered to be a problem in the ordinary sense as far as the mathematical community at large is concerned”. My only point is that there are other problems of current interest in set theory that surely can be so regarded.
My point here is to emphasize that contrary to some appearances, we are dealing with a logical subject through and through.
If this refers to set theory as a whole, then this claim is not right in light of the recent developoments in descriptive set theory. For example, Ben Miller recently showed that the Harrington-Kechris-Louveau generalisation of the Glimm-Effros Dichotomy, which originally used logical methods, can be proved without them. So there are ZFC-provable results coming out of set theory of mathematical interest making no use of logical methods.
Why accept large cardinals? I. The consistency hierarchy.
A distinction must be made between small large cardinals, those consistent with V = L, and large large cardinals, those which are not. The former can be justified using reflection, an intrinsic feature of the maximal iterative concept of set. But in my view there is no convincing argument, based on either intrinsic or extrinsic evidence, for the existence of large large cardinals. What you present in your Section 4 is evidence only for the consistency of large large cardinals, not for their existence. Aside from the more refined question of whether consistency strengths are linearly ordered, the fact remains that large cardinals provide an apparently cofinal “hierarchy” of consistency strengths. This is obviously of great importance for set theory. But it makes no distinction for example between the existence of large cardinals in V and their existence in inner models of V. (Indeed the initial indications of the HP are that they do indeed exist in inner models but not in V.)
You indicate the extrinsic evidence for large cardinal existence based upon their consequences for regularity properties in the projective hierarchy. As I have argued, this is a very weak argument, as it is based on an extrapolation from the 1st projective level to the higher levels; but there are analogous extrapolations that are provably false, such as generalisations of Shoenfield absoluteness to the higher projective levels or even from Borel sets to sets in the descriptive set theory of generalised Baire Space ( instead of ).
Also note (see the Koellner quote you provide on Page 13), that although large cardinals prove PD, in the converse direction you only get that PD gives inner models for large cardinals, not their existence. So again this is an important hint that what is going on is that large cardinals may fail to exist in V but only in inner models and PD may fail in V but only hold in certain inner models which fail to contain all of the real numbers.
Two other claims have been made to justify large cardinal existence and AD in : Woodin has asserted that the only explanation for the consistency of large cardinals is their existence. He points out that this is the case, for example, for the totality of exponentiation on the natural numbers (I agree) and analogously for large cardinal axioms. But I think Woodin is making a simple mistake here: It is indeed hard to imagine a coherent explanation for exponentiation to be consistently total without being actually total. But this is simply because , where the question is formulated, has no proper inner models. But V can have many inner models and for this reason the natural explanation for the consistency of large cardinals without their existence is provided by their existence in inner models. No such argument is available for statements of strength regarding the totality of functions on .
There can be no question that for the same [mathematical] community, the proposed raising of these large cardinal axioms to the status of ‘truths’ alongside the accepted informal principles of set theory is indeed deeply problematic.
I would only add that this is deeply problematic for some members of the set theory community as well. That is of course not to detract from the beauty and importance of the Martin-Steel Theorem, a fine piece of mathematics that can only be properly judged by researchers with experience with large cardinals. The result should not be diminished in importance by others who lack such experience, simply because it is not clear that large cardinal axioms are true. Indeed the connection between “truth” in set theory and what is central to the mathematical development of the subject is not obvious and is in need of clarification (I plan to prepare a paper on this with my philosophical colleagues).
In your Section 6 you discuss two programmes, -logic and the Inner Model Programme. In my view, the latter is not worthy of much discussion, as it is still just a set of unverified conjectures, despite it having been launched by Dodd and Jensen about 40(?) years ago.
-logic is clearly more developed (despite the unverified -conjecture), but it suffices to note that this programme achieves nothing more than a sophisticated analysis of set-generic absoluteness. The highlights are Woodin’s absoluteness of the theory of for set-generic extensions from large cardinals and Viale’s absoluteness of the the theory of for stationary-preserving set-generic extensions from large cardinals + . As Cohen implicitly asked: What does this have to do with the analysis of set-theoretic truth? It is only a beautiful mathematical theory. Indeed, if one wants a solution to CH using the concept of set-genericity the solution is straightforward: absoluteness with absolute parameters for ccc forcing extensions (a parameter is absolute if it is uniformly definable in cardinal-preserving extensions). This principle is consistent and implies that the continuum is very large.
Obviously this is not a solution to the continuum problem and in my view the only problem is that it hinges on the purely technical notion of “ccc forcing extension”.
Let me now briefly explain what the HP is about (see the attached Chiemsee slides or my paper with Arrigoni on my website for more details) and why it addresses the concerns you raise. Two of your quotes are relevant:
But one cannot say that it [CH] is a definite logical porblem in some absolute sense unless the systems of models in question have been singled out in some canonical way.
(Gödel) Probably there exist other axioms based on hitherto unknown principles … which a more profound understanding of the concepts underlying logic and mathematics would enable us to recognize as implied by these concepts.
Briefly: I do not think that CH can be resolved using intrinsic features of the concept of set. Instead I create a context, the Hyperuniverse = The set of countable transitive models of ZFC, in which one can compare different universes of set theory. This comparison evokes intrinsic features of the set-theoretic universe which could not otherwise be expressed, such as the maximality of a universe in comparison with other universes. These features are formulated as mathematical criteria and the universes which satisfy these criteria are regarded as “preferred universes of set theory”. First-order statements which hold in all preferred universes become candidates for new axioms of set theory, which can be tested against set-theoretic practice. Ideally if these candidates in addition are supported by extrinsic evidence then a strong case can be made for their truth.
Thus the idea behind the programme is to make no biased assumptions based on mathematical concepts like genericity, but rather to select preferred pictures of V based on intrinsic philosophical principles such as maximality (another is “omniscience”). The challenge in the programme is to arrive at a stable set of criteria for preferred universes based on such principles. This will take time (the programme is still quite new). Also the mathematics is quite hard (for example sophisticated variants of Jensen coding are required). The current status is as follows: The programme suggests that small large cardinals exist, large large cardinals exist in inner models and CH is very false (the continuum is very large). But there are many loose ends at the moment, both philosophical and mathematical. It is too early to predict what the long-term conclusions will be. But it is clear to me that a solution to the continuum problem is quite possible via this programme; indeed there is a proposed criterion, the Strong Inner Model Hypothesis which will lead to this outcome. A serious mathematical obstacle is the difficulty in showing that the SIMH is consistent.
In summary: My view is that CH will be seen as a definite logical problem if it can be resolved by examining intrinsic features of preferred universes of set theory. In other words, I see a realistic scenario for contradicting the claim you make in the title of your paper. At the same time, I cannot yet claim with confidence that the programme will resolve CH; it may only lead to a small set of possible extensions of ZFC, which I believe will still be of significant interest. It is simply too early to say what will happen, as the choices of motivating phiolsophical principles for preferred universes, the formulation of mathematical criteria to instantiate these principles and the theorems required to establish consistency of the desired criteria are all important future challenges to be met. But I remain optimistic about the prospect of resolving ZFC-undecidable propositions like CH via the programme.
With best wishes and many thanks for including me into the discussion,