For the sake of clarification in the discussion, I’d like to restate the main views in my paper regarding definite/indefinite mathematical and logical problems and what I take to be the inherent vagueness of the concept of arbitrary subset of an infinite set, be it the natural numbers, the real numbers, etc. I find it simplest to quote myself.
At the beginning of the paper, I wrote:
I want to begin by distinguishing mathematical problems in the direct, or ordinary sense from those in the indirect, or logical sense. This is a rough distinction, of course, but I think a workable one that is easily squared with experience. Although the Continuum Hypothesis (CH) in any of its usual forms is prima facie a mathematical problem in the ordinary sense, it has become inextricably entwined with questions in the logical (i.e., metamathematical) sense. I shall argue that for all intents and purposes, CH has ceased to exist as a definite problem in the ordinary sense and that even its status in the logical sense is seriously in question….
Mathematicians at any one (more or less settled) time find themselves working inmedia res, proceeding from an accepted set of informal concepts and a constellation of prior results. The attitude is mainly prospective, and open mathematical problems formulated in terms of currently accepted concepts present themselves directly as questions of truth or falsity. Considered simply as another branch of mathematics, mathematical logic (or metamathematics) is no different in these respects, but it is distinguished by making specific use of the concepts of formal languages and of axiomatic systems and their models relative to such languages. So we can say that a problem is one in the logical sense if it makes essential use of such concepts. For example, we ask if such and such a system is consistent, or consistent relative to another system, or if such and such a statement is independent of a given system or whether it has such and such a model, and so on. A problem is one in the ordinary sense simply if it does not make use of the logical concepts of formal language, formal axiomatic system and models for such. Rightly or wrongly, it is a fact that the overwhelming majority of mathematicians not only deal with their problems in the ordinary sense, but shun thinking about problems in their logical sense or that turn out to be essentially dependent on such. Mathematicians for the most part do not concern themselves with the axiomatic foundations of mathematics, and rarely appeal to logical principles or axioms from such frameworks to justify their arguments. …. 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 speak here of mathematics in the ordinary sense and mathematical logic as ongoing enterprises, and the judgment as to whether a problem is of the one sort or the other is to some extent contextual. The history shows that CH ceased to be a mathematical problem in the ordinary sense in 1904-1908, but it took a long while for people to realize that. As far as I can tell from the contributors to the discussion, except possibly for Bob Solovay (see also below) this has been accepted in the discussion.
Now, the further question whether a mathematical problem is definite or indefinite involves personal judgment to some extent. But I expect when we go down the list of Hilbert’s problems or the Millennium problems, there will be substantial agreement as to whether a mathematical problem is definite (or definite relative to the background state of knowledge and efforts) or not (it might be programmatic, for example). So, from the point of view of 1900, CH is a definite problem, but in our current eyes, it is no longer. This is not a philosophical judgment but simply an assessment of the subject then and now.
The matter is different for logical problems. In sec. 6 of the paper, I return to the question of the status of CH as a logical problem. I wrote:
Clearly, it can be considered as a definite logical problem relative to any specific axiomatic system or model. But one cannot say that it is a definite logical problem in some absolute sense unless the systems or models in question have been singled out in some canonical way.
I can see that there could well be differences of opinion as to whether my criterion in terms of canonicity is the right one to take, and even if it is taken, that there can be greater differences in judgment as to whether a logical problem is definite or not (compared to the assessments above of mathematical problems). In the paper, I examined two approaches to CH as a logical problem, the -logic approach and the inner model program. My conclusion was that neither of these yet meets the criterion to situate CH as a definite logical problem. In the discussion, both Hugh and Sy have presented what they claim to be definite logical problems that are relevant to CH as a logical problem, but differ in their assessments of these. I have not formed a final view on these matters, but am thus far not convinced by either of them. However, I intend to take their arguments into serious consideration in the final version of the paper. (I have also pointed out earlier that there could well be other proposals for such that ought to be considered.) Part of the differences between Hugh and Sy concern the weight to be given to “intrinsic” vs. “extrinsic” evidence. Those terms are no more definite than “definite” and “indefinite”, and also involve matters of judgment. I have questioned whether Sy’s use of “intrinsic” is a useful extension of Gödel’s and suggest that perhaps another term in its place would be more revealing of his claims.
In the final section 7 of the paper proper, I raised what I call the “duck” problem:
We saw earlier that for all intents and purposes, CH has ceased to be a definite mathematical problem in the ordinary sense. It is understandable that there might be considerable resistance to accepting this, since the general concepts of set and function involved in the statement of CH have in the last hundred years become an accepted part of mathematical practice and have contributed substantially to the further development of mathematics in the ordinary sense. How can something that appears so definite on the face of it not be? In more colloquial terms, how can something that walks like a duck, quacks like a duck and swims like a duck not be a duck?
I go on to say that “of course there are those like Gödel and a few others for whom there is no “duck” problem; on their view, CH is definite and we only have to search for new ways to settle it …” But here I take “definite” in the sense that it “has determinate truth value” in some platonistic sense. Thanks to Bob’s remarks, I’m glad that I can class him among the few others. In view of Geoffrey’s appeal to “full” third order semantics over the natural numbers, I would so classify him too, but he might have reasons to resist.
The “duck” problem is a philosophical problem, not a question of what is definite or not as a mathematical or logical problem in the ongoing development of those subjects. And as a confirmed anti-platonist, I have had to grapple with it. In part because of all the circumstantial evidence discussed in the body of the paper concerning the problematic status of CH, my conclusion was as follows.
I have long held that CH in its ordinary reading is essentially indefinite (or “inherently vague”) because the concepts of arbitrary set and function needed for its formulation can’t be sharpened without violating what those concepts are supposed to be about.
Again, here, the question of whether something is “indefinite” is evidently different from its use in the body of the paper in assessing the status of CH as a mathematical and logical problem. I shall have to emphasize that in the final version of the paper. Also the notions of definiteness and indefiniteness brought up in the appendix are philosophically motivated and have to be distinguished as such.
Finally, some (Harvey?) say that what is “inherently vague” is itself “inherently vague”. On the contrary, I explain above exactly in what sense I am taking it. That is why we can agree that sharpening of the concept of arbitrary set to that, e.g., of constructible set, or set constructible over the reals, etc., violates what that concept is supposed to be about. I can’t prove that no such sharpening is possible, but that is my conviction and have to leave it as it lays.
PS: In my view, the side discussion raised by Harvey and pursued by Geoffrey as to the methodology and the philosophy of the natural sciences–as interesting as that may be in and of itself–is not relevant to the issues here.