Re: Paper and slides on indefiniteness of CH

Dear Sy,

In answer to your questions below, it seems to me that my work has philosophical significance in several ways. First, it shows that the reach of Quine’s (and perhaps Putnam’s) indispensability argument is extremely limited (for whatever that’s worth). Secondly, I believe it shows that one can’t sustain the view from Galileo to Tegmark that mathematics (and the continuum in particular) is somehow embedded in nature. Relatedly, it does not sustain the view that the success of analysis in natural science must be due to the independent reality of the real number system.

My results tell us nothing new about physics. And indeed, they do not tell us that physics is somehow conservative over PA. In fact it can’t because if Michael Beeson is right, quantum mechanics is inconsistent with general relativity; see his article, “Constructivity, computability, and the continuum”, in G. Sica (ed.) Essays on the Foundations of Mathematics and Logic, Volume 2 (2005), pp. 23-25. It just tells us that the mathematics used in the different parts of physics is conservative over PA.

Finally, to be “quite happy with ZFC” is not the same as saying that there is a good philosophical justification for it.


Re: Paper and slides on indefiniteness of CH

On Fri, Aug 29, 2014 at 2:50 AM, Harvey Friedman wrote:

​I developed some alternative conservative extensions of PA and HA called ALPO and B. The former, “analysis with the limited principle of omniscience”, was based on classical logic low down, and constructive logic higher up, and the latter for “Bishop” which was based on constructive logic. If I recall, both systems accommodated extensionality, which demanded additional subtleties in the conservation proofs. Both of these systems, if I recall, had substantially simpler axioms, and of course, there is the obvious issue of how important an advantage it is to have extensionality and to have simple axiomatizations.

There are different ways of isolating weak formal systems in which substantial portions of actual mathematics can be formalized; they each have advantages and disadvantages. I am of course familiar with your systems ALPO and B and appreciate your fine conservation results for those as having real metamathematical interest. But the price you pay to insure extensionality seems to be that they both employ (as you say) partial or full intuitionstic logic and thus are not as readily useful to represent current mathematics in which non-constructive arguments are ubiquitous.

And if it is just constructive mathematics that one wants to represent, Bishop has shown that extensionality is a red herring: every “natural” mathematical kind carries an appropriate “equality” relation, and functions of interest on such objects are supposed to preserve those relations. But the objects themselves can be interpreted as being explicitly given and the functions of interest are given by underlying computable operations. That is a way to have one’s constructive cake and eat it too. Incidentally, the model theorist Abraham Robinson also promoted thinking in terms of equality relations instead of extensionality for quite different reasons.

Simplicity is not really relevant here. The question was, what mathematical notions and principles concerning them are indispensable to (current mathematized) science? The answer provided by my system W is: no more than what can be reduced to PA. Formal systems that are reduced to PA like your fragments of set theory or fragments of 2nd order arithmetic may be formally simpler on the face of it, but one has to see what it takes to actually check to see what part of mathematics can be done on that basis. I contend that the simpler the formal system the less direct is that verification. My system W makes use of a less familiar formalism, but it is more readily used than others I have seen to carry that out. Witness the notes to which i have referred: “How a little bit goes a long way. Predicative foundations of analysis.”

The real issue you want to deal with is the formalization of the actual mathematics.

Well, the only issue I dealt with in response to Hilary was the question of the formalization of scientifically applicable mathematics.

There is a “new” framework to, at least potentially, deal with formalizations of actual mathematics without prejudging the matter with a particular style of formalization. This is the so called SRM = Strict Reverse Mathematics program. I put “new” in quotes because my writings about this PREDATE my discovery and formulation of the Reverse Mathematics program. SRM at the time was highly premature. I’m just now trying to get SRM off the ground properly.

The basic idea is this. Suppose you want to show that a body of mathematics is “a conservative extension of PA”. Closely related formulations are that the body of mathematics is “interpretable in PA” or “interpretable in ACA_0″.. You take the body of mathematics itself as a system of mathematical statements including required definitions, and treat that as an actual formal system. This is quite different than the usual procedures for formalizing actual mathematics in a formal system, where one has, to various extents, restate the mathematics using coding and various and sundry adjustments. SRM could well be adapted to constructive mathematics as well. Actually, it does appear that there is merit to treating actual abstract mathematics as partially constructive. When I published on ALPO and B, I did not relate it to SRM ideas. ​

I remember your talking about the idea of SRM before you turned to RM. That was ages ago (c. 1970?). I am totally skeptical of this because the only way that an actual body of mathematics can be treated as a formal system is to be given as a formal system to begin with, and no mathematics of interest has that character (proof checking systems notwithstanding) Moreover, we understand different formalizations, e.g. of elementary arithmetic, as being essentially the same even though differing in formal details, because we understand them in terms of their intended meaning.

Consider analysis: say there are 100 textbooks on functional analysis, all covering essentially the same material. Which one is “the” text for your SRM? Ditto for every other part of mathematics.

​There is a research program that I have been suggesting that I haven’t had the time to get into – namely, systematically introduce physical notions directly into formalisms from f.o.m.

I don’t foresee any problems with that.

I conjecture that something striking will come out of pursuing this with systematic imagination. ​

Let’s see.

​I am curious as to where your anti-Platonist view kicks in. I understand that you reject \textsf{Z}_2 per se on anti-Platonist grounds. Presumably, you do not expect to be able to interpret \textsf{Z}_2 in a system that you do accept? Perhaps the only candidate for this is Spector’s interpretation? Now what about \Pi^1_1\textsf{-CA}_0? This is almost interpretable in $\textsf{ID}_{<\omega}$ and interpretable just beyond. So you reject \Pi^1_1\textsf{-CA}_0 but accept the target of an interpretation? What about \Pi^1_2\textsf{-CA}_0? How convincing are ordinal notation systems as targets of interpretations — or more traditionally, their use for consistency proofs?

First, as a mathematician (specializing in logic and related topics), my work doesn’t hew in a direct way to my philosophy. I use current everyday set theory (sets, set-theoretical operations, cardinals, ordinals) like most every other mathematician. But one of my aims is to investigate what is really needed for what, and to see whether that has philosophical significance.

Second, proof theory does not make the consistency of this or that formal system any more convincing than what one was reasonably convinced of before. (See my article, “Does proof theory have a viable rationale?”) But the reduction of subsystems of classical analysis to constructive theories of iterated inductive definitions from below is significant for a generalized Hilbert’s program. In any case, I don’t have a red line.

Here is my view. There are philosophies of mathematics roughly corresponding to a lot of the levels of the interpretation hierarchy ranging from even well below EFA (exponential function arithmetic) to perhaps j:V_{\lambda+1}\to V_{\lambda+1} and $j:V \to V$ without choice, and perhaps beyond. These include your philosophy. Most of these philosophies have their merits and demerits, their advantages and disadvantages, which are apparent according to the skill levels of the philosophers who advocate them. I regard the clarity of the associated notions as “continuously” degrading as you move up, starting with something like a 3 x 3 chessboard.

I decided long ago that the establishment of the following Thesis – which has certainly not yet been fully established – is going to be of essential importance in any dialog. Of course, exactly what its implications are for the dialog are unclear, and it may be used for unexpected or competing purposes in various ways by various scholars – just like Gödel’s first and second incompleteness theorems, and the Gödel/Cohen work on AxC and CH.

THESIS. Corresponding to every interesting level in the interpretation hierarchy referred to above, there is a \Pi^0_1 sentence of clear mathematical interest and simplicity. I.e., which is demonstrably equivalent to the consistency of formal systems corresponding to that level, with the equivalence proved in EFA (or even less). There are corresponding formulations in terms of interpretations and conservative extensions. ​

Furthermore, the only way we can expect the wider mathematical community to become engaged in such issues (finitism, predicativity, realism, Platonism, etcetera) is through this Thesis.

I am not out to get mathematicians generally engaged in such issues; it is a rare mathematician who does (Weyl, Brouwer, Hilbert, Bishop). But even if one is out to do that, I don’t think your Thesis (to whatever extent that may be verified) will serve to engage them any more in that respect.


Re: Paper and slides on indefiniteness of CH

Dear Hilary,

Thank you for bringing my attention to Ch. 9 of your book, Philosophy in an Age of Science, and especially to its Appendix where you say something about my work on predicative foundations of applicable analysis. I appreciate your clarification in Ch. 9 of the relation of your arguments re indispensability to those of Quine; I’m afraid that I am one of those who has not carefully distinguished the two. In any case, what I addressed in my 1992 PSA article, “Why a little bit goes a long way. Logical foundations of scientifically applicable mathematics”, reprinted with some minor corrections and additions as Ch. 14 in my book, In the Light of Logic, was that if one accepts the indispensability arguments, there still remain two critical questions, namely:

Q1. Just which mathematical entities are indispensable to current scientific theories?, and

Q2. Just what principles concerning those entities are need for the required mathematics?

I provide one answer to these questions via a formal system W (‘W’ in honor of Hermann Weyl) that has variables ranging over a universe of individuals containing numbers, sets, functions, functionals, etc., and closed under pairing, together with variables ranging over classes of individuals. (Sets are those classes that have characteristic functions.) While thus conceptually rich, W is proof-theoretically weak. The main metatheorem, due to joint work with Gerhard Jäger, is that W is a conservative extension of Peano Arithmetic, PA. Nevertheless, a considerable part of modern analysis can be developed in W. In W we have the class (not a set) R of real numbers, the class of arbitrary functions from R to R, the class of functionals on such to R, and so on. I showed in detail in extensive unpublished notes from around 1980 how to develop all of 19th c. classical analysis and much of 20th c. functional analysis up to the spectral theorem for bounded self-adjoint operators. These notes have now been scanned in full and are available with an up to date introduction on my home page under the title, “How a little bit goes a long way. Predicative foundations of analysis.” The same methodology used there can no doubt be pushed much farther into modern analysis. (I also discuss in the introduction to those notes the relationship of my work to that of work on analysis by Friedman, Simpson, and others in the Reverse Mathematics program.)

Now it is a mistake in your appendix to Ch. 9 to say that I can’t quantify over all real numbers; given that we have the class R of “all” real numbers in W, we can express various propositions containing both universal and existential quantification over R. Of course, we do not have any physical language itself in W, so we can’t express directly that “there is a [physical] point corresponding to every triple of real numbers.” But we can formulate mathematical models of physical reality using triples of real numbers to represent the assumed continuum of physical space, and quadruples to represent that of physical space-time, and so on; moreover, we can quantify over “nice” kinds of regions of space and space-time as represented in these terms. So your criticism cannot be an objection to what is provided by the system W and the development of analysis in it.

As to the philosophical significance of this work, the conservation theorem shows that W is justified on predicative grounds, though it has a direct impredicative interpretation as well. When you say you disagree with my philosophical views, you seem to suggest that I am a predicativist; others also have mistakenly identified me in those terms. I am an avowed anti-platonist, but, as I wrote in the Preface to In the Light of Logic, p. ix, “[i]t should not be concluded from … the fact that I have spent many years working on different aspects of predicativity, that I consider it the be-all and end-all in nonplatonistic foundations. Rather, it should be looked upon as the philosophy of how we get off the ground and sustain flight mathematically without assuming more than the structure of natural numbers to begin with. There are less clear-cut conceptions that can lead us higher into the mathematical stratosphere, for example that of various kinds of sets generated by infinitary closure conditions. That such conceptions are less clear-cut than the natural number system is no reason not to use them, but one should look to see where it is necessary to use them and what we can say about what it is we know when we do use them.” As witness for these views, see my considerable work on theories of transfinitely iterated inductive definitions and systems of (what I call) explicit mathematics that have a constructive character in a generalized sense of the word. However, the philosophy of mathematics that I call “conceptual structuralism” and that has been referred to earlier in the discussion in this series is not to be identified with the acceptance or rejection of any one formal system, though I do reject full impredicative second-order arithmetic and its extensions in set theory on the grounds that only a platonistic philosophy of mathematics provides justification for it.


Re: Paper and slides on indefiniteness of CH

Dear All,

Thanks, first, to Peter Koellner, for his contribution to the discussion.  I will only address the first part of it concerning the relation to the EFI Project of my paper that initiated the present discussion here.  My draft paper for the EFI project to which Peter refers was entitled “Is the Continuum Hypothesis a definite problem?”, let’s refer to it as (F1); the draft paper that initiated this discussion is entitled “The continuum hypothesis is neither a definite mathematical problem nor a definite logical problem”; call it (F2). [Links to both of these can be found on my home page at #95 and #115, resp.] In my view, (F2) is a considerable revision of (F1), so last spring I sent it to a group of colleagues for comments.

That was not intended to open an online discussion and it only became such when Sy Friedman made an email response to (F2) with cc’s to all those on my list that he expanded to include some further names; since then others have added still more names.  It is not at all what I expected or was initially seeking, namely individual comments on (F2) to me alone.  In the process, the discussion widened considerably and following it has indeed become something like “engaging with a hydra.”  Speaking for myself, I have already benefited from it w.r.t. my original purpose even though I have not been able to find time to absorb all the ins and outs of the discussion. In the meantime, I still welcome any individual personal comments on (F2) that people might like to make without getting involved in the online discussion.

The format of the EFI Project to which Peter refers consisted of a series of individual lectures at Harvard over the academic year 2011-2012.  The discussion of each was primarily limited to the faculty and students attending the given lecture, though online exchange was solicited.  There was a short meeting over the Labor Day weekend of 2013 to which all the speakers were invited; unfortunately, neither Tony Martin nor I could attend, so I cannot report on the discussion there.  Peter reports a great deal of convergence of views on the “search for new axioms”.  I wonder about that specifically with respect to the status of CH, since, e.g. under the multiverse approaches of Hamkins and Steel among others, it has no absolute status.  In any case, it seems to me that the discussion here shows that there is a greater divergence of views in our community than reflected in the results of the EFI project.

This is not to diminish at all the importance of the EFI project.  I think this was very important and we have Peter to thank for organizing it and for his critical examination of various of the contributions, mine among them. That required a great deal of energy, time and thought for which we should be very grateful.  And we should very much welcome the prospect of seeing “a final volume containing the papers, commentary, and accounts of the exchanges.”

Peter has provided a link to his acute criticisms of (F1) and says that they apply equally well to (F2). However, (F2) was revised to meet  those criticisms (evidently not to his satisfaction) and contains some essentially new and, in my view, quite important points not addressed in them.  A certain amount of that has emerged in our ongoing online discussion and I will not go over them again here. Instead, I hope those of you who are interested will judge all this for yourself.

In the second part of Peter’s message, not duplicated below, he has also raised specific questions about the material of the Appendix to (F2), material which did not occur at all in (F1).  I shall respond to those separately when I have more time.


Re: Paper and slides on indefiniteness of CH

Dear Sy,

So many questions (not just yours), so little time.

Here, some answers that I will try to keep brief.


On Wed, Aug 20, 2014 at 2:49 AM, Sy David Friedman wrote:

Dear Sol,

Many thanks for the updated summary of your paper in progress on the indefiniteness of CH. Of course your distinction between mathematical and logical definiteness is both important and useful. And I appreciate your willingness to take the HP into account when preparing a future version of your paper. But even your updated summary has left me confused on a few important points and I’d like to ask if you would be willing to clarify them.

1. You still maintain that whether CH qualifies as a problem in the logical sense is “seriously in question”. Surely we agree that the status of CH as a logical problem has not yet been established, and I think you agree that it is possible that at some point in the future it will be established. But what is not clear to me is why you feel that the potential for establishing CH as a problem in the logical sense in the future is “seriously in question”.

That was a direct quote from the draft. I have since said in the discussion that I will acknowledge the positions of both you and Hugh that CH is already (or nearly so) a definite logical problem, though you disagree as to what that is. I have also said that from what I have seen, I am so far not convinced of either, but will need to study these more closely to reach a more definite (yes!) view of the matter.

For example, take the following quote of yours, which suggests to me that you and I are in fact thinking along the same lines:

Clearly, it [CH] 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.

Now of course the word “canonical” is very strong, but if you substitute it with “intrinsic” then you arrive at the Hyperuniverse Programme. Precisely what I am trying to do is determine whether ZFC-undecidable propositions like CH can be decided by moving from the class of all models of ZFC to those which have been singled out in some justifiable way. (I know that you don’t like the word “intrinsic”; more about this in point 2 below.) Now doesn’t the HP therefore give you a degree of optimism about establishing CH as a definite logical problem and challenge your use of the phrase “seriously in question”?

I have said that my canonicity criterion could be up for dispute. In any case, in view of HP, I may want to allow, along with supposedly canonical models as candidates for the criterion, a somehow canonical class of models, all of which agree on CH.

Looking further in your updated summary there is another quote of yours which could explain your use of the phrase “seriously in question”:

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.

As I understand it (please correct me) your valid point is that by for example taking “set” to mean “constructible set” we have violated the intrinsic feature of “maximality”, a feature which the concept of set is meant to exhibit. (Aside: I can then well imagine that on similar grounds you would hesitate to accept an axiom called V = Ultimate-L! But perhaps Hugh will clarify that this need not even imply V = HOD, so it should not be regarded as an anti-maximality statement.) But why would “singling out the models in question” either in a canonical or just intrinsic way constitute a violation of what the concept of set is supposed to be about? This is a very different method of “sharpening” than what is done when adopting V = L, and indeed unlike the latter could remain faithful to the feature of “maximality”.

The point in the last section of the draft is: If CH is not a definite mathematical problem and its status as a logical problem is seriously in question, can we look to philosophy to explain why these are so? My answer by the “inherent vagueness” of the concept of arbitrary subset of an infinite set is given in terms of my anti-platonistic philosophy and more particularly in terms of what I call conceptual structuralism.

Re criteria of “maximality” I take that to be another way of saying “arbitrary.”

Btw, in answer to Harvey, I don’t think that the general concept of being “inherently vague” is inherently vague, at least not relative to the concept of vagueness. Some concepts were vague (e.g. mechanical method) but then could be sharpened. A vague concept is inherently vague if there is no way of sharpening it without violating what it is supposed to be about. For example, the concept of heap is inherently vague.

2. My use of the word “intrinsic”

In my most recent e-mail to Pen I tried to be much more precise about my use of the phrase “intrinsic features of the universe of sets”:

These are those practice-independent features common to the different individual mental pictures [that members of the set theory community have of the universe of sets], such as the maximality of the universe of sets.

Indeed this is very different from Gödel’s use of the term, as I am phrasing it in terms of shared mental pictures, a superfluous move for a Platonist. But it seems to me that it’s OK to use a word as long as it is carefully defined. If you still feel that I should use a different word then I will take your recommendation seriously. In any case, this is a very useful notion for my purposes as I am building a theory of truth that draws almost exclusively on features of the universe of sets which are “intrinsic” in the sense above.

Even if you explain your use of “intrinsic” in carefully defined terms, it muddies a long-standing distinction in our subject stemming from Gödel’s article on CH, namely between new axioms that are accepted for intrinsic reasons (i.e. the same reasons that led us to accept those of ZFC to begin with) and those accepted for extrinsic reasons. Now you are wanting to use intrinsic not as a property of axioms but as a property of features of the universe of sets (however conceived).

I also would say that my notion of “intrinsic”, although not entirely “sharp” is significantly “sharper” than the notion of “definite” as it is being used in this discussion.

Let’s see.

3. Mathematicians’ attitudes regarding problems in the logical sense

You say:

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.

On July 31 I responded as follows:

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.

Of course there is also the use of inaccessibles in Grothendieck’s work. (The fact that with extra effort one can get by without them does not render them irrelevant to the fundamental aims of mathematicians.)

So a case can be made, albeit not yet totally convincing, that what you assert above is not correct. What do you think?

I said in the paper (last par. sec. 1) that “there are borderline cases, to be sure” and gave examples. I don’t know enough about the work of Farah et al. to be able to say what should be said about them. In the case of Grothendieck, he proposed assuming arbitrarily large universes for a foundation of category theory in set theory, without specifically referring to the axioms of set theory. So then set-theorists pointed out that that was equivalent to assuming arbitrarily large inaccessibles. Grothendieck’s proposal thus falls under borderline cases.

4. Why does your paper focus on CH? Would your views be the same if CH were to be replaced by other ZFC-undecidable problems in 3rd order number theory, and if so, which ones? As far as problems in 2nd order number theory are concerned, do you take the position that they cannot be inherently vague because they are typically settled by LC axioms?

In contrast to SH, for example, CH stands out for historical reasons (Cantor’s first real problem, Hilbert’s first, etc.) and for prima facie conceptual simplicity given an understanding of the real numbers and the general concepts of set and function.

On Wed, Aug 20, 2014 at 5:59 AM, Harvey Friedman wrote:

I neglected to ask Sol for his take on the status of consistency questions. I assume Sol does not regard them as definite mathematical problems, but perhaps Sol regards them as definite logical problems?

  1. \text{Con}(\textsf{PA}) – I am guessing that Sol regards this as a solved logical question.
  2. \text{Con}(\textsf{Z}_2)
  3. \text{Con}(\textsf{Z})
  4. \text{Con}(\textsf{ZFC})
  5. \text{Con}(\textsf{ZFC}+\text{various large cardinal hypotheses}).
If these are definite logical problems, what are the prospects for solutions?

PA is consistent because it is true of our conception of the natural numbers, which is a definite conception obtained by reflection from the core structure with zero and successor in order to adjoin addition and multiplication. (That is a philosophical “solution” of the problem of Con(PA) from the point of view of conceptual structuralism, not a logical or mathematical solution.)

As for the rest, they are all prima-facie definite logical problems, but Gödel’s theorem is discouraging as to “solutions”. Some may be convinced by relativized Hilbert’s programs, but those don’t get us very far into \textsf{Z}_2. The combinatorial equivalents to the consistency (or 1-consistency) statements that have been obtained over the years by you and others are prima facie definite mathematical problems, but I don’t see what would count as their solutions except belief in the logical statements to which they are equivalent. I have no reason to expect inconsistency of any ones you list, modulo what goes into the last.


Re: Paper and slides on indefiniteness of CH

Dear all,

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 \Omega-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.

Re: Paper and slides on indefiniteness of CH

Thanks, Harvey, for trying to get the discussion focused back to the point of departure, namely my contentions that CH is neither a definite mathematical problem nor a definite logical problem [as of now].  As far as I can tell, no one is contending that it may still be considered to be a definite mathematical problem. As I wrote at the beginning of sec. 6 of my paper, I wrote that “[CH] 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 agree with what you say in point 1.  One proposal that has been explicitly offered by Hugh is to establish the proposition, V = Ulimate-L, though it is not clear in what sense it would mean to establish that.  It sounds like a candidate for a canonical proposition, with the paradigmatic V = L in mind. But what a wealth of difference:  it is certainly a very complex (and sophisticated) statement, I suppose even for set-theorists, and surely, as you say, for logicians generally and philosophers of mathematics.  By contrast (point 2) Sy’s HP program has its appeal when considered in general terms, but like you re point 3, I would like to see something much more definite.  And when we have that, the question will be, what would it mean to establish whatever that is?

In both these cases, if the proposed “solution” fails, CH is left in limbo.

Perhaps there are other proposals for asserting CH as a definite logical problem; unless I’ve missed something, none has been put forward in this discussion.  But in any case, the same kinds of questions would have to be raised about such.

Re 4, first, I hope “blackboxing” does not get accepted as a verb.  Second: “Good mathematics” and “good set theory”: only the experts can judge what constitutes that. And the practice of very good mathematical, logical and set-theoretical work will go on whether or not it has a clear foundational purpose. But there are certain problems where what one is up to cries out for such a purpose, with CH standing first in line, making as it does the fundamental concepts and methods of set theory genuinely problematic.

Perhaps we can also have a meeting of minds about “mental pictures” (point 5).  I’ve written a lot about that; in particular, I have referred to my article “Conceptions of the continuum” (, #83). I now look forward to seeing what you have to say about such.


Re: Paper and slides on indefiniteness of CH

Dear Sy,

I’m very pleased that my paper has led to such a rich exchange and that it has brought out the importance of clarifying one’s aims in the ongoing development of set theory. Insofar as it might affect my draft, I still have much to absorb in the exchange thus far, and there will clearly be some aspects of it that are beyond my current technical competence. In any case,  I agree it would be good to bring the exchange to a conclusion with a summary of positions.

In the meantime, to help me understand better, here is a question about HP: if I understand you properly, if HP is successful, it will show the consistency of the existence of large large cardinals in inner models. Then how would it be possible to establish the success of HP without assuming the consistency of large large cardinals in V?  If so, isn’t the program circular?  If not, it appears that one would be getting something from nothing.


Re: Paper and slides on indefiniteness of CH

Dear Sy,

There is no retreat from my view that the concept of the continuum (qua the set of arbitrary subsets of the natural numbers) is an inherently vague or indefinite one, since any attempt to make it definite (e.g. via L or an L-like inner model) runs counter to what it is supposed to be about. I talk here about the concept of the continuum, not the supposed continuum itself, as a confirmed anti-platonist.  Mathematics in my view is about intersubjectively shared (human) conceptions of idealized structures, not any supposed such structures in and of themselves.  See my article “Conceptions of the continuum” (Intellectica 51 (2009), 169-189).

I can’t have claimed that I have established that CH is neither a definite mathematical problem nor a definite logical problem, since one can’t say precisely what such problems are in either case.  Rather, as workers in mathematics and logic, we generally know one when we see one.  So, the Goldbach conjecture and the Riemann Hypothesis (not “Reimann” as has appeared elsewhere in this exchange) are definite mathematical problems.  And the decidability of the first order theory of the reals with exponentiation is a definite logical problem.  (Logical problems make use of the concept of formal language and are relative to models or axioms.) Even though CH has the appearance of a definite mathematical problem, it has ceased to be one for all intents and purposes because it was long recognized that only logical considerations could be brought to bear to settle it, if at all.  So then what would make it a definite logical problem? Something as definite as: CH is true in L.  I can’t exclude that some time in the future, some model or axiom system will be produced that will be as canonical in nature for some concept of set as L is for the concept of hereditarily predicatively definable set.  But I’m not holding my breath either.

I don’t know whether your concept of set-theoretical truth can be assimilated to Maddy’s A-realism, but in either case I see it as trying to have your platonist cake without eating it.  It allows you to accept CH v not-CH, but so what?