# 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.

Best,
Sol

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.

Best,
Sol

# 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.

Best,
Sol

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

It seems obvious to me that the usual fundamental explanation of what a set is, is not itself sufficient to even quite set up the axioms of ZFC, although it can be reasonably argued that it is. In any case, it does not appear sufficient to go beyond ZFC, or at least go beyond ZFC by much.

So this means that more than the usual fundamental explanation of what a set is is requires to do anything going well beyond ZFC. It appears that MUCH MORE is needed to deal with the continuum hypothesis, and also some other really natural things like Borel’s conjecture (all strongly measure 0 sets of reals are countable), which follows from CH, and Suslin’s hypothesis (is every complete dense linear ordering without endpoints with the countable chain condition isomorphic to the real line?). All four propositional combinations of  SH and CH are consistent with ZFC.

Now anything really useful going beyond the usual fundamental explanation of what a set is is going to be radically more sophisticated than the usual fundamental explanation of what a set is. At least that is conventional wisdom. This has all the trappings of inherent vagueness.

I sidestepped the issue of just what inherent vagueness is, as here we have two distinguished scholars — Sol and Geoffrey — who are at complete opposites ends on the issue of whether set or even set of natural numbers is inherently vague.

You can think of my saying that “CH research is not a relatively promising area of foundational research” as a poor man’s version of saying that CH is inherently vague.

Is “inherently vague” inherently vague? I think so. In fact, let me make the same move. Research on “inherent vagueness” is not a relatively promising area of foundational research.

Let me come at this from a little different perspective. For most people, it is very very hard to separate the issue of inherent vagueness from the phenomena that “there is no apparent path to determining the truth value of basic assertions about the notion”. We are so used to being able to settle things about very very concrete situations – like the integers up to 1 million, generally even without a computer, but nothing wrong with a computer for the sake of this discussion. Are we just drunk with success, or is this the trappings of inherent vagueness? In the pragmatic line I am taking, it doesn’t make any difference. My coin of the realm is “promising foundational research”.

Harvey