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:
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.
3. Mathematicians’ attitudes regarding problems in the logical sense
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:
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 . 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.