# 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” (http://math.stanford.edu/~feferman/papers.html, #83). I now look forward to seeing what you have to say about such.

Best,
Sol

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

Thanks for your helpful comments on my draft, “The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem,” and especially for bringing your Hyperuniverse Program (HP) to my attention.  I had seen your 2013 article with Arrigoni on HP back then but had not taken in its point.  I have now read it as well as your Chiemsee slides, and will certainly take it into account in the final version of my paper.

I’m glad that we are in considerable agreement about my fundamental argument that one must distinguish mathematical problems in the ordinary sense from logical problems, and that as of now what I claim in the title is true, even taking HP into consideration.  Is my title misleading since it does not say “as of the time of writing”? The reader will see right away in the abstract and the opening section that what I claim does not exclude the possibility that in the future CH will return as a definite mathematical problem [quite unlikely] or that it will somehow become a definite logical problem.

This is not the place to respond to your many interesting comments on the draft, nor on the substance of the HP and your subsequent exchange with Woodin.  But I would like to make some suggestions regarding your terminology for HP (friendly to my mind).  First all, it seems to me that “preferred models” is too weak to express what you are after.  How about, “premier models” or some such?  (Tapping into the Thesaurus could lead to the best choice.) Secondly, I’m not happy about your use of “intrinsic evidence for set-theoretic truth” both because “intrinsic evidence” is commonly used to refer to the constellation of Gödel’s ideas in that respect (not the line you are taking) as opposed to “extrinsic evidence”, and because “set-theoretic truth” suggests a platonistic view (which you explicitly reject).  I don’t have anything to take its place, but it reminds me of the kinds of methodological maxims that Maddy has promoted, so perhaps a better choice of terminology can be found in her writings in place of that.

Best,
Sol

# Paper and slides on indefiniteness of CH

Dear all,

Here are two attachments as pdf files.

The first is a paper entitled, “The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem”; it is a revision of my 2011 Harvard EFI project lecture.

The second consists of the slides for a recent lecture here, “An outline of Rathjen’s proof that CH is indefinite, given my criteria for definiteness.”