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.