It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.
Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously.
Surely doing serious set-theoretic mathematics with the hope of resolving CH isn’t a mere ‘philosophical discussion’!
I agree, and I did not mean to imply that the discussion was only philosophical. But my belief is that there are at most 3 or 4 set-theorists actually engaged in the attempt to resolve CH. Resolving CH was certainly never my goal; I got into the HP to better understand large cardinals and internal consistency, with no particular focus on CH. But as this thread began with Sol’s paper on CH, I have been naturally talking about what the HP could offer to that problem. (In any case you already know my views on CH: There will never be a Type 1 solution, we don’t know if there will be a Type 2 solution and I expect a Type 3 refutation.) But if CH motivates Hugh to do good set theory then that is valuable. The motivation fo the HP is much broader than the continuum problem.
In any case, for the record, only the foundational goal figured in my case for the methodological principles of maximize and unify. The goal of resolving CH was included to illustrate that I wasn’t at all claiming that this is the only goal of set theory. Your further examples will serve that purpose just as well:
The goals I’m aware of that ST-ists seem to really care about are much more mathematical and specific, such as a thorough understanding of what can be done with the forcing method.
It would be interesting to ask other set-theorists (not Hugh or I) what the goals of set theory are; I think you might be very surprised by what you hear, and also surprised by your failure to hear “solve CH”.