I think it’s fair to say that contemporary set theory also has the goal of resolving CH somehow.
No, Type 1 considerations (ST as a branch of math) are not concerned with resolving CH, that is just something that a handful of set-theorists talk about. The rest are busy developing set theory, independent of philosophical concerns. Both Hugh and I do lots of ST for the sake of the development of ST, without thinking about this philosophical stuff. Philosophers naturally only see a small fraction of what is going on in ST, for the simple reason that 90% of what’s going on does not appear to have much philosophical significance (e.g. forcing axioms).
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’!
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.