Thank you for your rich message. I had thought that, on Sy’s understanding, the HP isn’t really a multiverse view, but a way of discovering new things about V (like an answer to CH). If we’re to understand it as a true multiverse view, that’s a different matter.
I’d like to ask one preliminary question on behalf of the advocate of the ‘universe view’. You write:
Now, there surely are reasons to believe that the universe-view has better prospects within the foundations of set theory. Some of them might be related to … more general concerns related to the foundations of mathematics as more safely couched within a single-universe rather than a plural-universe framework.
You may have something more sophisticated in mind, but this could be read as a simple worry about the ‘foundational’ role of set theory. A universer might think that set theory arose with the proliferation of pure mathematics, for many reasons, but partly to certify the coherence of new structures and to provide a single arena for all those new structures to be studied in relation to one another — and she might think that it continues to play that role today. (In a recent ASL talk, even Vladimir Voevodsky, advocate of ‘univalent foundations’, assigned this role to set theory, as the theory ‘used to ensure that the more and more complex languages of the univalent approach are consistent’ (from the abstract in the BSL), or ‘at least as consistent as set theory’ (from the slides).) It appears that having one standard theory of sets is a requirement for playing this role: when the algebraist asks whether or not there’s a so-and-so, we look to see whether you can prove there’s a (surrogate for) a so-and-so in our accepted theory of sets. And perhaps this lends itself to a universe-view.
So I’m wondering, on your multiverse picture, how this would work. You might say to the algebraist: there’s a so-and-so if there’s one in one of the universes of the multiverse. Or you might say to the universer that her worries are misplaced, that your multiverse view is out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.
Is it the latter?