I’d have thought that a true “multiverser” would want to replace all talk of “V” –understood as the universe of [absolutely] all ordinals(and sets, etc.)–with some more benign term, such as some very large, (perhaps maximally) fat, transitive model of (here a ref to ZFC + some very large card axioms).
I was imagining the multiverser saying that there isn’t just one universe, there are a bunch — with some account to tell us what universes there are and what they’re like. None of them is V.
But, for most mathematical purposes outside the higher reaches of set theory itself, I thought that it wouldn’t matter. Several messages back, ref was made to how a group theorist, for instance, might choose. But couldn’t either view accommodate any new axiom that might possibly matter to a group theorist? That seems to be the case with my modal version of multiverse, in which the possible structures are, up to isomorphism, linearly ordered by “end-extension”.
This is why I asked Claudio if a potentialist (like you) counts as a multiverser. In practice, it doesn’t seem there’s a lot of difference between your potentialist multiverser and a universer who says: there’s a single fixed universe, but we can’t describe it completely; we have to keep adding more large cardinal axioms. If the algebraist comes to the set theorist in his foundational role and asks a question turns out to hinge on, say, inaccessibles (as apparently in Wiles’ original proof), you’d say, ‘no problem, what you want lives in this end-extention’, and my universer would say, ‘no problem, there are inaccessibles’.
But I was imagining that Claudio’s multiverse would be more varied that that. So I floated a couple of possibilities:
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.
(Claudio seemed to opt for the second, but ultimately rejected it; I’m not sure what he thinks about the first.)
On the first, for our simple example, the multiverser would presumably say pretty much what your potentialist says: here, work in this universe with inaccessibles. For this to work, the multiverser would need to give us a theory of what universes there are. For your simple height potentialist, perhaps we have this, but the more varied multiverser would owe us such an account. (I would have asked about that if Claudio had gone for this option.)
Matters get a little harder when the algebraist is after something dicier. Suppose he wants a definable (projective) well-ordering of the reals. My universer might say: well, there’s isn’t such a thing, but if you restrict yourself to thinking inside L, you can have one there; just be sure that all the other apparatus you need is available there, too. Would your potentialist want to say something like that?
PS: To be honest, I have the uneasy feeling that there’s something off in this way of thinking about the foundational goal, but I don’t know what it is.