On Mon, Oct 20, 2014 at 1:29 PM, Penelope Maddy wrote:
Dear Claudio and Sy, … I’m at a loss.
I’m also at a loss, for similar reasons, and have been for a while. 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).
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”. The V theorist can almost certainly model the possibilities as set-models inside V. (One of the few things s/he can’t do is respect the unrestricted Extendability Principle, applied in context of second-order logic (or logic or plurals), since, as Kreisel “complained” decades ago, responding to Putnam’s “Mathematics without Foundations”, V is a counterexample!)