Pen, in a sense you’re right, the hyperuniverse “lives” within V (I’d rather say that it “originates from” V) and my multiverser surely has a notion of V as anybody else working with ZFC.
OK, but now I lose track of the sense in which yours is a multiverse view: there’s V and within V there’s the hyperuniverse (the collection of ctms). Any universer can say as much.
(I confess this is a disappointing. I was hoping that a true multiverser would be joining this discussion.)
I’m not sure that Sy entirely agrees with me on this point, but to me HP implies an irreversible departing from the idea of finding a single, unified body of set-theoretic truths. Even if a convergence of consequences of H-axioms were to manifest itself in a stronger and more tangible way, via, e.g., results of the calibre of those already found by Sy and Radek, I’d be reluctant to accept the idea that this would automatically reinstate our confidence in a universe-view through simply referring back such a convergence to a pristine V.
Now I’m confused again. Here’s the formulation you agreed to:
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.
Though embracing a single universe is the most straightforward way of pursing unify, I was taking you to be pursuing it in a multiverse context (not to be embracing ‘a pristine V’). Fine with me.
But now that you’ve clarified that you aren’t really a multiverser, that you see all this as taking place within V, why reject unify now? And if you do, what will you say to our algebraist?
Moreover, HP, in my view, constitutes the reversal of the foundational perspective I described above (that is, to find an ultimate universe), by deliberately using V as a mere inspirational concept for formulating new set-theoretic hypotheses rather than as a fixed entity whose properties will come to be known gradually.
So there’s a sense in which you have V and a sense in which you don’t. If V is so indeterminate, how can the collection of ctms within it be a well-defined object open to precise mathematical investigation?
Has this brief summary answered (at least some of) your legitimate concerns?
I very much appreciate your efforts, Claudio but the picture still isn’t clear to me. A simple, readily understandable intuitive picture can be an immensely fruitful tool, as the iterative conception has amply demonstrated, but this one, the intuitive picture behind the HP, continues to elude me.