With Claudio’s permission I will reply to your latest message (and also take the opportunity to make some further remarks).
I took Claudio to be proposing an interpretation of the HP program different from yours, one that’s an explicit multiverse program. No? If not, then I guess we return to the place where I still don’t quite grasp what the HPer is up to. The universer is studying V. The HPer is studying …
And indeed in the HP we have a “single-universe view” (the “ideal V”) which is analysed via a multiverse construct (the Hyperuniverse). It is a hybrid, and that may have caused confusion, for which I apologise. Another added ingredient is the consideration of “thickenings” in quotes, in addition to the lengthenings of the height potentialist. Both are reduced to the mathematical study of the Hyperuniverse (the reduction to the Hyperuniverse”, a bit more about that below).
… an ‘ideal V’? What is that?
The Hyperuniverse is also an ideal construct, it is defined within the ideal V. Conversely, truth in the ideal V is clarified through an analysis of its associated (ideal) Hyperuniverse. There is a dynamic interplay, a “dualism” as Claudio explained. There is no thick ontology for the ideal Hyperuniverse just as there is none for the ideal V.
The hyperuniverse is also ideal: the collection of ctms inside ‘ideal V’, right? What work is ‘ideal’ doing here? Why not just V and the collection of ctms in V?
1. I can acknowledge that a single-universe view (with no multiverse considerations) is the most obvious view for the foundations of Set Theory. But I don’t think it has better prospects than a multiverse view (or “hybrid” view as in the HP), since it has been quite obvious for a long time that independence is extensive in ST (Set Theory) and in my view this makes any single universe view that exceeds TR (Thin Realism) quite useless. What good is a single universe if we don’t know what form it takes and can only imagine a wide range of possibilities for that form?
Sure, the ordinary universer has a long way to go in figuring out the features of V. But the aim remains a single theory of sets, which is the most natural way of satisfying the foundational goal. My point is just that this goal generates a methodological maxim I once called ‘Unifiy': go for one accepted theory of sets if at all possible. (It might turn out to be impossible, given other goals, but it doesn’t seem to me that that has happened yet.)
What Claudio and I established was that, on his understanding of the HP, the HP has embraced the methodological maxim of Unify — not in the universer’s way, in a different way — so I was ready to move on to the next question: how to understand the ontology of the HP.
As for the Thin Realist, she’s beholden to those extrinsic payoffs. Until they abound, she sticks to her simple universe understanding.
“HP identifies a core model-theoretic construct, that is, c.t.m., as the only constituent of multiverse [thin] ontology. Further, mathematical and logical, reasons for this choice have been explained at length by Sy, but I wish to recall that the main (and, to some extent, remarkable) fact is that we do not lose any information about set-theoretic truth by making this choice.”
I hope that this point has finally come across. Pen, Hugh and Harvey have each asked how I make the reduction to ctm’s and I can only ask them to please re-read what I have said about this at great length and through great effort in this exchange. The HP analyses Maximality through the study of certain very particular properties of ctm’s, but unlike what both Hugh and Harvey have tried to claim, it is much more than the study of ctm’s. The ctm’s are just the mathematical tool needed.
I realize this is frustrating for you, but what you’ve said so far about the ‘reduction to ctms’ hasn’t yet produced understanding (for me) or conviction (for Hugh or Harvey). I’m still stuck, as above, on figuring out where the ctms live, what makes everything ‘ideal’, and so on. They have other concerns.