I think we are approaching the point where a summary of this discussion is in order. The mathematical dust has largely settled — as far as the program as it currently stands is concerned –, thanks to Hugh’s contributions. But there is one major remaining matter of a more philosophical nature that still remains unclear to me — it has to do with my original question of whether you are an actualist or a potentialist and ultimately with the picture that forms the backdrop of your program. To get clear on this matter I will have to recapitulate a good part of this discussion. Please bear with me.
In response to my original question, on Sept. 14 you wrote:
I am a radical potentialist, indeed you might say a Skolem-worshipper! (Remember what Pen quoted from my article with Tatiana: “V is a product of our own”!) Indeed my view is that there is no real V, but instead a huge wealth of different “pictures of V”. Given any such picture P of V, let denote the universe depicted by P; there are pictures P* of V such that is a rank-initial segment of , a proper inner model of , or (here’s the radical Skolem-worshipping) even countable in ! So yes, it is a given for me that you can lengthen or thicken a picture of V, in fact you can make it countable!
Now here is a possible source of confusion: Sometimes one fixes an initial picture P and corresponding universe as a reference picture; in that case one can talk about cardinality, in reference to . But of course itself is countable from the perspective of a bigger , so there is no absolute notion of “countable”, only a relativised one.’
In response to this — with focus on the line “there is no absolute notion of “countable”, only a relativized one” — on Sept. 15 I responded:
I thought: “He can’t mean this! Look. If everything is countable from the perspective of an enlargement and if enlargements always exist then everything is countable. (Of course it can be countable from a local perspective — when one has one’s blinkers on and looks no further — but given the tenet that it countable from a higher perspective it follows that it is ultimately countable.) But if everything is countable — or if “there is no absolute notion of `countable’, only a relativized one” — then how can he be understanding CH? This whole exchange was sparked with the presentation of a new and promising approach to questions like CH, one that promised to reinvigorate “intrinsic justifications” to the point where they could touch questions like CH. But now it seems that on this approach the straightforward sense of CH has evaporated. Indeed it seems that set theory has evaporated!”
Your view, as described above, is indeed like that of Skolem. But Skolem (rightly) took this view to involve a rejection of set theory. And yet you don’t. You seem to want to have it both ways: reject an absolute notion of countability and say something about CH (beyond that it has no meaning).
This got me greatly confused.
But then in the outline of the HP program that you sent on the same day things changed. For in that outline you speak of mental pictures of “the universe V of all sets” and you write: “But although we can form mental pictures of other universes, the only such universes we can actually produce are wholly contained within V, simply because V by its very definition contains all sets.” So now you appear to be an actualist and not a potentialist at all. (Of course you are a potentialist with regard to the little V’s — the countable transitive models of ZFC — but we are all potentialists with regard to those, trivially.)
So, which is it: Are you a potentialist or an actualist?
On Sept. 15 you responded:
OK, now to radical potentialism: Maybe it would help to talk first about something less radical: Width potentialism. In this any picture of the universe can be thickened, keeping the same ordinals, even to the extent of making ordinals countable. So for any ordinal alpha of V we can imagine how to thicken V to a universe where alpha is countable. So any ordinal is “potentially countable”. But that does not mean that every ordinal is countable! There is a big difference between universes that we can imagine (where our becomes countable) and universes we can “produce”. So this “potential countability” does not threaten the truth of the powerset axiom in V!
At that point I thought: “OK, I think I am getting a grip on the picture: Sy distinguishes between extensions that “actually” exist and extensions which “potentially” (or “virtually”) exist. When talking about extensions that actually exist (lengthenings and thickenings) he doesn’t use scare quotes but when talking about extensions that do not actually exists but only potentially (or virtually) exist he uses scare quotes — as, for example, when in the context of width-actualism he speaks of “thickenings”.”
Let’s take stock: (1) In the case of countable transitive models of ZFC we all agree that there are actual lengthenings and thickenings (no scare quotes). And we can agree that there is always such an actual extension in which any given model is seen to be countable. (2) In the context of with-actualism there are actual lengthenings but only virtual thickenings (“thickenings”, with scare quotes — the “imaginary” extensions). And we can agree, via Jensen coding through class forcing, that there is always such a virtual thickening (“thickening”, with scare quotes — an “imaginary” extension) in which the model is “seen” to be countable. (3) But in distinguishing your radical potentialism from width actualism + height potentialism you must endorse actual lengthenings and actual thickenings and, moreover, such actual extensions in which any given transitive model of ZFC (whether countable or not) is (actually) seen to be countable. But then everything is ultimately countable, as I pointed out and you rejected.
This got me greatly confused. It seemed we were back to where we started.
I doubted that until on 21 Oct you wrote:
Extreme Multiverse view: We have no single V but a wealth of different possible V’s. This wealth is so wealthy that any particular V can be thickened or lengthened (no quotes!) and shockingly, made countable by going to a larger V. So there is no absolute notion of cardinality, only distinct notions of cardinality within each of the possible V’s. OK, now when talking about maximality of a possible V we simply mean that lengthening or thickening V will not reveal new properties that we couldn’t already see in V. (Note: One could go further and even look at blowups of V which see V as countable, but mathematically this doesn’t seem to add much.) Then when we talk of a first-order statement like not-CH being a consequence of maximality we mean that it holds in all of the possible V’s which are maximal.
Frankly speaking, the Extreme Multiverse View is my own personal view of things and gives the cleanest and clearest approach to studying maximality. That’s because it allows the freedom to make all of the moves that you want to make in comparing a possible V to other possible V’s.
Note that the multiverse described above looks exactly like the Hyperuniverse of a model of ZFC. In other words, the Extreme Multiverse View says that whether or not we realise it, we live in a Hyperuniverse, and we are kidding ourselves when we claim that we have truly uncountable sets: Some bigger universe looks down at us and laughs when she hears this, knowing perfectly well that we are just playing around with countable stuff.
In the second sentence you emphatically indicate that there are no scare quotes — these lengthenings and thickenings actually exist — there is “no absolute notion of cardinality”.
So we are back to where we started, to the view I thought you held all along.
I thus repeat my earlier point:
But now it seems that on this approach the straightforward sense of CH has evaporated. Indeed it seems that set theory has evaporated! Your view, as described above, is indeed like that of Skolem. But Skolem (rightly) took this view to involve a rejection of set theory. And yet you don’t. You seem to want to have it both ways: reject an absolute notion of countability and say something about CH (beyond that it has no meaning).
I hope you don’t repeat your earlier response:
So any ordinal is “potentially countable”. But that does not mean that every ordinal is countable! There is a big difference between universes that we can imagine (where our aleph_1 becomes countable) and universes we can “produce”. So this “potential countability” does not threaten the truth of the powerset axiom in V!
Because if you do we will be caught in a loop…
I suspect your view has changed. Or not changed. In any case, what is your view?