Re: Paper and slides on indefiniteness of CH

Dear Pen and Geoffrey,

I don’t really understand what you are after with this multiverse vs. single-universe (augmented with height potentialism) discussion. As I see it:

If we are talking about ST in terms of its role as a foundation for or subfield of mathematics (Types 1 and 2) then we needn’t trouble ourvselves with this discussion of universes for ST and can hang our hats on what axioms of set theory are advantageous for the development of set theory and mathematics, as was done with AC and the Axiom of Infinity, for example. With Thin Realsim, who cares about our conception of the universe of sets if it’s fixed by set-theoretic and mathematical practice?

But if we are interested in the maximality of the set-concept, more specifically the maximality of the universe of sets in height and width, then indeed this multiverse vs. single-universe discussion kicks in. With regard to the HP I see two different ways to handle this, one with an extreme multiverse view (resulting from radical potentialism/Skolem worshipping) and the other from a single-universe view (stroked a bit by height potentialism):

1. 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.

Now I do know that some (all?) of you don’t dig this way of doing things (do you feel insecure not knowing that there is a “true uncountable”?). I don’t see your point but I’m an accomodating guy, and so:

2. Single-Universe view (peppered by height potentialism). OK, now we want to understand the maximality of “the real V” (whatever the hell that is; OK, I’ll try to put a hold on my sarcasm). How are we going to do this, since “the real V” has all of the sets so it is totally obvious that it is “maximal”, as there are no alternatives! Well, Pen and Geoffrey have thrown us a bone and told us that it’s OK to think about lengthening V (height potentialism), quoting Hilary’s theological claim (just teasing, Hilary). Good, then we can make all sorts of moves with maximality in height by saying things like: Well surely we can lengthen V to a model of ZFC (after all, all rank initial segments of V can be so lengthened, why not V itself?), but if we can do that then surely we can lengthen that lengthening again to a 2nd model of ZFC, again and again, and then apply reflection to say that first-order statements about V which hold in such lengthenings also hold about some V_alpha inside a coresponding lengthening of V_\alpha, … If you push hard enough you get a very strong height maximality which corresponds to Friedman-Honzik #-generation.

But what do we do about width maximality? Well, we had this discussion already, it involves V-logic. To reduce it to a sound-bite: Width maximality can be expressed in terms of “thickenings” (we need quotes!), a concept we can handle without thickening V but by just lengthening V a bit. All this stuff about IMH, $\textsf{SIMH}^\#$, unreachability, width reflection, strong absoluteness, Cardinal Maximality, … can be handled in this way.

Fine. Now the next move I made was to say that actually you don’t need to tie your hands by staying so close to “the real V”, because the mathematics you are doing would work equally well if you replace “the real V” by a countable transitive model of ZFC (“reduction to the Hyperuniverse”). You don’t need to trash “the real V” for this, I was just saying that the mathematical results would be the same if you were to do that, and then we get ourselves into a context where we can make all the moves we want to make in the comparison of universes with no awkwardness, as the universes we need for the comparisons are genuinely available.

In other words, the mathematical analyses of Maximality via the Extreme Multiverse and Single-Universe views are the same!So pick your ontology or lack thereof as you like it, it’s not going to change things mathematically!

Now I can be even more accomodating. Some of you doubters out there may buy the way I propose to treat maximality via a Single-Universe view (via lengthenings and “thickenings”) but hide your money when it comes to the “reduction to the Hyperuniverse” (due to some weird dislike of countable transitive models of ZFC). OK, then I would say the following, something I should have said much earlier: Fine, forget about the reduction to countable transitive models, just stay with the (awkward) way of analysing maximality that I describe above (via lengthenings and “thickenings”) without leaving “the real V”! You don’t need to move the discussion to countable transitive models anyway, it was just what I considered to be a convenience of great clarification-power, nothing more!

Is everybody happy now? You can have your “real V” and you don’t need to talk about countable transitive models of ZFC. What remains is nevertheless a powerful way to discuss and extract consequences from the maximality of V in height and width. Of course you will make me sad if you block the move to ctm’s, because then you strip the programme of the name “Hyperuniverse Programme” and it becomes the “Maximality Programme” or something like that. I guess I’ll get over that disappointment in time, as it’s only a change of name, not a change of approach or content in the programme. But I’ll still be disappointed, please feel sorry for me.


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