# Re: Paper and slides on indefiniteness of CH

Dear HP-ers, HP-worriers, and friends,

In this thread (which I confess has been moving pretty quickly for me; I’ve read it all but do apologise if I’m revisiting some old ground) we’ve seen that the key claim is that there is a deep relationship between countable transitive models and some V, either real, ideal, or sat within a multiverse. I have a few general worries on this that if assuaged will help me better appreciate the view.

I’m going to speak in “Universey” terms, just because its the easiest way I find for me to speak. Indeed, when I first heard the HP material, it occurred to me that this looked like an epistemological methodology for a Universist; we’re using the collection of all ctms as a structure to find out information (even probabilistic) about V more widely. If substantive issues turn on this way of speaking, let me know and I’ll understand better.

Let’s first note that in the wake of independence, it’s going to be a pretty hard-line Universist (read “nutty Universist”) who asserts that we shouldn’t be studying truth across models in order to understand V better. Indeed the model theory gives us a fascinating insight into the way sets behave and ways in which V might be. However, its then essential to the HPers position that it is the “truth across ctms” approach that tells us best about V, rather than “truth across models” more generally. I see at least two ways this might be established:

A. Ctms (and the totality of) are more easily understood than other kinds of model.

B. Ctms are a better guide to (first-order) truth than other kinds of model.

I worry that both A and B are false (something I came to worry in the context of trying to use the HP for an absolutist).

A.1. It would be good if we could show two things to address the first question:

A.1.1. The Hyperuniverse is in some sense “tractable” in the sense that we can refer to it easily using fairly weak resources.
A.1.2. The Hyperuniverse is in some sense “minimal”; we only have the models we need to study pictures of V. There’s no extraneous subject matter confusing things.

The natural way to assuage A.1.1. for someone who accepts something more than just first-order resources is to provide a categoricity proof for the hyperuniverse from fairly weak resources (we don’t want to go full second-order; it’s the very notion of arbitrary subset we’re trying to understand). I thought about doing this in ancestral logic, but this obviously won’t work; there are uncountably many members of the Hyperuniverse and the downward LST holds for ancestral logic. So, I don’t see how we’re able to refer to the hyperuniverse better than just models in general in studying ways V might be.

(Of course, you might not care about categoricity; but lots of philosophers do, so it’s at least worth a look)

Re: A.1.2 The Hyperuniverse is not minimal. For any complete, maximal, truth set T of first-order sentences consistent with ZFC, there’s many universes in H satisfying that truth set. So really, for studying “first-order pictures of V” there’s lots in there you don’t need.

So, I’d like to hear from the HPers the sense in which we can more easily access the elements of H. One often hears set theorists refer to ctms (and indeed Skolem hulls and the like) as nice’, managable’, “tractable”. I confess that in light of the above I don’t really understand what is meant by this (unless it’s something trivial like guaranteeing the existence of generics in V). So, what is meant by this kind of talk? Is there anything philosophically or epistemically deep here?

On to B. Are ctms a better guide to truth in V than other kinds of model? Certainly on the Universist picture it seems like the answer should be no; various kinds of construction that are completely illegitimate over V are legitimate of ctms; e.g. \alpha-hyperclass forcing (assuming you don’t believe in hyperclasses, which you shouldn’t if you’re a Universist). Why should techniques of this kind produce models that look anything like a way V might be when V has no hyperclasses? Now maybe a potentialist has a response here, but I’m unsure how it would go. Sy’s potentialist seems to hold that it’s a kind of epistemic potentialism; we don’t know how high V is so should study pictures on which it has different heights. But given this, it still seems that hyperclasses are out; whatever height V turns out to have, there aren’t any hyperclasses. If one wants to look at pictures of V, maybe it’s better just to analyse the model theory more generally with standard transitive models and a ban on hyperclass forcing?

[A note; like Pen I have worries that one can't make sense of the hybrid-view. The only hybrid I can make sense of is to be epistemically hyperuniversist and ontologically universist. I worry that my inability to see the real' potentialist picture here is affecting how I characterise the debate.]

Anyway, I’m sympathetic to the idea that I’ve missed a whole bunch of subtleties here. But I’d love to have these set to rights.

With Best Wishes,

Neil.

P.S. I’ve added my good friend Chris Scambler to the list who was interested in the discussion. I hope this is okay with everyone here.

P.P.S. If there are responses I’ll try to reply as quick as I can, but time is tight currently.

# Re: Paper and slides on indefiniteness of CH

Dear Sy and friends,

Firstly, let me just say that I’ve found this thread really interesting, and I’m very keen to keep the discussion together (though the technical subtleties sometimes elude me, it’s great to see the thought process).

But I fear that height actualism is not dead; surely there must be even a few Platonists out there, and for such people (they are not “nuts”!) I’d have to work a lot harder to make sense of the HP. Is the Height Actualism Club large enough to make that worth the effort? It would help a lot to know how the height actualists treat proper classes: are they all first-order definable? And how do they feel about “collections of proper classes”; do they regard that as nonsense?

I think this very much depends on who you talk to. For example, lots of height actualists like to render proper class talk using plural reference (given in Boolos and developed in Uzquiano). Again, however, people differ on whether all “pluralities” should be first-order definable.

Another route some have taken is to regard class talk as simply shorthand for talk for first-order satisfaction in $V$, and associated formalisations of such talk in NBG (without putting words in people’s mouths, this is I believe Peter Koellner’s position, but I’m prepared to be corrected on this). This is obviously all first-order definable.

Still another way is to take Leon Horsten and Philip Welch’s approach, and regard proper classes as mereological fusions of sets that lie outside the scope of our first-order quantifiers. Again, whether or not you think such things are first-order definable depends on taste.

Finally, there’s the view that proper classes should be understood as “properties” or some other similar intensional notion. Pen’s already mentioned her 1983 paper, but it also pops up in some of Øystein Linnebo’s work from his (I beieve) pre-potentialist days (e.g. “Sets, Properties, and Unrestricted Quantification”).

Similarly whether or not one should have collections of proper classes is also going to depend on who you talk to. I’d say most actualists reject collections of proper classes, and this maps loosely on to the philosophical views above; properties/mereological fusions are fundamentally different kinds of entities from sets, and you can’t (for reasons that differ between authors) take collections of them. Similarly, in the plural case, this turns on the legitimacy of super-plural quantification, a hotly debated topic in the Philosophy of Language, Logic, and Set Theory (see, for example, the different views presented by Hanoch Ben-Yami’s “Higher-Level Plurals versus Articulated Reference, and an Elaboration of Salva Veritate” and Linnebo and Nicholas’ “Superplurals in English”). I’d say most actualists who take the plural route reject super plural quantification (usually by arguing that it is more ontologically committing than plural quantification).

There are some who have embraced collections’ of proper classes, however. In his PhD thesis Hewitt allows for finite order superplural reference and some have been tempted to use collections of proper classes in the pursuit of a class-theoretic interpretation of Category Theory (I think Muller has something like this in “Sets, Classes, and Categories”, but it’s a while since I read that – I should check the reference). Of course, the extent to which this is coherent is fiercely contested!

In short, when talking about “Actualism” we face exactly the same problem as when talking about “Potentialism”; there’s a whole gamut of positions referred to by the term. Constructing general arguments for and against these kinds of positions, and analysing the extent to which a view that falls under one of the two labels can account transparently for a piece of set-theoretic discourse is thus rather tricky.

Very Best,
Neil