I’m thinking of writing something about ontological indeterminacy and the continuum hypothesis, and this post is basically a request for any background stuff I should know about.

Here are some of the questions I’m interested in. Assume that we have a world with continuum many atoms. One might wonder whether there are some atoms in that world such that (a) there are uncountably many of them, and (b) there are fewer of them than there are atoms in the world. Here’s a proposed answer to that question: It is metaphysically indeterminate. There is, in some deep sense, no fact of the matter about whether there are, or are not, such atoms.

I don’t much like metaphysical indeterminacy, so I don’t much like that answer. But I’m not sure there’s an obvious and clear counterargument to it. Hopefully when I start seriously thinking/reading about this, I’ll come up with a clear counterargument! Any suggestions for where I should start such reading would be much appreciated.

Here are two related questions.

Could it be contingent whether there are such atoms as described above? That is, might there be two worlds, alike in their distribution of atoms (and for that matter in the properties those atoms have) but unlike in terms of which pluralities of atoms exist?

If we assume unrestricted composition, we can reask the last two questions about objects. So the first question becomes, could it be indeterminate whether there is an object with uncountably many, but fewer than continuum many, atomic parts? And the second becomes, could it be contingent whether there is an object with uncountably many, but fewer than continuum many, atomic parts?

On a slightly different note, there’s another question about vagueness and composition that kicks in at the ‘top’ of the set-theoretic hierarchy.

Lewis believed that the union of some sets, if it existed, was their fusion. He also believed in unrestricted composition. Since it isn’t always true that some sets have a union, he inferred that there are proper classes that are not sets, and which are the fusions of sets that lack a union.

Here’s an alternative position to Lewis’s. Set-theoretic union just is fusion, as applied to sets. If some sets have a union, that’s their fusion. If they don’t have a union, they don’t have a fusion. I think the alternative position has some attraction (it lets us have an unrestricted version of the axiom of pairing, for instance, and it gives us a closer connection between mereology and set theory), but for now I’m just interested in some questions about this position, not about its truth.

So the same two questions arise. Could it be indeterminate whether Lewis’s position, or this alternative position, is correct? And could it be contingent whether Lewis’s position, or this alternative position, is correct? Any readers have advice on where I should look for guidance?

Hi Brian,

My first thought on your first question was that there may well be some atoms such that it is indeterminate whether there are continuum many of them, though it is determinate that there are uncountably many, and there may well be some atoms such that it is determinate that there are not continuum many of them, though it is indeterminate whether there are countably many. The idea would be that there is a determinate set of atoms, but what is indeterminate is just whether a function exists pairing those atoms with the reals (in the first case) or the naturals (in the second case).

Of course, that’s only if someone likes indeterminacy. I don’t see any reason why mathematical statements that are independent of the axioms (or the mereological statements tied to them) should be better candidates for indeterminacy than anything else.

You might want to look at Bruno Whittle’s paper on cardinality. As I recall, he suggests that “size” or “number” of a set is an intrinsic property, but cardinality is an extrinsic property, and therefore Cantor’s identification of the two is wrong, even though it was mathematically fruitful. So all your worries about vague existence become more like my example above – though I suppose those do again vague existence, at a higher level.

John Steel has often made a point that there’s a way for the Continuum Hypothesis to be indeterminate (assuming that Woodin’s project doesn’t work out). Basically what it would take is two equally good set theories that have all the same “generic absoluteness” properties (that is, Cohen’s method of forcing can’t be used to affect the truth values of many statements, while holding the theory fixed) and where each model of one is a forcing extension of a model of the other. Then it could be indeterminate whether the word “set” refers to the objects of one theory or the other – either way, the word “set” refers to one and “name of a set in the forcing extension” refers to the other, and each is an equally good candidate.

Of course, maybe this is just a way to have semantic indeterminacy here with no metaphysical indeterminacy.

Also, I don’t know of anywhere that he has detailed this potential possibility in writing – but I’ve heard him say it in conversation several times.

Hi Brian,

I don’t have Parts of Classes to hand, but I don’t think your characterisation of Lewis is quite right. I think he claims that whenever you have some sets, you have the union of those sets – it’s just that sometimes that union is a proper class. Actually, he’s also going to claim that whenever you have a union of a set of sets, that is going to be a set. That’s the “axiom of union” you get in the standard formulations of ZF, I thought. I’m also not sure what you mean by “an unrestricted version of the axiom of pairing”. If you mean the axiom “for any two sets x and y, there is a set z such that all members of x and all members of y are members of z (and every member of z is a member of x or a member of y)”, with the quantifiers over sets unrestricted, then Lewis’s system already has that as a theorem too.

You do get closer to set theory if the set theory you have in mind is formulated with unrestricted plurals or with impredicative schemas. Will it be closer to vanilla ZF without plurals or impredicative schemas, if that’s what you had in mind? (If you didn’t have vanilla ZF in mind, then whether Lewis’s system is closer to “set theory” rather depends on which “set theory” is under discussion.)

On a different note, I seem to remember that Peter Forrest has suggested restricting composition for sets when there are more than set-many of them, but unfortunately I can’t give a reference right now – maybe when I get back to my books.

Ah, I can answer my own question from the second paragraph, and give what I think you meant by the “axiom of pairing” that resolves my perplexity in the first. My apologies for posting twice in a row.

If you have ZF with a weakening of extensionality to not rule out urelements, or ZFC likewise weakened, or ZFCU,

andfirst order quantifiers that are not implicitly restricted to sets, then you will be able to prove in Lewis’s system that it is false that for all x and for all y there is a set whose only members are x and y. I take it you want that to come out true, instead, in “set theory”, and it’s what you were calling the axiom of pairing. (It’s unfortunate that “axiom of pairing”, like a number of other names of axioms, is name used of more than one axiom.) That’s an axiom specified without plurals and without impredicative comprehension schemas, so it answers my question, and clears up my puzzlement about your claim concerning an “unrestricted axiom of pairing”.Sorry to clutter the comments!

“Could it be contingent whether there are such atoms as described above? That is, might there be two worlds, alike in their distribution of atoms (and for that matter in the properties those atoms have) but unlike in terms of which pluralities of atoms exist?”

I need to think about this some more, but I get the feeling that it would be very difficult for this to be contingent or indeterminate. Could there, for example, be some xx’s such that everything that’s in fact an xx exists in w but it’s not the case in w that that there are some yy’s such that: any z is a yy iff it’s an xx? It seems like you could get a contradiction from this if you had a version of plural comprehension; but I don’t know how to formally state this in a system where it can be necessitated.

I would have thought you could run the same considerations for polyadic second order quantification as you can for pluralities. So I’m thinking even the situation Kenny described couldn’t arise. I.e. a situation where it’s contingent/indeterminate what size a given plurality has not because it’s contingent/indeterminate which pluralities exist, but because of contingency in which relations exists.

Daniel,

The second comment is what I had in mind. Lewis’s position is incompatible with a version of the Axiom of Pairing that is consistent with ur-elements, and quantifies over them. Since there are things that don’t have singletons, there will be some pairs of things that don’t have a pair set.

That’s obviously not a terrible result for Lewis. Giving up this version of the Axiom of Pairing isn’t exactly giving up on standard mathematics! But it is an interesting feature of his theory.

Andrew,

You’re right, there is something very odd about the idea that the version of the continuum hypothesis I have in mind could be contingent. For it to be contingent, something like the following would have to be true: there are some things, and it isn’t necessarily true that if each of them were to exist, then they would exist.

That’s undoubtedly surprising! Is it impossible though? I think that’s harder to say. I suspect that anything we say around here will have some very surprising consequences.

Kenny,

Thanks for those suggestions. I hadn’t looked at Whittle’s papers, and they look really interesting.

I agree, by the way, that we shouldn’t think there’s going to be any metaphysical indeterminacy around here. I’m just worried that some of the arguments I’ve used against metaphysical indeterminacy in the past don’t generalise to these cases, so I’m on the lookout for better ones!

Actually, I was thinking it was more the failure of plural comprehension than the non-distributivity of “exists” that is bizarre. It seemed to me that most people who think there’s no matter of fact about CH don’t usually think that there’s no matter of fact whether there are some things which are all and only the F’s. This would be an additional surprising consequence.

My formal claim, which I haven’t checked, is that (i) the first order barcan formula and its converse, (ii) plural comprehension (iii) some other plausible sounding principles like necessitation and plural generalisation entail the second order barcan formula and it’s converse. Once you had that it would seem like CH or it’s negation would have to be determinate/necessary.

I’m not entirely sure if this is relevant to this discussion, but Tim Williamson has some stuff on second order modal logic and the second order barcan formulae which might be useful:

http://www.philosophy.ox.ac.uk/__data/assets/pdf_file/0011/9479/Bernepaper.pdf