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?

*Posted by Brian Weatherson at 12:32 pm*