On the cardinality of worlds

For every way things might have been there is a possible world where they are that way. What does that tell us about the number of worlds?

If we identify ways things might have been ("propositions") with sentences of a particular language, or with semantic values of such sentences, the answer will depend on the language and will generally be small (countable). But that's not what I have in mind. It might have been that a dart is thrown at a spatially continuous dartboard, and each point on the board is a location where the dart's centre might have landed. These are continuum many possibilities, although they cannot be expressed, one by one, in English.

Suppose the number of worlds is k, for some cardinal k. This seems wrong, for familiar combinatorial reasons: couldn't there be k objects each of which either has or lacks some intrinsic property, for a total of 2^k possibilities? More simply, couldn't there be k objects, for every cardinal k, so that the number of possibilities exceeds every cardinal? Arguably, these objections can also be raised to the view that the worlds form a "proper class". I also find the whole idea of proper classes a bit fishy, so I'd rather avoid this commitment. (The standard cumulative picture doesn't have proper classes.)

So it looks like there are more worlds than fit in any set or class. We can still talk about the totality of worlds using plurals or the resources of primitive higher-order logic. Quantifiers over propositions could be replaced by second-order quantifiers, and names for propositions by predicates (true of worlds). This is close to Frege's later views, but it is inconvenient in practice.

Another option is to adjust the job description for possible worlds so that we don't need an absolute space of worlds, fixed once and for all. Maybe we can do with an infinite hierarchy of ever-expanding sets of worlds. Or maybe we can convince ourselves that "possible" is always relative to something -- an agent, a language, a modeling purpose -- and that these relative possibilities always form a set. For example, if our topic is epistemic possibility for agents like us, then it is plausible that there are (inexpressible) cardinals k and k' such that we cannot in principle distinguish between the possibility that there are k objects and the possibility that there are k' objects. I.e., we could not possibly have an attitude that is satisfied under one condition and not under the other. Infinitely smarter agents might be able to draw these distinctions, but must we allow for agents that can draw all the distinctions?

Perhaps there is yet another option. Consider the possibility that there are continuum many objects (spacetime points, say). Call this hypothesis H. Above I assumed that for every cardinal k, there is a distinct possibility H_k that there are exactly k objects. Which of these possibilities is identical to H? Arguably, while it is true that the continuum has some cardinality or other, there is no fact of the matter about which cardinality it has: our conception of sets does not settle the Continuum Hypothesis, nor is it settled by some determinate, unique structure in Platonic heaven. The system composed of ZFC plus the Continuum Hypothesis is just as good a mathematical system as ZFC plus the negation of CH, or ZFC plus the axiom that the continuum has cardinality aleph_2 (or aleph_3, or aleph_omega, etc.) So, on the one hand, the hypothesis H settles how many things there are (continuum many); but on the other hand, there is no fact of the matter about whether H says that there are aleph_1, or aleph_2, or aleph_omega many things. Conversely, the hypothesis that there are aleph_so-and-so many things doesn't really settle how many things there are -- for example, whether there are continuum many.

Similarly, if there were a distinct possibility for every hypothesis about the cardinality of objects, it would be highly relevant to the space of possibilities whether there are large cardinals, or extensions of the cardinals to proper classes. But real possibilities aren't created or annihilated by such arbitrary choices.

In general, the idea is that the infinite cardinals aren't objectively determinate measures of size, so that for every way things might have been there is a fact of the matter about the cardinality (if any) of objects which would then have existed. Without this assumption, can we still argue that the worlds exceed every set?