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I'm trying to learn about modal logic and the metaphysics of modality, and there's something that has been bugging me about what I've read so far: are there set-theoretical issues with supposing we can refer to every possible world? I'm not worried about models with finitely many worlds, but some of the modal metaphysics I've read defines the set of possible worlds (i.e, W) as something like "a set that contains all and only the maximally consistent sets of propositions". This makes me nervous for two reasons:

Reason 1: If w is a possible world then, for every subset s of w, the proposition "s is a subset of w" is true (and therefore should be an element of w). This makes it seem like w contains an element for every subset of itself. But if that were true, then the power set of w would have a cardinality equal to or less than w, which is impossible.

Reason 2: It seems like it would be possible for the world to consist of a point particle in R^3.Because the point particle exists in a real number space, there are an uncountable number of places such a particle could be. Does this imply that the set of possible worlds is uncountable? And, if it is uncountable, is that a problem?

Any clarification of these issues would be greatly appreciated. Thanks.

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  • Note that something like "the set of all maximally consistent sets of propositions" is actually quite small, from a set-theoretic standpoint anyways, as long as "proposition" means something reasonable: e.g. if by "proposition" we mean (say) "first-order sentence in the language of set theory," then there are only countably many of those and so our desired set is ... exactly as big as the set of real numbers. This is only problematic if we're skeptical of uncountable sets in the first place. (Note that your first issue arises from not being careful about what "proposition" means.)
    – Noah Schweber
    Sep 21, 2020 at 3:45
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    If w is a set of propositions then "s is a subset of w" is not in w. The propositions are meant to be about world's objects and relations, not about other propositions or their sets, they are not second order. There is no problem not only if W is uncountable, but if it is not even a set but a proper class. A simple example is the class of all models of a formal theory, which may contain models of any cardinality, and hence not be a set. Nothing in modal talk requires dealing with all possible worlds as a collection, so it need not be a set (but usually is).
    – Conifold
    Sep 21, 2020 at 7:01

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Looking at the "maximal consistent sets of propositions" idea for concreteness, the key issue is definitional: what exactly does "proposition" mean here?

If we try to play fast and loose with this, we do indeed run into problems - in particular, if we allow ourselves to refer to arbitrary worlds and sets of worlds and etc. in propositions we wind up with "too many propositions." But we already know that playing fast and loose with notions in this area gets us in trouble (e.g. consider "The least natural number not definable using fewer than ten billion English words").

Instead, this sort of model makes sense after a precise notion of "proposition" has been specified. For example, we might take as our worlds the maximal consistent sets of first-order sentences in the language of arithmetic. Now there's no size paradox at all.

Re: your second concern, we do indeed wind up with uncountably many worlds under this approach, but mathematically there's nothing wrong with that - the uncountable crops up all over the place and it's just not that problematic. If however we want to keep things countable, one way to do so is to impose a complexity constraint: allow only those maximal consistent sets which are "simple" in some precise sense. If all we're concerned about is countability we have tons of options - e.g. we could restrict attention to the maximal consistent sets of first-order sentences which are hyperarithmetic. This is a ridiculously loose constraint and permits basically everything you'll ever run across outside of mathematical logic, but is still strict enough to keep things countable.

(It's worth noting that imposing a too narrow constraint here can be problematic in a new way: if for example we restrict attention to only the recursive maximal consistent sets of first-order sentences in the language of arithmetic, by Godel's incompleteness theorem we won't have any world which resembles classical arithmetic, and if we restrict attention to the arithmetically definable maximal consistent sets we'll get lots of things kind of like actual arithmetic but we won't get the true theory of arithmetic, per Tarski.)

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