## Is there a set of all possibilities?

This question can be rephrased thus: is there a set of all logically possible worlds? Thinking of worlds as maximally consistent sets of propositions yields set-theoretic paradoxes, which, it seems to me, we can get out of by treating them, as I have been in the constant habit of doing for a while now, as maximally specific propositions. Taking the set of all logically possible worlds doesn’t run directly into any set-theoretic paradoxes, for worlds are not like propositions, and cannot be multiplied indefinitely by taking the power set of the set of all possible worlds. However, the question may be contaminated in a more subtle way with the set-theoretic problem raised by the fact that there is no set of all true propositions.

Consider that for any two logically possible worlds, the sufficient reason for their being non-identical is at-least-one proposition whose truth value(s) differ(s) from one to the other. However, the number of propositions which may exchange one truth value for another is not infinite, but indefinite, since at no world is there a set of propositions (true or false). Since there are indefinitely many propositions, it appears as though there are an indefinite number of possibilities, which then in turn implies that there is no set of all possibilities. This then implies in turn that there is no set of all possible worlds.

The answer seems to be that there is not a set of all possibilities. However, this seems uncomfortable if we adopt even modest versions of modal realism (by modest I mean something short of Lewis’ extreme modal realism). What kind of solution could there be?

Here is a one I have been thinking of. We could say, it seems to me, that a logically possible world is either a predicate bearer, or a collection of predicate bearers (I am biased toward the latter). Suppose a logically possible world is a collection of predicate bearers; all propositions either express that to some subject belongs some predicate, or express truths which supervene on these. Perhaps relations are predicates too. That would seem to cut the head off the snake, so to speak, by making it irrelevant that propositions can be multiplied indefinitely, since the set of predicates affirmable of some collection of predicate bearers, whether finite or infinite, seems to exist (i.e., is not indefinite), so long as we distinguish predicates from propositions and recognize that for some subject S to have predicate Pr may be expressible by an innumerable number of propositions {P1, P2,… Pn}. This solution, if coherent, would also advance the cause of substance-realism.