A common question asked in Philosophy of Logic is as follows:  In virtue of what can a proposition asserting either the possibility, impossibility, or necessity of something be said to be true or false?  A theory of modality is needed to explain it.  Some, like David Lewis, claim that all possible worlds exist.  However, the metaphysics of this claim is outlandash.  Therefore, I have a more reasonable account.

A possible world is a set of internally consistent propositions.  By internally consistent I mean that there is not a contradiction, nor is it possible to prove from those propositions, a contradiction.  Therefore, when one says that some proposition P is possible, they are asserting that there is either a set, T (or one can construct a set, depending upon your philosophy of mathematics) that P is a member of T, and ~P is disjoined from T.  A necessary proposition Q, is such that for any set T, Q is a member of T.  The concequence of this, is that every possible world is infinite, since there are infinitely many true propositions.  A contradiction is any set T, which as P as a member unioned with a set T' which has as its member, ~P.  This union is strictly forbidden.

Now, why would I choose to modal a theory of modality within Set Theory?  For the simple reason that all of mathematics can be reduced to set theory.  Numbers are given set theoretic definitions [the number 1, for example, is defined as the set of all singletons, the number 2 as the set of all doubles, and so on].  Hence, the success mathematics has with set theory can be extended to an explanation of where modal logic propositions derive their truth values from.