Worlds as complete novels

Abstract

1. Patrick Grim has argued that there can be no set of all true propositions; for every set having, say, the number 3 as a member, there is the true proposition that 3 belongs to the set, and for every other set the true proposition that 3 doesn't belong to it, so there are at least as many true propositions as sets, making the class of truths a proper class. (In more detail; assume standard, Zermelo-Frankel, set theory, formulated in a language allowing quantification over propositions as well as sets, with Extensionality adjusted to allow propositions or other urelements to be members of sets, and allowing the vocabulary of our theory of propositions as well as set-theoretic jargon to occur in instances of the axiom schemes. The argument sketched above defines a one-one mapping between all the sets and some of the truths. Specify some arbitrary set as corresponding to any true proposition not in the range of this mapping and we have a function from truths onto the sets, so a set of all truths would, by Replacement, yield a set of all sets, something whose existence is refutable in ZF) He concludes from this that possible worlds cannot be construed as complete sets of propositions ('complete novels' in the jargon of the subject); the actual world, after all, would on this construal have to be the set of all true propositions, and every other possible world would be an equally, inadmissably, large set. Since this and similar conceptions of possible worlds have seemed attractive to many philosophers concerned with the metaphysics of modality (etc.), it is worth examining Grim's argument carefully. The propositions correlated with the sets in my version of the argument above are all necessary truths; a set is defined by its members, and so any set including (excluding) the number 3 does so essentially. Necessary truths, however, are in a certain sense the ones that don't have to be mentioned in specifying a possible world; they are true at every world, and so a world is fully determined by the contingent propositions true at it. Could we, then, define a world as the set of these contingent propositions? Grim notes that this was suggested to him by a correspondent, and quickly refutes the suggestion by defining a one-one mapping from the sets to true contingent propositions; pick some arbitrary contingent truth, and take its conjunctions with the necessary truths correlated with the sets earlier. These conjunctions will all be true (having two true conjuncts), contingent (having one contingent conjunct), and distinct (having different necessary conjuncts).