Saturday, April 01, 2006

Iterative Modal Theory

David Armstrong argues that there cannot be a set of all possible worlds:
Given a world with n wholly distinct elements, we can combine the elements to form 2^n possible worlds. Then we can construct a single world having non-overlapping parts parts, with a part to duplicate each of the 2^n worlds. And since this procedure is available for any set of possible worlds, there is no all-embracing set of such worlds.

The obvious parallel is with set theory. We know that there is no set of all sets. Nevertheless, there is an iterative procedure which shows us that, given a set, we can go on to form a higher-order set with a higher cardinality than the original set. Iterative set theory, which is contradiction-free as far as we know, provides a respectable way of talking about sets.

What we must accept, therefore, is an iterative conception of possible worlds. Given any world, in particular the actual world, Combinatorial principles deliver further worlds. But any attempt to form the set of all such worlds is defeated by a procedure which uses the given set to form worlds outside the set.

-- D. Armstrong, A Combinatorial Theory of Possibility, p.29.

He goes on to say that we can still speak of "all" possible worlds, so long as the universal quantifier is allowed to range over broader entities than sets (e.g. perhaps there is a "class" of all possible worlds).



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