The iterative conception of Set

The two expressions “The cumulative hierarchy” and “The iterative conception of sets” are usually taken to be synonymous. However the second is more general than the first, in that there are recursive procedures that generate some illfounded sets in addition to wellfounded sets. The interesting question is whether or not the arguments in favour of the more restrictive version—the cumulative hierarchy—were all along arguments for the more general version. The phrase “The iterative conception of sets” conjures up a picture of a particular set-theoretic universe—the cumulative hierarchy—and the constant conjunction of phrase-with-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively then the result is the cumulative hierarchy. In this paper I shall be arguing that this is a mistake: the iterative conception of set is a good one, for all the usual reasons. However the cumulative hierarchy is merely one way among many of working out this conception, and arguments in favour of an iterative conception have been mistaken for arguments in favour of this one special instance of it. (This may be the point to get out of the way the observation that although philosophers of mathematics write of the iterative conception of set, what they really mean—in the terminology of modern Computer Science at least—is the recursive conception of sets. Nevertheless, having got that quibble off my chest I shall continue to write of the iterative conception like everyone else.) Thanks are due to my Auckland students, to Flash Sheridan, Fred Kroon, Isaac Malitz and Herb Enderton. 1 The Cumulative Hierarchy There is a celebrated observation of Quine’s [11]: “No entity without identity” which throws down a challenge to theory-designers everywhere. If you lack a satisfactory identity criterion for widgets then you cannot use first-order predicate calculus with equality to theorise about them; that is to say, you are unable to treat them formally. In particular any story about what sets are had better include a chapter in which we learn how to tell when two sets are the same set and when they are different. The cumulative hierarchy gives an entirely satisfactory response to this challenge. Two sets are identical if every member of the one is identical to a member of the other, and vice versa. Since ∈ is wellfounded in the cumulative hierarchy, this regress must terminate; then the fact that it terminates gives us an unequivocal and intelligible criterion for identity between sets. I argued in [7] that it is precisely this feature of the cumulative hierarchy that makes it so attractive. I don’t know who was the first person to make this point: when making it in [7] I assumed I was merely giving routine expression to an uncontroversial common understanding. Certainly one of the—uncontroversial—