Challenges to the Iterative Conception

Abstract

The main purpose of the chapter is to defend the iterative conception against three objections. The first objection, the missing explanation objection, is that if the iterative conception is correct, one cannot explain the intuitive appeal of the Naive Comprehension Schema. The chapter provides plausible explanations of this fact which are compatible with the correctness of the conception. The second objection, the circularity objection, is that the iterative conception presupposes the notion of an ordinal, and since ordinals are treated in set theory like certain kinds of sets, this means that the conception is vitiated by circularity. The chapter shows that this objection can be defeated by constructing ordinals using a trick that goes back to Tarski and Scott or dispensing with the notion of well-ordering altogether in the formulation of the conception. The third objection, the no semantics objection, is that the iterative conception prevents us from giving a semantics for set theory. The chapter defends the approach that this problem can be overcome by doing semantics in a higher-order language. The chapter concludes by discussing the status of the Axiom of Replacement on the iterative conception.