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 Naïve 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.

This chapter offers some concluding remarks. It provides an overview of some of the central features of the conceptions of set encountered in the book. It then discusses to what extent the findings of the book are compatible with some form of pluralism about conceptions.

The chapter introduces the iterative conception, according to which every set appears at one level or another of the mathematical structure known as the cumulative hierarchy, as well as theories based on the conception. The chapter presents various accounts of the iterative conception: the constructivist account, the dependency account and my own minimalist account. It is argued that the minimalist account is to be preferred to the others. A method – which I call inference to the best conception – is then described to defend the correctness of the iterative conception so understood. This method requires one to show that the iterative conception fares better than other conceptions with respect to a number of desiderata on conceptions of set. This provides additional motivation for exploring alternative conceptions of set in the remainder of the book.