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.

The chapter explains what conceptions of set are, what they do and what they are for. Following a tradition going back to Frege, concepts are taken to be equipped with criteria of application and, in some cases, criteria of identity. It is pointed out that a concept need not settle all questions concerning which objects it applies to and under what conditions those objects are identical. This observation is used to characterize conceptions: a conception settles more questions of this kind than the corresponding concept. On this account, a conception sharpens the corresponding concept. This view is contrasted with two other possible views on what conceptions do: one according to which they provide an analysis of what belongs to a concept, and one according to which they are possible replacements for a given concept. The chapter concludes with a discussion of the uses of conceptions of set. Along the way, the naïve conception of set, which holds that every condition determines a set, is introduced and a diagnosis of the set-theoretic paradoxes is offered. According to this diagnosis, the naïve conception leads to paradox because it requires the concept of set to be both indefinitely extensible and universal.

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 discusses the naïve conception of set and criticizes attempts to rehabilitate it by modifying the logic of set theory. The focus is on the proposal that the Naïve Comprehension Schema – which formally captures the thesis that every condition determines a set – is to be saved by adopting a paraconsistent logic. Three strategies for doing so are distinguished: the material strategy, the relevant strategy and the model-theoretic strategy. It is shown that these strategies lead to set theories that are either too weak or ad hoc or give up on the idea that sets are genuinely extensional entities.