The chapter considers attempts to modify the Naïve Comprehension Schema according to syntactic notions and in particular the technical notion of stratification used in Quine's New Foundations. This and related theories (such as New Foundations with Urelemente) are examined and their philosophical underpinnings discussed and assessed. It is argued that, contra what has often been suggested in the literature, one can describe a conception of set – the stratified conception – which is well motivated whilst incorporating the idea that the pathological properties are those whose syntactic expression does not satisfy the stratification requirement. However, it is argued that this conception is best seen as a conception of objectified properties or extensions rather than as a conception of sets as combinatorial collections. Understanding New Foundations and cognate systems as theories of objectified properties leads to a further development of the diagnosis of the set-theoretic paradoxes offered in the first chapter and allows one to deal with some of the standard objections to set theories based on stratification.

This chapter is the first of two exploring the idea that the core thought of the naïve conception – that there is an intimate connection between sets and properties – can be preserved as long as we build into the conception the idea that certain properties are pathological and, for this reason, do not determine a set. The chapter first uses a result from Incurvati and Murzi (2017) to show that restricting attention to those properties that do not give rise to inconsistency will not do. It then focuses on the limitation of size conception of set, according to which the pathological properties are those that apply to too many things. Various versions of the doctrine are distinguished. The chapter also discusses what it calls the definite conception, according to which the pathological properties are the indefinitely extensible ones. It is argued that the limitation of size fails to provide a complete explanation of the set-theoretic paradoxes. The definite conception faces the same problem and, in addition, it is unclear whether it has the resources to develop a reasonable amount of set theory.

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.