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Published online by Cambridge University Press: 09 January 2020
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.
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