A theory of finite sets can be formulated in such a way that it is quite analogous to arithmetic. Hence much of what we already said about natural numbers will have application in this chapter. In this chapter, however, we shall treat some questions and issues that are specific to constitution of the consciousness of finite sets. Questions about the intuition of finite sets are of interest in their own right but this chapter can also be viewed as providing part of an account of the notion of “multitudes” as objects that are intuited in Husserl’s conception of numbers as “determinate multitudes”. In PA Husserl tended to treat the concepts of “multitude” and “set” as synonymous, claiming that the concepts were undefinable. Hence, what is needed philosophically, as in the case of any of the primitive concepts of mathematics, is an analysis of their origins. In this chapter we shall be speaking only about finite multitudes, and we shall again be applying the Kantian strategy. Thus we shall be claiming that there is knowledge of finite sets, that we do make determinations about finite sets, and then asking how this is possible. An analysis of origins provides the answer.
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