7 - The Graph Conception

Summary

The chapter presents and discusses the graph conception of set. According to the graph conception, sets are things depicted by graphs of a certain sort. The chapter begins by presenting four set theories, due to Aczel, which are formulated by using the notion of a graph. The graph conception is then introduced, and a historical excursion into forerunners of the conception is also given. The chapter continues by clarifying the relationship between the conception and the four theories described by Aczel. It concludes by discussing four objections to the graph conception: the objection that set theories based on graphs do not introduce new isomorphism types; the objection that the graph conception does not provide us with an intuitive model for the set theory it sanctions; the objection that the graph conception cannot naturally allow for Urelemente; and the objection that a set theory based on the graph conception cannot provide an autonomous foundation for mathematics. It is argued that whilst the first two objections fail, the remaining two retain their force.