The treatment of Cantorian sets by Bourbakian abstract set-theory
Abstract sets, that is, sets with "ur-elements", elements with no individuality other than being members of the set in question, have been around from the start, for Dedekind, Cantor, and others. They became of importance for Bourbaki, who insists that the identity of a mathematical object lies in its structural relations to other objects, rather than in an intrinsic identity. Lawvere's first-order theory of the category of sets (FOTCS) and his subsequent topos theory are decisive steps towards a set-theory that allows only abstract sets. I have introduced a simple formal system of abstract set-theory based on dependent types for which Bourbaki's requirement "all properties must be invariant under isomorphisms" holds true as a meta-theorem in the strong ("parametric") form demanded by Bourbaki (in Lawvere's FOTCS, as Colin McLarty has shown, the weaker non-parametric version is true only). This time I want to emphasize the inclusiveness of the new abstract set-theory, not shared by topos theory in a sufficiently natural way, by explaining that, in abstract set theory, present-day epsilontic Cantorian set-theory can be formulated as a theory of a particular Bourbakian "species of structures".