Abstract
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".