  Isomorphism is central to the structure of mathematics and has been
formalized in various ways within dependent type theory. All previous
treatments have done this by replacing quantification over sets with
quantification over groupoids of some form --- categories in which every
morphism is an isomorphism. Quantification over sets is replaced by
quantification over standard groupoids in the groupoid model, by quantification
over infinity groupoid in Homotopy type theory, and by quantification over
morphoids in the morphoid model. Here we give a treatment of isomorphism based
on the intuitive notion of sets as collections without internal structure.
Quantification over sets remains as quantification over sets. Isomorphism and
groupoid structure then emerge from simple but subtle syntactic restrictions on
set-theoretic language. This approach more fully unifies the classical ZFC
foundations with a rigorous treatments of isomorphism, symmetry, canonicality,
functors, and natural transformations. This is all done without reference to
category theory.
