In a related vein, it's interesting that in math there exist things which are indistinguishable, yet not identical. Perhaps the simplest example of this is i and -i. Another simple example is the non-identity elements in the Klein 4-group (the isometry group of a rectangular solid having unequal sides). ------------------------------------------------------------------------------ -------------------------------------------- But back to the absence of canonical correspondences: Let V(p) be a 1-dimensional vector space over F_p (the field of size p prime), and V(p)* = Hom(V(p),F_p) its dual vector space. It's amusing that there's no canonical isomorphism between V(p) and V(p)* iff p > 2. Question: Is the absence of a canonical isomorphism between an arbitrary vector space V and its dual V* equivalent to the Axiom of Choice? --Dan