A relation S is said to be
determined up to isomorphism by relations R with respect to a theory K if for all
models A1,A2 of K,A1 restricted to R is isomorphic to A2 restricted to R implies
A1 is isomorphic to A2. In this paper simple necessary conditions for S to be
determined up to isomorphism by R are given. These are applied in set theory to
show there are (nonstandard) models of set theory with isomorphic ordinals and
nonisomorphic constructible sets. The isomorphism on the ordinals may be taken to
preserve many familiar arithmetic functions on the ordinals as addition,
multiplication and exponentiation.
