The paper considers structures of recursive similarity type whose universes are the natural numbers. there are many examples where such a structure has good properties, from the recursive viewpoint, while an isomorphic copy has bad properties. in this paper, "good" is taken to mean recursive (r.e., "s" 0 $$subscript$$"a" ) relative to the atomic diagram of the structure and a single forcing construction is used to show, for several different notions, that where good properties are inherited by all isomorphic copies, there is always an obvious reason, expressible in terms of the recursive infinitary properties of the abstract structure.
Ash, C., Knight, J., Manasse, M., & Slaman, T. (1989). Generic copies of countable structures. Annals of Pure and Applied Logic, 42(3), 195–205. https://doi.org/10.1016/0168-0072(89)90015-8