F U N D a M E N T a Mathematicae Recursive Expansions

Let A be a recursive structure, and let ψ be a recursive infinitary Π2 sentence involving a new relation symbol. The main result of the paper gives syntactical conditions which are necessary and sufficient for every recursive copy of A to have a recursive expansion to a model of ψ, provided A satisfies certain decidability conditions. The decidability conditions involve a notion of rank. The main result is applied to prove some earlier results of Metakides–Nerode and Goncharov. In these applications, the ranks turn out to be low, but there are examples in which the rank takes arbitrary recursive