Pairs of Recursive Structures

A number of recursive constructions can be based on the following: given structures U n and B n and for an arbitrary Π 0 α set S construct recursive structures C n , uniformly in n, such that C n ≅{U n if n∈S, B n if n∈S. We concentrate initially on the special case where U n and B n are independent of n. So we ask for which structures U and B of the same type and which α<ω 1 CK the following statement is true. «For every Πα 0 set S there are structures C n , recursive uniformly in n, such that C n ≅{U if n∈S, B if n¬∈». In this paper, we obtain some general recursive-syntactical conditions for such statements