A recursive structure consists of a recursive underlying set vb;vb;, and recursive relations and operations on vb;vb;. A new relation on is called formallyΣ0α if it has a definition as a certain type of infinitary formula involving the given relations on the structure. It is called intrinsicallyΣ0α if the corresponding relation in any isomorphic recursive structure forms a Σ0α set (or set of ordered tuples).
In [3] it is shown that given a condition guaranteeing a certain amount of extra decidability in the structure these two notions coincide for the Σ01 (recursively enumerable) case. Here we deal with the analogous result for the general case, for α a constructive ordinal. We give a direct (infinite injury) construction for the case α = 2, together with several examples, which demonstrate that the decidability conditions required are satisfiable in natural examples. Then, applying a theorem of C.J. Ash [1,2], we deal with the general case.
