Intrinsically Hyperarithmetical Sets

The main result proved in the paper is that on every recursive structure the intrinsically hyperarithmetical sets coincide with the relatively intrinsically hyperarithmetical sets. As a side eeect of the proof an eeective version of the Kueker's theorem on deenability by means of innnitary formulas is obtained. 1. Introduction One of the main achievements of the classical recursion theory is the classiication of certain sets based on the complexity of their deenitions. So, we have complexity classes of sets organized in hierarchies, as the arithmetical hierarchy, the hyper-arithmetical hierarchy, the analytical hierarchy, etc. All these hierarchies classify sets of natural numbers or sets of reals (usually considered as subsets of the Baire space). A natural problem is to obtain generalized versions of the classical hierarchies , which will work for subsets of the domains of arbitrary abstract structures. There are two approaches to this problem. The rst one, called internal, is based on a direct development of recursion theory on abstract structures, as is done by Moschovakis 11, 12]. The second approach, called external, uses enumerations of the abstract structures. Let A be a denumerable abstract structure. Assume that a subset A of the domain of A is xed and suppose that for every enumeration f of A the set f ?1 (A) belongs to the same classical complexity class C relative to the atomic diagram of f ?1 (A). Then we have evidence to think that A belongs to the complexity class C on A and say that A is relatively intrinsically C on A. The