The position of index sets of identifiable sets in the arithmetical hierarchy

Abstract

We prove that every set of partial recursive functions which can be identified by an inductive inference machine is included in some identifiable function set with index set in Σ 3 ∩ Π 3 . An identifiable set is presented with index set in Σ 2 ∩ Π 3 but neither in Σ 2 nor in Π 2 . Furthermore we show that there is no nonempty identifiable set with index set in Σ 1 . In Π 1 it is possible to locate this king of set. In the last part of the paper we show that the problem to identify all partial recursive functions and the halting problem are of the same degree of unsolvability.