Recursion in a quantifier vs. elementary induction

Abstract

Spector [21] proved that a relation R on ω is Π11 if and only if it is (positive elementary) inductive on the structure 〈ω, +, ·〉; Kleene [8] showed that R is Π11 if and only if it is semirecursive in the type 2 object E. These two “constructive” characterizations of the Π11 relations have led to the independent study of (positive elementary) induction and recursion in E on an arbitrary structure, as natural generalizations of the theory of Π11 relations on (ω, +, · 〉.The theory of (positive elementary) induction on an arbitrary structure was developed by Moschovakis in his book Elementary induction on abstract structures (EIAS); one of the most important theorems there is a generalization of the classical theorem of Spector [20] and Gandy [3] about the Π11 relations on ω: a relation R is inductive on an acceptable structure . if and only if there is a formula φ(Y, x) of the language of such that:where HYP is the collection of hyperelementary relations on .Moschovakis [17] and Kechris and Moschovakis [6] showed how to develop the theory of recursion in higher types as a chapter in the general theory of inductive definability. This approach to recursion in higher types makes it possible to use methods from the theory of inductive definability in studying recursion in E.In this paper we establish some results about recursion in E on a structure which bolster the naturalness of this theory and contribute to its comparison with (positive elementary) induction.