A representation of recursively enumerable sets through Horn formulas in higher recursion theory

We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model A by some Horn theory, where A can be any model of a higher recursion theory, like primitive set recursion, αrecursion, or β -recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set A is recursively enumerable in models of the above mentioned recursion theories.