Diagonalization in Double Frames

Abstract

We consider structures of the form (Φ, Ψ, R), where Φ and Ψ are non-empty sets and \({R\subseteq \Psi\times \Phi}\) is a relation whose domain is Ψ. In particular, by using a special kind of a diagonal argument, we prove that if Φ is a denumerable recursive set, Ψ is a denumerable r.e. set, and R is an r.e. relation, then there exists an infinite family of infinite recursive subsets of Φ which are not R-images of elements of Ψ. The proof is a very elementary one, without any reference even to e.g. the \({S_{n}^{m}}\)-theorem. Some consequences of the main result are also discussed.