Frames

Abstract

The method of frames originated in Nerode [1961]. There frames were used to extend recursively enumerable relations on the natural numbers to Dedekind $\mathbb{S}$-RETs (isols). The main effect of the method is to obtain effective properties of Dedekind $\mathbb{S}$-RETs by approximating them effectively by using finite sets. The main application is the reduction of questions about Dedekind $\mathbb{S}$-RETs to questions about natural numbers. The method generalizes to all the categories we have considered but the results are most striking (and most obviously the generalizations of earlier results) in the case where the categories involved have dimension. However, in many other cases a weaker property (the automorphism extension property) allows us to obtain full analogues of Nerode’s [1961] results. But first we need the machinery.