On the divergence of extension procedures in isol theory

Abstract

We show that the Myhill and Nerode extensions begin to disagree on the domain of the Nerode extension at a point in the arithmetical hierarchy ⩽ Δ 3 0 \leqslant \Delta _3^0 . This disagreement, at level Δ 3 0 \Delta _3^0 , goes hand in hand with a certain way in which the Myhill extension fails, at Δ 3 0 \Delta _3^0 , to commute with composition.