Böhm's theorem, Church's delta, numeral systems, and Ershov morphisms

Abstract

In this note we work with untyped lambda terms under β-conversion and consider the possibility of extending Böhm's theorem to infinite RE (recursively enumerable) sets. Böhm's theorem fails in general for such sets V even if it holds for all finite subsets of it. It turns out that generalizing Böhm's theorem to infinite sets involves three other superficially unrelated notions; namely, Church's delta, numeral systems, and Ershov morphisms. Our principal result is that Böhm's theorem holds for an infinite RE set V closed under beta conversion iff V can be endowed with the structure of a numeral system with predecessor iff there is a Church delta (conditional) for V iff every Ershov morphism with domain V can be represented by a lambda term. © Springer-Verlag Berlin Heidelberg 2005.