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 $\mathcal{V}$ 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 $\mathcal{V}$ closed under beta conversion iff $\mathcal{V}$ can be endowed with the structure of a numeral system with predecessor iff there is a Church delta (conditional) for $\mathcal{V}$ iff every Ershov morphism with domain $\mathcal{V}$ can be represented by a lambda term.

Keywords

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