The range property fails for H

Abstract

We work in $\lambda{\mathcal{H}}$, the untyped $\lambda$-calculus in which all unsolvables are identified. We resolve a conjecture of Barendregt asserting that the range of a definable map is either infinite or a singleton. This is refuted by constructing a $\lambda$-term $\Xi$ such that $\Xi M=\Xi {\mathtt I} \iff \Xi M\neq \Xi \Omega$. The construction generalizes to ranges of any finite size, and to some other sensible lambda theories.