Taming the wild ant-lion; a counterexample to a conjecture of Böhm

Abstract

In Barendregt (1984), Corrado Böhm conjectured that every adequate (Barendregt 1984, 6.4.2 (ii)) numeral system of normal combinators has normal successor, predecessor and zero test. In this note, we give a counterexample to this conjecture. Our example is shown to have no normal zero test. Böhm has informed us that Intrigila (1994) has given an example with no normal successor. Our strategy, in terms of the ant-lion paradigm, is to pry open the trap so wide that it enters its active state before its jaws are shut.