Many more predecessors: A representation workout

Abstract

Abstract From the outset, lambda calculus represented natural numbers through iterated application. The successor hence adds one more application, and the predecessor removes. In effect, the predecessor un-applies a term—which seemed impossible, even to Church. It took Kleene a rather oblique glance to sight a related representation of numbers, with an easier predecessor. Let us see what we can do if we look at this old problem with today’s eyes. We discern the systematic ways to derive more predecessors—smaller, faster, and sharper—while keeping all teeth.