Some Results on Numeral Systems in $\lambda$-Calculus

Abstract

In this paper we study numeral systems in the $\lambda\beta\eta$-calculus. With one exception, we assume that all numerals have normal form. We study the independence of the conditions of adequacy of numeral systems. We find that, to a great extent, they are mutually independent. We then consider particular examples of numeral systems, some of which display paradoxical properties. One of these systems furnishes a counterexample to a conjecture of Böhm. Next, we turn to the approach of Curry, Hindley, and Seldin. We dwell with the general problem of obtaining their results with the additional requirement of nonconvertibility of numerals. In particular we solve a problem that they left open. Finally, we give the first example of an adequate unsolvable numeral system without a test for zero in the usual sense, thus solving a problem of Barendregt and Barendsen.