The þ-function in λ-K-conversion

Abstract

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ-K-conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T(n) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ(T, r) is convertible to the formula q representing the least natural number q, not less than r, for which T(q) conv 0.2 The method depends on finding a formula Θ with the property that Θ conv λu·u(Θ(u)), and consequently if M→Θ(V) then M conv V(M). A formula with this property is,The formula þ will have the required property if þ(T, r) conv r when T(r) conv 0, and þ(T, r) conv þ(T, S(r)) otherwise. These conditions will be satisfied if þ(T, r) conv T(r, λx·þ(T, S(r)), r), i.e. if þ conv {λptr·t(r, λx·p(t, S(r)), r)}(þ). We therefore put,This enables us to define also a formula,such that (T, n) is convertible to the formula representing the nth positive integer q for which T(q) conv 0.