Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 120 / 3 / a / Solution

The Church numeral corresponding to the natural number $n$ isA function $g:{\mathbb{N}}^k\to \mathbb{N}$ is a lambda-definable function if some closed lambda term $G$ satisfiesfor all natural numbers $n_1,\dots ,n_k$.

DefineThen beta reduction givesTherefore the successor function is lambda-definable; this is the lambda definition of the successor function.