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

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.

A combinator is a lambda term without free variables. It is a fixed-point combinator whenfor every lambda term $F$.

The fixed-point theorem for the untyped lambda calculus states that every untyped lambda term $F$ has a fixed point up to beta equivalence. PutOne beta reduction giveswhich proves the theorem. Equivalently,is a fixed-point combinator.

Apply the theorem to the lambda term $\mathrm{Succ}$. Its fixed point $Y\mathrm{Succ}$ is a nonnormalizing lambda term satisfying $Y\mathrm{Succ}{\equiv }_{\beta }\mathrm{Succ}(Y\mathrm{Succ})$; it is not a Church numeral. The definition of a lambda-definable function describes the representing term only on Church-numeral inputs, so it does not turn this syntactic fixed point into a natural number $n$ satisfying $n+1=n$.

By assumption the set of combinators is recursively enumerable. Finite beta reduction sequences, and hence finite certificates of beta equivalence, are also recursively enumerable.

For a closed term $Y$, choose a fresh variable $f$. The term $Y$ is a fixed-point combinator exactly whenIndeed, substitution then gives the required equivalence for every $F$, and the forward direction follows by taking $F=f$. Dovetail the enumeration of closed terms with all finite beta-equivalence certificates. Whenever a certificate of the displayed equivalence is found, output $Y$. This enumerates exactly the fixed-point combinators, proving recursively enumerable fixed-point combinators.